Vorlesungsskript Version - Home INFN Milanovacchini/mqa/lecture_notes.pdf · 2018-09-28 · analysis of course relies on the premise that the statistical experiments are reproducible,

Advanced quantum mechanics

Bassano Vacchini

Università degli Studi di MilanoINFN Sezione di Milano

[email protected]:/www.mi.infn.it/~vacchini

Vorlesungsskript Version: December 27, 2014

Epilogue 3

Table of contents

I Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1 Quantum mechanics as quantum probability . . . . . . . . . . . . . . . . . . . . . . . . . 9Classical probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Quantum probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25E�ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Dispersion-free states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Maximal entropy state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Positive operator-valued measure . . . . . . . . . . . . . . . . . . . . . . . . 37Naimark's dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Coexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Position observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Position and momentum observable . . . . . . . . . . . . . . . . . . . . . . 51

4 Transformations of states and observables . . . . . . . . . . . . . . . . . . . . . . . . . 55Complete positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Operations and measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 62Operations and state preparation . . . . . . . . . . . . . . . . . . . . . . . . 63Mathematical characterization of operations . . . . . . . . . . . . . . . . . 65Trace distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Von Neumann instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Measurement models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

II Open quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5 Open quantum systems as composite systems . . . . . . . . . . . . . . . . . . . . . . . 85Reduced dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Bipartite states and entanglement . . . . . . . . . . . . . . . . . . . . . . . 88Schmidt decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Pure entangled states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Mixed entangled states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Puri�cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5

Positive maps and entanglement detection . . . . . . . . . . . . . . . . . . 92Parametrization of quantum maps . . . . . . . . . . . . . . . . . . . . . . . 94Canonical basis and Bloch basis representation . . . . . . . . . . . . . . . 96Choi-Jamiolkowski isomorphism . . . . . . . . . . . . . . . . . . . . . . . . 101

6 Dynamics of open quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Quantum dynamical semigroups . . . . . . . . . . . . . . . . . . . . . . . 105Structure of the generator . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Projection operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Standard projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Correlated projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Projected equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 113Nakajima-Zwanzig master equation . . . . . . . . . . . . . . . . . . . . . 115Weak coupling master equation . . . . . . . . . . . . . . . . . . . . . . . . 117Quantum optical master equation . . . . . . . . . . . . . . . . . . . . . . 124Bloch equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Wiener-Khintchine theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 135Multi-time correlation functions . . . . . . . . . . . . . . . . . . . . . . . . 138Resonance �uorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6 Table of contents

Part I

Quantum mechanics

``These are my principles. If you don't like them I have others.''

(Groucho Marx)

1 Quantum mechanics as quantum probability

Quantum mechanics is a probability theory for the description of statistical experiments,also valid at a microscopic level. To highlight this fact we �rst address the general for-mulation of a statistical theory, which is relevant for the description of physical systemsin which the outcome of experiments has to be described by a probability distribution.

The general structure of a statistical theory can be outlined considering two basicsets, the convex set of states S and the set of observables B, so that for any p 2 S andany X 2 B one can construct a probability measure �pX de�ned on the �-algebra of theBorel sets over R, let us denote it by B(R), such that for any M 2 B(R) the number�pX(M) provides the probability that X takes values in M if the state is p. The whole

analysis of course relies on the premise that the statistical experiments are reproducible,and the frequencies with which the di�erent outcomes appear are well de�ned. Thisstructure based on three basic elements, the set of states and observables and a proba-bility formula to combine each state with each observable to provide a probability distri-bution, is realized in two di�erent mathematical framework in the classical and quantumcase. It is interesting to notice that the formalization of the two theories, by Kolmogorovand von Neumann respectively, took place around the same time.

Classical probability

In the classical case, as realized by Kolmogorov, probability theory is naturallyembedded within abstract measure theory. Just to recall, in abstract measure theoryone has:

� a set and a �-algebra F (that is a collection of subsets of closed under com-plementation and countable union)

� a measure � de�ned on F and taking values in R+[f1g such that

¡ �(;)= 0

¡ �-additivity holds: �([iAi)=P

i �(Ai) provided Ai\Aj= ; for 8i=j.

�The triple (; F ; �) is called a measurable space, and given a real measurable func-

tion f , that is a function such that f¡1(M) is a measurable set in F for all open setsM 2 B(R), one can consider integrals of the form

RMfd� for M 2 F . If we further ask

the measure to be normalized in the sense that �() = 1, we call it a probability mea-sure. Denoting such measure with p we say that the triple (; F ; p) is a probabilityspace. As suggested by Kolmogorov this is the natural setting to describe a probabilitytheory in the classical setting, so that a probability model is �xed by specifying a mea-sure space, the space of elementary events, a �-algebra on this space characterizing themeaningful events, and a probability measure on it.

1 Quantum mechanics as quantum probability 9

In the Kolmogorov model of classical probability theory one starts from a measurablespace and considers the following Kolmogorov construction of states and observables togive a statistical theory. The set of states S is identi�ed with the convex set of proba-bility measures on F and the set of observables B is identi�ed with the set of measur-able functions seen as random variables. The probability formula which provides the dis-tribution for the possible values of the random variable X given the state p is given by

�pX(B) =

ZX¡1(B)

dp

= p(X¡1(B)):

Also the mean values can be written in terms of integrals

hX ip =ZXdp;

similarly for higher moments and in general for a measurable function of the observable

hf(X)ip =Zf(X)dp:

�

Consider a point particle. The measure space is the usual phase-space, the proba-bility measure can be expressed via a probability density f(x; p), i.e. a positive andnormalized element of L1(R3 � R3), and observables are described as random variablesgiven by real functions X(x; p) in L1(R3�R3).�

Exploiting the canonical duality relation between L1 and its dual L1 mean valuesare given by

hX if =ZR3�R3

d3xd3pX(x; p)f(x; p):

Any observable taking values in R de�nes a probability measure on this space accordingto the formula

�fX(M)=

ZX¡1(M)

d3xd3pf(x; p)

where M is a Borel set in the outcome space R of the observable. In particular theexpectation value of any observable, i.e. random variable, is obtained from the verysame probability density. Considering the case of the position observable X(x; p) = xone can check that the probability measure �f

x can be expressed through the densitycorresponding to the marginal

fx(x) =ZR3d3pf(x; p):

�Let us make a few remarks on the logic underlying these models. We call events the

elements of the �-algebra F over which � is de�ned, and note that the structure of the�-algebra naturally provides a Boolean algebra, so that one has the basic elements ofclassical logic. Indeed if the events A and B are in F one can also consider the eventsA [ B, A \ B and AC, while ; and can be identi�ed with the impossible and certainevent respectively.�

10 Table of contents

The notion of inclusion, union and intersection then allow to consider on the �-algebra a natural structure of lattice.�

Indeed a lattice is a set in which one has partial order, join and meet operations (_and ^ respectively) which are associative and commutative. The lattice is said to becomplete if it has a minimum and a maximum element (0 and 1), orthocomplemented iffor any element of the lattice a one �nds a complement a0 such that a _ a0 = 1, and dis-tributive if join and meet are distributive the one with respect to the other

a^ (b_ c) = (a^ b)_ (a^ c):

A lattice with these properties is said to be a Boolean algebra. As shown above a �-algebra of subsets of a set is a Boolean algebra, and conversely it can be shown thatany Boolean algebra is isomorphic to a lattice of subsets of a set. The outcomes of trialsor experiments naturally have a lattice structure, but distributivity as we shall see is acrucial property, satis�ed by classical events but not by quantum events.

We are now in the position to point to three features of classical probabilistic modelwhich are not shared by quantum probability.

In the classical case the logic of events identi�ed with the subsets of the �-algebraprovides as shown a Boolean algebra, which implies in particular the distributivity of themeet and join operation. This fact allows to consider the decomposition of an event interms of elementary events. This is no more true in quantum mechanics, as one seesconsidering the following suggestive example. Let us introduce the following incompat-ible experimental outcomes within a double slit experiment: 1) a=interference pattern;2) b=passage through slit one; 3) b0=passage through slit two. One then has

a^ (b_ b0) =/ (a^ b)_ (a^ b0);

since the l.h.s. is equal to a, but since a is incompatible with both b and b0 the r.h.s. isequal to the empty set. The point is that in the quantum case a natural lattice of eventsis given by the set of orthogonal projectors on a given Hilbert space, which despite beingcomplete and orthocomplemented, is not distributive. One can directly check violationof distributivity by a suitable choice of projectors. This fact can be rephrased statingthat in quantum mechanics there are no elementary events.��

In the classical case the convex set of states is in particular a simplex, that is eachstate can be uniquely demixed in terms of extremal points of the set. We recall that theextremal points of a convex set are those elements which cannot be expressed as nontrivial convex combination of other elements in the set. E.g. in the phase-space exampleintroduce above extremal elements are given by measures concentrated in one point.This is no more true in quantum mechanics, as we will show by means of example lateron.�

Another feature of the classical case no long recovered in the quantum setting isgiven by the fact that the set of random variables over a measure space is a commuta-tive algebra under point wise multiplication, while as well known quantum observablesdo generally not commute.�


To highlight these facts we consider another example complementary to the previousone.

We consider a classical statistical N level system, so that we take = f1; :::; Ng andthe �-algebra F is given by the power set of . The generic state can be identi�ed witha probability vector, so that

S =

(p=(p1; :::; pN) j pi> 0;

Xi=1

N

pi=1

):

Note that introducing the extremal points of this convex set as (ei)j = �ij, so that e.g.e1=(1; 0; :::;0), for each state one has the convex decomposition

p =Xi=1

N

piei;

whose uniqueness follows from the fact that the extremal points also provide a linearbasis.

Real random variables corresponding to observables can be identi�ed with N -tuplesof real numbers

B = fx=(x1; :::; xN) jxi2Rg;

and their commutativity under pointwise multiplication is immediately apparent. Theprobability formula then simply reads

�px(fxig) = pi:

The mean values according to their de�nition are given by

hxip =Xi=1

N

xi�px(fxig)

=Xi=1

N

xi pi

Let us now consider a quantum N -level system, whose Hilbert space is CN. The set ofstates is

S = fN �N hermitianmatricesS jS> 0;TrS=1g;

while for the observables

B = fN �N hermitianmatricesg:

Exploiting the expression of a hermitian matrix X

X =Xj=1

N

xjEj ;

where fxjg are the eigenvalues, and fEjg provide an orthogonal resolution of the iden-tity, one can express the probability formula as

�SX(xi) = TrSEi;


which provides a probability distribution over the spectrum of the possible valuesattained by the observable X. In quantum mechanics this is assumed to be the proba-bility distribution of X provided the system is in the state S. The mean values can beobtained according to their de�nition

hX iS =Xj=1

N

xj�SX(fxjg)

=Xj=1

N

xjTrSEj

= TrSX;

thus recovering a simple formula in which the random variable X directly appears.

�Within this quantum probabilistic model one recovers the classical probabilistic

model in dimension N considering suitable subsets of states and observables. This showsthat a classical probabilistic model can be imbedded in a quantum one, but not vicev-ersa. To this aim consider the subset of S given by the states diagonal in a given basis

Sdiag =

8<:0@ p1 ::: 0

�� 0 ::: pN

1A�� pi> 0;Xi=1N

pi=1

9=;;and note that this is a convex subset. On the same footing consider the subset of B

Bdiag =

8<:0@ x1 ::: 0

�� 0 ::: xN

1A�� xi2R

9=;;which is closed under matrix multiplication. In this case the quantum probability for-mula reduces to the classical one

�SX(xi) = TrSEi

= pi

= �px(fxig)

and similarly for the mean values

hX iS =Xj=1

N

xj�SX(fxjg)

=Xj=1

N

xjpj

=Xj=1

N

xj�px(fxig):

�


Quantum probabilityAs a starting point we stress the fact that quantum mechanics arises in order to pre-

dict the outcomes of statistical experiments. Thus, despite the original evidence to lookfor quantum mechanics was in well-known but more elaborated experiments such as thestudy of black body radiation and atomic spectra, having in mind to identify the basicbuilding blocks of the theory one is led to consider the most simple statistical experi-ments, in terms of which all other experiments can be formulated. Such experiments aregiven by single particle statistical experiments in which a single microsystem is preparedby a preparation apparatus which feeds without feedback a registration apparatusaccording to the simple scheme depicted below as Ludwig's kisten

[preparationapparatus] [registrationapparatus]:

The experiment is of statistical nature in that only the frequencies and not the singleoutcomes are reproducible. Note that in this spirit the state of the system is the mathe-matical representative for the concrete preparation through which a certain system hasbeen obtained.

The basic question to be answered is how to correctly obtain the statistics of theoutcome of the experiment. In analogy with the classical case, which mathematicalobjects describe the preparation procedure (state), the registration procedure (observ-able) and how we combine them (probability formula) to obtain the probability measureon the outcomes of the experiment. Note that actually these mathematical objects willcharacterize whole equivalence classes. We will point to these mathematical objects in abottom up approach, starting from the usual formalism of quantum mechanics.

�

2 States

In the standard presentation one is given the Hilbert space of the system H, and statesare described by vectors 2H with unit norm. Given the Hilbert space one also has thelattice of projection operators, which as discussed below can be taken as events eP , iden-tifying each projection P with the associated subspace MP . Given an event and a statevector one can consider the probabilities

p = jjP jj2

corresponding to the norm of the projected vector, to be interpreted as the probabilitythat the event eP is veri�ed if the state is . The event is certain if 2MP and impos-sible if 2MP

?.

Considered an observable in the standard sense of a bounded self-adjoint operatorB 2B(H), assuming B takes values in R and denoting by M 2B(R) a generic set in theBorel �-algebra on the real line, one has the following expression for the probability for-mula

� B(M) = h jEB (M) i

= jjEB (M) jj2;


which gives the probability to obtain outcomes for the measurement of B in the set M .Here EB (�) denotes the projection-valued measure associated to the self-adjoint oper-ator. Note that the last line has been obtained relying on idempotency of EB (M). Notefurther that starting from the projection-valued measure upon integration one obtainsall the relevant information for a statistical description, such as mean values and othermoments of the probability distribution. Assuming for simplicity that B has a nondegenerate pure point spectrum, so that

B =Xj

bjPubj;

with Pubj= jubjihubj j one dimensional projections, one has

EB (M) =X

fj jbj2M gPubj

and in particular EB (fbjg)=Pubj. Mean values are de�ned according to

hBi =ZRxd� B(x)

=Xj

bj� B(fbjg)

=Xj

bjh jEB (fbjg) i

= h jXj

bjEB (fbjg) i

= h jB i;

so that mean values can be most directly expressed as expectation value of the self-adjoint operator. Note that to obtain this result we have not exploited idempotency ofthe projections EB (�). Considering however the second moment one has

hB2i =ZRx2 d�

B(x)

=Xj

bj2� B(fbjg)

=Xj

h jbj2EB (fbjg) i

= h j X

j

bjEB (fbjg)

!�Xi

biEB (fbig)

� i

= h jB 2 i;

so that idempotency is now a crucial property in order to express the second moment ofthe probability distribution of the observable B as expectation value of the square of theoperator. On the same premises one has

hf(B)i =ZRf(x)d� B(x)

= h jf(B) i;

and in particular

(�B) 2 = hB2i ¡hBi 2 :

2 States 15

Having in mind that a state should describe a concrete preparation procedure, let usconsider how to obtain a pure state and insofar such a state corresponds to a generalpreparation. A preparation corresponding to a pure state can be obtained in the veryspecial situation in which we measure a complete set of commuting observables A, �xinga unique common eigenvector, so that =ua.

This is not the most general state, since the set of pure states is not closed underconvex mixtures. Let us mix two such preparation procedures corresponding to out-comes given by common eigenvectors ua and ub with frequencies N 1/N = � and N 2/N =(1¡�) respectively, with N 1+N2=N . Relying on conditioning and using the rela-tion

P (x) =Xy

P (x; y)

=Xy

P (xjy)P (y)

we have

P (B 2M) = P (B 2M jua1)N 1

N +P (B 2M jua2)N 2

N= �hua1jEB (M)ua1i+(1¡�)hua2jEB (M)ua2i;

which cannot be written as jjEB (M) jj2 with a state vector. A similar result can beobtained for the mean values starting from

hX i =Xx

xP (x)

=Xx;y

xP (x; y)

=Xx;y

xP (xjy)P (y)

=Xy

�Xx

xP (xjy)�P (y);

so that

hBi = hua1jBua1iN 1

N + hua2jBua2iN 2

N :

We are thus led to introduce an operator of the form

� = �jua1ihua1j+(1¡�)jua2ihua2j

which allows us to write

P (B 2M) = Tr �EB (M)hBi = Tr �B

and in general

hf(B)i =Zf(x)d��E

B(x)

with

��EB(M) = Tr �EB (M):


Let us recall what we mean by trace of an operator (if it exists). Given a SONC fung inH one has by de�nition

TrA =Xn

hunjAuni;

so that the trace is a linear mapping, further satisfying invariance under a cyclic trans-formation, namely

TrAB = TrBA:

An operator like � is called a statistical operator. We have been led to such an expres-sion starting from a simple analysis of states as representatives of a preparation proce-dure. One is led to considers statistical operators also when analyzing measurements inquantum mechanics, or more generally bipartite systems, namely systems described on aHilbert space with a tensor product structure. Also the analysis of the classical limit inthe description of a macroscopic system is best performed considering statistical opera-tors.

We now want to better specify the functional space in which a general statisticaloperator can be described, thus characterizing its properties. Indeed instead of mixturesof two one-dimensional orthogonal projections as in the previous example one can obvi-ously consider an higher number of terms as well as a mixture, even a continuous one, ofnon-orthogonal states, which thus cannot be read as classical mixture of mutually exclu-sive events.

Let us start giving a precise meaning to the trace operation. Let A2B(H) be a posi-tive operator, namely h jA i> 0 for all 2H, and consider the series of positive num-bers

TrA =Xn

hunjAuni;

for a given basis fung. If the series converges the result is independent of the choice ofbasis, indeed considered another basis fvmg we haveX

n

hunjAuni =Xn

kA1/2unk2

=Xn

�Xk

��hvkjA1/2uni��2�

=Xk

�Xn

��hunjA1/2vki��2�

=Xk

kA1/2vkk2

=Xk

hvkjAvki:

The independence from the choice of basis justi�es the de�nition. For a generic operatorA2B(H) not necessarily positive, one considers the positive operator

jAj = AyAp

;

and if TrjAj <1 one says that A is a trace-class operator, then also Tr A is �nite andindependent of the basis. The space T (H) of trace-class operators on a given Hilbertspace turns out to be a Banach space with respect to the trace norm de�ned for A 2T (H) as

kAk1 = TrjAj:

2 States 17

The space of trace-class operators can also be obtained as closure with respect to thetrace norm of the set of degenerate operators, that is operators of the form

A =Xi=1

n

juiihvij;

where the sums run over a �nite index set and fuig, fvig are linearly independent fami-lies. The space T (H) is also a bilateral ideal of B(H), that is to say invariant under leftand right multiplication, namely 8A2B(H), 8T 2 T (H) one has AT ; TA 2 T (H) and inparticular

kAT k1 6 kAkkT k1:

Another important bilateral idel of B(H) is given by the space of Hilbert-Schmidt opera-tors. Given A 2 B(H) one can consider the quantity

Pn hunjA

yA uni, whose value asshown above does not depend on the basis. If this quantity is �nite the operator is inthe Hilbert-Schmidt class, and the expression

kAk2 = TrAyAp

actually de�nes a norm which makes the space of Hilbert-Schmidt operators HS(H) aBanach Space. It is the closure with respect to the Hilbert-Schmidt norm of the set ofdegenerate operators. Again it is a bilateral ideal of B(H), indeed 8A 2 B(H), 8T 2HS(H) one has AT ; TA2HS(H) and in particular

kAT k2 6 kAkkT k2:

Most importantly the space of Hilbert-Schmidt operators HS(H) is also a Hilbert spacewith respect to the following scalar inner product

hA;Bi = TrAyB;

well de�ned because the product of two Hilbert-Schmidt operators is trace-class. In thissetting the Cauchy-Schwarz inequality becomes

jTrAyB j 6 kAk2kBk2

with A;B 2HS(H), to be compared with

jTrAyB j 6 kAkkBk1

with A 2 B(H) and B 2 T (H). The latter relationship can be better understood notingthat B(H) = T 0(H), namely the space of bounded operators is the dual space withrespect to the space of trace class operators, that is the map B!Tr(B � ) is an isometricisomorphism of B(H) onto T 0(H). The space HS(H) is self-dual since it is a Hilbertspace.

Let us recall another subspace of B(H), let us call it C(H), which is the space ofcompact operators, closure of the set of degenerate operators with respect to the uni-form norm. One has in particular

B(H)�C(H)�HS(H)�T (H);

and accordingly

k�k6 k�k26 k�k1:

The equal sign in the inclusion between di�erent spaces is obtained when the underlyingHilbert space is �nite dimensional, so that the distinction among the di�erent spaces dis-appears.


�Let us now come back to the duality relation among the di�erent spaces.Suppose �rst dimH = n, then one can consider the canonical basis in T (H) given by

the matrices fEijg with (Eij)kl = �ik�jl, and write any T 2 T (H) as T =P

i; j=1n TijE

ij.We now show that any continuous linear functional f de�ned on T (H) can be identi�edwith a matrix B 2B(H). Indeed de�ne

f(Eij) = Bji;

we have 8T 2T (H)

f(T ) = f

0@ Xi; j=1

n

TijEij

1A= Xi; j=1

n

Tijf(Eij)

=Xi; j=1

n

TijBji=TrTB;

so that the linear functional f can be identi�ed with the matrix B, and the duality rela-tion with the trace operation.

In the case of a generic Hilbert space given a �xed B 2B(H) we have recalled that

TrBT

is a well de�ned complex number 8T 2T (H), and furthermore

jTrBT j 6 kBkkT k1;

so that each bounded operator can be identi�ed with a linear continuous functional onthe space T (H) exploiting the trace operation, and also the opposite can be shown tohold. One has therefore indeed B(H) = T 0(H). Note however that in the in�nite dimen-sional case one has T (H)�B 0(H), as can be shown along the same lines, the inclusion ishowever strict.

Considering self-adjoint operators one has the important result:Theorem Let A 2 B(H), self-adjoint and compact, then there exists a SONC fung

of eigenvectors of A with corresponding eigenvalues fang, s. t. the spectral representa-tion reads

A =Xn

anjunihunj:

In particular the eigenvalues fang form a l0(C) sequence, so that limn!1an=0, and theeigenspaces of the non zero eigenvalues are �nite dimensional.

If A2HS(H) the eigenvalues fang form a l2(C) sequence, so thatP

n janj2= kAk22<

1; if A 2 T (H) the eigenvalues fang form a l1(C) sequence, so thatP

n janj = kAk1 <1.

We can now introduce the set of states in the quantum probabilistic setting, �xed bythe choice of Hilbert space and given by the convex set of statistical operator

S(H) = f�2T (H)j�� 0;Tr�=1g:

The convexity of the space S(H) implies that any convex combination of elements ofS(H) is still in the set, i.e.

f�ig2S(H); �i> 0;Xi=1

n

�i=1 )Xi=1

n

�i�i2S(H):

2 States 19

Actually it is enough to consider mixtures of two elements only, that is convexity isimplied by

�1; �22S(H); 06�6 1 ) ��1+(1¡�)�22S(H);

as follows from the simple identityP

i=1n �i�i= �1�1+ (1¡ �1)

��2

1¡�1�2+ :::+ �n

1¡�1�n�.

Since trace class implies compact and positive implies self-adjoint we have the followingfundamental representation of a statistical operator, which provides a orthogonal decom-position of the statistical operator

� =Xn

pnP'n

=Xn

pnj'nih'nj;

where note that due to the choice of one-dimensional projection P'n the pn are generallynot distinct. Indeed the f'ng do form a SONC and the fpng provide a probability dis-tribution: pn> 0,

Pn pn= 1. Such a decomposition always exists, though it is in general

not unique; moreover � also admits non-orthogonal decompositions. We will discuss thispoint at length later on.

PurityLet us �rst give some information on the structure of the set S(H). We call pure

states the extremal points of this convex set, i.e. those that cannot be written as a nontrivial convex mixture. It turns out that pure states are given by one-dimensional pro-jections. In fact the following three statements are equivalent:

i. � is pure

ii. � is a one-dimensional projection

iii. Tr �2=1.

i!ii) Since � is a statistical operator one can consider its orthogonal decomposition,which being extremal must read � = p1j'1ih'1j, but due to the contraint on the traceone has p1=1, so that � is a one-dimensional projection.

ii!i) Suppose as an absurd P = ��1 + (1 ¡ �)�2 with 0 < � < 1. Then for all ' 2f g? one has 0 = �h'j�1'i + (1 ¡ �)h'j�2'i, so that the contributions at the r.h.s.have to be zero, and therefore the support of �1; �2 reduces to f g and one must have�1= �2=P .

ii!iii) Obvious.iii!ii) Suppose as an absurd that Tr �2= 1 but � admits a non trivial decomposition

�=��1+(1¡�)�2, then

1 = Tr �2=�2Tr �12+(1¡�)2Tr �22+2�(1¡�)Tr �1�26 �2+(1¡�)2+2�(1¡�)jTr �1�2j6 1

but since due to Cauchy-Schwarz jTr �1�2j6 Tr �12p

Tr �22p

6 1, this is actually a collec-tion of identities. In particular the Cauchy-Schwarz inequality is saturated, so that wehave �1= c�2, and the constant c has to be one due to the trace constraint.

It is therefore useful to introduce the quantity

P(�) = Tr �2;


called purity if the state. One has the upper bound Tr �2 6 k�kk�k1 = k�k 6 k�k1 = 1and therefore

06P(�)6 1;

as immediately follows from the properties of the eigenvalues of a statistical operator,indeed note that given an orthogonal resolution for � one also has P(�)=

Pn pn

2.We recall the following properties of the purity

a) P is a convex map: P(��1 + (1 ¡ �)�2) 6 �P(�1) + (1 ¡ �)P(�2) 6 max fP(�1);P(�2)g

b) P is invariant under unitary transformations: P(U�U y)=P(�)

c) P(�)= 1 if and only if � is a pure state.

The extremal states thus coincide with one-dimensional projections, but do they coin-cide with the boundary of the convex set? The answer is no provided dimH > 3, as wenow show. Let us �rst consider the peculiar and well-known structure of states in C2.

If dimH = 2, the Hilbert space is isomorphic to C2, and a generic state can bewritten as

� = 12(1+a ��);

with a 2R3, kak6 1, and 1

2p f1;�g an orthonormal basis in B(C2) =M2(C) according

to the Hilbert-Schmidt scalar product. This representation allows a one to one mappingbetween points in the sphere S2= fa2R3; kak6 1g and statistical operators in C2. Thegeometry of the sphere re�ects in this case the geometry of the quantum states. This iscalled Bloch-Sphere (Poincarè sphere in optics). Here the boundary coincides with theset of extremal states, and each non extremal state can be demixed in terms of twoextremal states. This is not generally true.

The set S(H) is a convex subset of the real normed space Ts(H) of self-adjoint traceclass operators. We can characterize the border of a convex set in an intrinsic way asfollows. We introduce its interior S~(H) as the subset of the elements � such that 8� 2S(H) there exists a �> 1 such that

(1¡�)�+��2S(H);

that is any line segment in S(H) having � as one of the endpoints can be prolongedbeyond � without leaving S(H). We de�ne then the boundary as @S(H) = S(H)nS~(H),that is �2 @S(H) i� there exists � 2S(H) such that 8�> 1

(1¡�)�+��2/ S(H):

This condition can also be formulated saying that there exists � 2S(H) such that 8"> 0

�+ "(�¡�)2/ S(H):

To connect this intrinsic de�nition of boundary to the speci�c structure we are consid-ering, we observe that if � belongs to the boundary 8� > 0 there exists a trace classoperator T� such that T� 2/ S(H) but k� ¡ T�k1< �. Indeed consider T� = � + �

3(� ¡ �),

then k� ¡ T�k16 2

3� < �, since for any couple of statistical operators �; w we have k� ¡

wk16 2. If � has a zero eigenvalue, it belongs to the boundary. Indeed let ' 2 H be aneigenvector corresponding to the zero eigenvalue, then considering T�= �¡ 1

2� j'ih'j, we

have that T� is not positive and k�¡T�k1< �.

2 States 21

�The converse holds true only in �nite dimensions. Supposing dimH > 3 consider a

orthogonal triple f'1; '2; '3g and the statistical operator

� = �P'1+(1¡�)P'2;

which is not pure, however �'3 = 0, so that it has a zero eigenvalue and thereforebelongs to the boundary. Thinking about Euclidean space, in a sphere pure states coverall the boundary, in a box only vertices are pure, other points on the boundary can bedemixed.�

EntropyFor a stochastic experiment the information content related to a certain event is a

measure for the uncertainty related to its appearance before we know it, or equivalentlyof the information gained upon learning its value. In information theory the informationcontent of a certain news/event is related to the number of binary questions necessary toascertain it. Suppose the sample space is a set with N = 2n elements !1; :::; !N, whichprovide the elementary events. An element of can then be characterized by a sequenceon n zeros or one. A realization of the stochastic event has an information content of Iequal to n = log 2N bits. More generally we set I = log2N , even if N is not a power oftwo. Suppose the elementary events have equal probability pi = 1/N , then the informa-tion of the realization of the event !i has information content I(!i) = log2 (N) = ¡log2 (pi), and following Shannon we adopt this quantity as information content of anevent taking place with probability pi for a distribution which is not necessarily uniform.The average information gained by a stochastic experiment measuring events with prob-ability pi is thus quanti�ed by the Shannon entropy

H(fpig) = ¡Xi=1

N

pi log2 (pi):

This quantity speci�es the amount of resources (bits) needed to store the informationassociated to a stochastic experiment.�

In the quantum setting the corresponding quantity is given by the von Neumannentropy, given by

S(�) = ¡kBTr�log�

which does not depend on the decomposition considered for �, and where kB denotes theBoltzmann constant. Since for any � 2 S(H) one can consider an orthogonal decomposi-tion in terms of pure states, we can write

� =Xi

piP'i

S(�) = ¡kBXi

pilog pi:

In particular

S(�) > ¡log (maxfpig);


so that the entropy is larger in the presence of many small eigenvalues, while it is equalto zero if the state is pure. In the case of a �nite dimensional Hilbert space withdimH=n we have the bounds

06S(�)6 logn1n6P(�)6 1;

saturated for the maximally mixed statistical operator corresponding to � = 1

n1. Note

that P and S introduce di�erent orders on the set of statistical operators.We have the following properties for the entropy

a) S is a concave map: S(��1 + (1 ¡ �)�2) > �S(�1) + (1 ¡ �)S(�2) > min fS(�1);S(�2)g, and the equality only holds if �1= �2, that is entropy is strictly concave

b) S is invariant under unitary transformations: S(U�U y)=S(�)

c) S(�)= 0 if and only if � is a pure state

d) Klein's inequality: given two statistical operators �; w2S(H), such that ker (w)�ker (�) one has

S(�) 6 ¡kBTr �logw;

and the equality sign only holds if �=w.

��

Proof of a) We consider the concave function h(x) =¡x log x, for x> 0, put equal tozero in x=0. We further set

� = ��1+(1¡�)�2and consider the decomposition

� =Xj

�jPj

=Xj;k

�j jujkihujk j;

where the �j are the distinct eigenvalues of the mixture, �j= hujk j�ujki, and Pj the pro-jections on the relative eigenspace, with possible degeneracy characterized by the indexk. We have

S(�) = ¡kBTr�log�=¡kBXj;k

�jlog�j

= kBXj;k

h(�j)= kBXj;k

h(hujk j�ujki)

> kBXj;k

(�h(hujk j�1ujki)+ (1¡�)h(hujk j�2ujki));

and we have exploited once the concavity of h. We now exploit it once more observingthat for a concave function

h(hBi) > hh(B)i;

where B 2 B(H) is a self-adjoint operator and hBi=Tr(�B), with � a state. Indeed themean value of an observable is a convex mixture of its eigenvalues. More explicitly con-sidering the spectral decomposition for B

B =Xi

biPbi;

2 States 23

and using the spectral theorem together with concavity of h we have:

h(hBi) = h

Tr��Xi

biPbi

�!=h�X

i

biTr(�Pbi)�=h�X

i

pibi

�>Xi

pih(bi)=Xi

Tr(�Pbi)h(bi)=Tr��Xi

h(bi)Pbi�

= Tr(�h(B))= hh(B)i:

Inserting this result in the previous expression we come to

S(��1+(1¡�)�2) > kBXj;k

(�hujk jh(�1)ujki+(1¡�)hujk jh(�2)ujki)

= kB(�Trh(�1)+ (1¡�)Trh(�2))= �S(�1)+ (1¡�)S(�2):

Proof of b) Follows from the fact that the entropy only depends on the distribution ofthe eigenvalues, which is not changed by a unitary transformation. Equivalently one canobserve that thanks to the ciclic property of the trace

S(U�U y) = ¡kBTr(U�U ylog(U�U y))=¡kBTr(U�log�U y)=¡kBTr�log�:

Proof of c) If � is a pure state, all eigenvalues are zero, apart from one equal to unit,therefore S(�) = 0. Viceversa S(�) = 0 implies that the sum of positive terms ¡kBP

i pilog pi is equal to zero, so that they are all zero, which again implies all pi equalto zero, apart from one equal to unit, and again the state is pure.

Proof of d) The statement is equivalent to

Tr�logw¡Tr�log� 6 0:

Suppose

� =Xj

�j jujihuj j

w =Xk

wkjvkihvk j

we then have

Tr�logw¡Tr�log� =Xj

�j(huj jlogwjuji¡ log �j)

=Xj;k

�j jhuj jvkij2(logwk¡ log �j)

where the expression is well de�ned since ker (w)� ker (�), restricting the sum to the �jdi�erent from zero, and exploiting the inequality log y6 y¡ 1 for y > 0 we have

Tr�logw¡Tr�log� =Xj;k

�j jhuj jvkij2logwk�j

6Xj;k

�j jhuj jvkij2�wk�j¡ 1�

=Xj;k

jhuj jvkij2(wk¡ �j)

= 1¡ 1=0:


DecompositionsWe have shown that each statistical operator has at least an orthogonal decomposi-

tion in terms of pure, that is extremal, states, thanks to the spectral theorem. Actuallythis decomposition is unique i� all eigenvalues are distinct. In general one has manyorthogonal decompositions, also known as canonical convex decompositions. Thefreedom lying in the change of basis in the eigenspaces. The higher the degeneracy, thehigher the number of orthogonal decompositions. All refer to the same statistical oper-ator, and possibly describe equivalent ways to obtain the same statistical operator to beunderstood as mathematical representative of an equivalence class of preparation proce-dures.

To consider an extreme case let us work in the �nite dimensional Hilbert space C2,where statistical operators are in one to one relationships with the points on the Blochsphere, and consider the maximally mixed state proportional to the identity �= 1

21. We

introduce a basis fj!+i; j!¡ig, which can be understood as eigenvectors of a polariza-tion along a direction !, which we can take as z-axis. Exploiting this basis for any � 2[0; �] and �2 [0; 2�] we obtain a new basis

j!+(�; �)i = cos�2e¡i�/2j!+i+ sin

�2e+i�/2j!¡i

j!¡(�; �)i = ¡sin �2e¡i�/2j!+i+ cos

�2e+i�/2j!¡i;

corresponding to eigenvectors of the polarization in an arbitrary direction. One then hasthe following orthogonal decompositions

� = 121

= 12(P!++P!¡)

= 12(P!+(�;�)+P!¡(�;�));

but also the non-orthogonal

� = 14�

Z0

2�

d�Z0

�

d� sin� j!+(�; �)ih!+(�; �)j

= 14�

Z0

2�

d�Z0

�

d� sin�P!+(�;�):

That the latter representation holds can be checked directly or obtained observing thatthe only state invariant under rotation must be proportional to the identity, and theproportionality factor is �xed by the value of the trace.

Indeed one generally has an uncountable number of distinct convex decomposition interms of pure states, not necessarily orthogonal between them. In dimension n at mostn orthogonal pure states are required, but more are allowed. To show this consider thefollowing construction. Given an arbitrary statistical operator � and an arbitrary com-plete orthonormal system f'ig we consider the numbers

�i = h'ij�'ii= �1/2'i 2;

which are positive and sum up to one. For the non zero �i we introduce the vectors

i = �1/2'i�i

p ;

2 States 25

which are of norm one but generally not orthogonal. We then have

� =Xi

�ij iih ij:

In fact 8�2H one has

h�j�X

i

�ij iih ij��i =

Xi

��h�j�1/2'ii��2=Xi

��h�1/2�j'ii��2= h�j��i;

where we have exploited the fact that the f'ig form a SONC. Since the identity holdsfor arbitrary � and � is bounded, the equality for the diagonal matrix elements impliesthe equality of the operator expressions.

Given another SONC f'i0g one has another decomposition

� =Xi

�i0j i0ih i0j;

where again the vectors i0 and therefore the associated projections need not be orthog-onal. Of course one can also consider decompositions in terms of non pure states. Indeedgiven a decomposition

� = ��1+(1¡�)�2;

one can consider in�nite other decompositions of the form

� = ��1 +�1¡ �

�

��2;

with 0<�< �<1 and ��1 = ��1+(1¡ �)�2.The existence of in�nitely many decompositions, orthogonal or non orthogonal, for

the same statistical operator show that indeed these decompositions do not have a phys-ical meaning. They cannot be unveiled by any subsequent measurement performed on �,whose statistics for any observable is �xed by � only. The state is described but notcharacterized by the ensembles f�i; ig or f�i0; i0g. We cannot describe it as arising byclassical ignorance, quanti�ed by the probability distribution of the eigenvalues, of aknown set of alternative events described by extremal elements. Indeed the decomposi-tion is highly non unique and the di�erent projection are generally not orthogonal, sothat they cannot be identi�ed with exclusive alternatives.

The di�erent decompositions might describe di�erent possible preparation proceduresleading to the same state. Note that this preparation procedures might even be incom-patible, in the sense that they cannot be performed together. Coming back to the polar-ization example, an unpolarized beam can be obtained by equal mixture of beams polar-ized in opposite directions along the same axis. The speci�c choice of axis is not rele-vant, and there is no way, by means of measurement performed on the �nal beam to�nd along which axis the polarization was performed. All these preparation proceduresdo belong to the same equivalence class. Moreover these di�erent preparations areincompatible, in the sense that they cannot be performed together.

E�ects


Now that we have identi�ed statistical operators as the mathematical representativesof equivalence classes of preparation procedures, we move on to characterize the mathe-matical representatives of equivalence classes of registration procedures. We �rst con-sider registrations or measurements corresponding to the simplest events, given by adichotomic alternative. Such apparata provide yes-no answers, e.g. to the questionwhether a certain quantity belongs to a given interval. Such elementary events or basicbuilding blocks of an observable can be naturally identi�ed with maps

E:S(H) ! [0; 1]� E(�)

which preserve the convex structure of the set of states, i.e. they are a�ne

E(��1+(1¡�)�2) = �E(�1)+ (1¡�)E(�2):

We call e�ects such a�ne maps from S(H) to the interval [0; 1], and we now show thatthey can be identi�ed with the positive bounded operators in the interval between thezero and the identity operator.

Let us consider a self-adjoint operator B 2 B(H), B = By. As we have seen due toB(H)=T 0(H) the map

� 7! Tr�B

provides an a�ne functional. Noting that for � = P we have Tr�B = h jB i one hasthat Tr�B > 0 for all � i� B > 0, and in the same way considering Tr�(1 ¡ B) = 1 ¡Tr�B we have that Tr�B 6 1 for all � i� B 6 1. Therefore a self-adjoint bounded oper-ator 0 6 E 6 1 provides an a�ne functional on the convex space of states through theformula

E:S(H) ! [0; 1]� Tr�E:

The viceversa holds, so that elementary events or observables corresponding to yes-nomeasurements can be identi�ed with positive operators between zero and one.�

Theorem Given an a�ne map E:S(H) ! [0; 1] there exists a bounded self-adjointoperator E such that the relation

E(�) = Tr�E

holds, and 06E61.Proof. Let us �rst give the idea: we want to extend the a�ne functional E to a linear

bounded functional on the whole linear space T (H) to exploit the duality relationB(H) = T 0(H) and therefore identify the map with a bounded operator, which then hasto be between zero and one to adjust the range of the functional. To extend E to auniquely de�ned linear map on T (H) we proceed in three steps, adopting a standardprocedure: we �rst extend the map from the convex set to the cone of positive operatorsT+(H), then to the set of self-adjoint operators Ts(H), and �nally to the whole linearspace T (H).

We start with the a�ne functional E de�ned on the convex set S(H), and extend itto a functional E+ de�ned on the positive cone T+(H) by setting E+(0)= 0 and for T > 0

E+(T ) = TrT E�

TTrT

�;

2 States 27

homogeneous with respect to multiplication by a positive scalar s since

E+(sT ) = Tr sT E�

sTTr sT

�= sE+(T )

and furthermore additive since exploiting the fact that E is a�ne we have for S; T 2T+(H)

E+(S+T ) = Tr (S+T )E�

S+TTr (S+T )

�= Tr (S+T )E

�TrS

Tr (S+T )S

TrS+ TrT

Tr (S+T )T

TrT

�= Tr (S+T )

�TrS

Tr (S+T )E�

STrS

�+ TrT

Tr (S+T )E�

TTrT

��= E+(S)+E+(T ):

To extend E+ to Ts(H) we note that each self-adjoint operator can be expressed as asum of positive and negative parts, according to

T = T+¡T¡

with

T+ = 12(jT j+T )

T¡ = 12(jT j¡T )

and we can thus set for T 2Ts(H)

E r(T ) = E+(T+)¡E+(T¡):

Again one can check that the de�nition makes the functional homogeneous with respectto multiplication by a real scalar and additive within Ts(H). We show e.g. additivity.From the relations

S+T = (S+T )+¡ (S+T )¡

= S+¡S¡+T+¡T¡

and therefore

(S+T )++S¡+T¡ = (S+T )¡+S++T+;

we have

E+((S+T )++S¡+T¡) = E+((S+T )+)+E+(S¡)+E+(T¡)= E+((S+T )¡+S++T+)= E+((S+T )¡)+E+(S+)+E+(T+)

so that

E+((S+T )+)¡E+((S+T )¡) = E+(S+)¡E+(S¡)+E+(T+)¡E+(T¡)

and �nally the additivity relation

E r(S+T ) = E r(S)+E r(T ):


As a last step we extend to the whole T (H) noting that each T 2 T (H) can be writtenas linear combination of two self-adjoint operators according to T =TR+ iTI with

TR = 12(T +T y)

TI = 12i(T ¡T y)

and we can thus set for T 2T (H)

E c(T ) = E r(TR)+ iE r(TI):

Again one can check that the de�nition makes the functional homogeneous with respectto multiplication by a complex scalar and additive within T (H). We have for �2C

E c(�T ) = E c((�RTR¡�ITI)+ i(�RTR+�ITI))= E r(�RTR¡�ITI)+ iE r(�RTR+�ITI)= �RE r(TR)¡�IE r(TI)+ i(�RE r(TR)+�IE r(TI))= (�R+ i�I)(E r(TR)+ iE r(TI))= �E c(T ):

And further

E c(S+T ) = E c((SR+TR)+ i(SI+TI))= E r((SR+TR))+ iE r((SI+TI))= E r(SR)+ iE r(SI)+E r(TR)+ iE r(TI)= E c(S)+E c(T ):

The thus de�ned linear functional on T (H) is further bounded since due to E�

T

TrT

�6 1

we have

jE c(T )j = jE c(TR+¡TR¡+ iTI+¡ iTI¡)j6 jE c(TR+)j+ jE c(TR¡)j+ jE c(TI+)j+ jE c(TI¡)j6 TrTR

++TrTR¡+TrTI

++TrTI¡

= TrjTRj+TrjTI j= 1

2TrjT +T yj+ 1

2Tr��T ¡T y��

6 2kT k1:

But according to the duality relation B(H) = T 0(H) such bounded functional, which wedenote again with the letter E for simplicity, can be identi�ed with an operator in B(H),let us call it E, via

E(T ) = TrTE 8T 2T (H);

and since the functional restricted to S(H) has to take values in [0; 1], as shown aboveone must have 06E61. Furthermore note that for E1=/ E2 one has in general Tr �E1=/Tr �E2, so that di�erent operators do correspond to di�erent e�ects.

We are now in the position to de�ne the set which provides the mathematical repre-sentatives of equivalence classes of elementary preparation procedures, corresponding toyes-no answer in the measurement

E(H) = fE 2B(H); 06E61g;

2 States 29

which we call set of e�ects. We have the strict inclusions, even in the �nite dimensionalcase

P(H)�E(H)�B(H);

so that projections are examples of e�ects. We note that the set of e�ects is convex, andE 2 E(H) implies E? = (1 ¡ E) 2 E(H): Moreover the usual sum of linear operatorsde�nes a partial operation in E(H), since the sum of two e�ects E1; E2 is de�ned pro-vided E1+E261. E�ects which can be summed do not necessarily commute.�

In particular projections do coincide with its extreme elements. We stress that eachprojection is an extremal element of the convex set of e�ects, not only one dimensionalprojections. Let us show this fact. Take P 2 E(H) \ P(H), so that P = P 2, and supposeas an absurd that it can be demixed in a nontrivial way

P = �E1+(1¡�)E2;

with E1; E2 2 E(H). Taking ' 2MP?, that is, in the orthogonal complement of the eigen-

space of P , so that P'=0, we have

0= h'jP'i=�h'jE1'i+(1¡�)h'jE2'i>�h'jE1'i> 0;

so that E1' = 0 8' 2 MP?. At the same time 8 2 MP we have, from 1 ¡ P = �(1 ¡

E1)+ (1¡�)(1¡E2),

0= h j(1¡P ) i=�(1¡h jE1 i)+ (1¡�)(1¡h jE2 i)>�(1¡h jE1 i)> 0;

so that E1 and P coincides on a SONC and therefore E1 = P . Projections are thereforeextreme elements in the convex set of e�ects. Suppose viceversa that E 2 E(H) is not aprojection, so that E2=/ E, then it cannot be extremal, as we show providing an explicitdemixture. Set E1=E2, and E2= 2E ¡E2, both are e�ects since for 06E 6 1 we haveE26E and (1¡E2)= (1¡E)2, and we have the decomposition

E = 12(E1+E2):

We note that positivity of the e�ects is all that we need to comply with the statisticalinterpretation of quantum mechanics, so that given arbitrary statistical operator � ande�ect E one has the probability formula Tr�E, giving the probability that a state prepa-ration corresponding to � triggers the registration corresponding to the e�ect E. Wehave as required for the probabilistic interpretation 0 6 Tr�E 6 1, while idempotency ofE is not necessary and generally does not apply. By means of example we can charac-terize e�ects in H=C2 as follows. A generic operator on this space can be written as

E = 12("1+ e ��);

where in order to be self-adjoint one has " 2 R, e 2 R3 and 0 6 E 6 1 leads to the fol-lowing inequality for the two eigenvalues 1

2("�kek)

06 12("¡kek)6 1

2("+ kek)6 1;

implying kek6 1 and

kek6 "6 2¡kek:


So far we have considered the most general preparation procedure as given by a sta-tistical operator � 2 S(H), and we have thus identi�ed the set of operators E 2 E(H) asthe most general yes-no elementary measurement. The key point being that the two setsare one dual to the other. A natural question is whether we can take the reverse route.Taking e�ects as the space of basic events to be observed in a measurement, with itspartial sum operation, one can ask what is the expression of a generalized probabilitymeasure de�ned on this space. Indeed a state can be naturally identi�ed with a proba-bility measure on the space of observables, it assigns to each elementary event the prob-ability of its occurrence once �xed the preparation. Let us �rst better specify what wede�ne as generalized probability measure. We say that

�: E(H) ! [0; 1]E �(E)

is a generalized probability measure on the set of e�ects provided

i. 06 �(E)6 1ii. �(1)= 1

iii. �(P

i=1n Ei)=

Pi=1n �(Ei) provided

Pi=1n Ei61.

It turns out that such a generalized probability measure can be identi�ed with a statis-tical operator.�

Theorem (Busch, 2003) Let � be a generalized probability measure on the set ofe�ects E(H), then there exists a statistical operator � 2 S(H) such that �(E) = ��(E) =Tr�E for all E 2E(H).

The proof is not di�cult and is similar to the one of the previous theorem. The basicfact is again a duality relation, while B(H) = T 0(H), in the in�nite dimensional case onehas the strict inclusion B 0(H) � T (H), in particular T (H) can be identi�ed with thelinear functionals on B(H) which besides being continuous are also normal, which is yetanother regularity condition. Given this fact the idea is to show that � can be extendedto a functional with values in [0; 1] over the whole linear space B(H) � E(H), which isfurthermore continuous and normal. It can therefore be identi�ed with an element ofT (H), and the restrictions on the range of the functional imply that this element is astatistical operator.

One might wonder whether restricting the generalized probability measure to act onthe set of projections, as in the more standard formulation of quantum mechanics, leadsto consider more general states than statistical operators. The answer is negative, andcorresponds to a famous theorem by Gleason, whose proof is very di�cult. Let us �rstprecisely de�ne what we mean by such a measure on the lattice of projections. We saythat

�:P(H) ! [0; 1]P �(P )

is a generalized probability measure on the space of projections provided

i. 06 �(P )6 1ii. �(1)= 1

iii. �(P

i=1n Pi) =

Pi=1n �(Pi) provided PiPj = 0 for i=/ j, that is the projections are

orthogonal.

2 States 31

It turns out that if dimH > 3 such a generalized probability measure can be identi�edwith a statistical operator.�

Theorem (Gleason, 1957) Let � be a generalized probability measure on the set ofprojections P(H)�B(H) and dimH> 3, then there exists a statistical operator �2S(H)such that �(P )= ��(P )=Tr�P for all P 2P(H).�Dispersion-free states

To better grasp the di�erence between classical and quantum probability, as well asbetween projections and e�ects, let us discuss the issue of dispersion free states. We saythat a state in a statistical theory is dispersion free provided the variance of any observ-able calculated with this state is zero, so that each outcome is predicted with certainty.Consider in the classical case R as measure space, then the extremal elements of thespace of probability measures on R are those with support reduced to a point, be it x0,and they can be formally expressed through a density given by a Dirac delta functiond�(x) = �(x¡ x0)dx. One can immediately check that this measure has zero variance, sothat this state assigns to each random variable f(x) the deterministic value f(x0), thestatistical aspect of the description is washed out. For this class of pure extremal statesno observables exhibit a nontrivial statistics. On the other hand, still within the classicaldescription, consider a �nite sample space , so that as discussed above states are of theform p= (p1; :::; pN), and pure states are of the form e1= (1; 0; :::; 0). In this setting oneeasily identi�es the basic elementary observables corresponding to projections ande�ects. Projections are N -dimensional vectors q whose entries qj are either 0 or 1,e�ects are given by vectors f whose entries fj satisfy 06 fj 6 1. As a result if one con-siders generalized observables in the sense of e�ects, also in the classical case there areno dispersion free states. For every p one can �nd an e�ect f such that the probabilityformula which now simply reads f �p =

Pj=1N fjpj gives a value strictly di�erent from

either 0 or 1. However restricting to observables given by projections one still has dis-persion free states, coinciding with the pure states fejgj=1;:::;N , indeed in this case q�ponly takes the values 0 or 1.

Let us now consider the quantum case. It immediately appears that taking e�ects forthe description of elementary observations, there are no dispersion free states. Indeedgiven any state �2S(H) one can consider the e�ect E= 1

2�, and the probability formula

gives the result TrE�= 1

2Tr �2< 1, strictly less than one as follows from the property of

purity. One might still ask if there exist dispersion free states in quantum mechanicswhen restricting to the set of projections. To show that this is not the case supposethere exist � such that

hA2i = hAi2

for any self-adjoint operator A2B(H), or equivalently

Tr�A2 = (Tr �A)2:

Of course for an observable A with a non empty point spectrum one can certainly �nd astate such that the dispersion of A is equal to zero, just by taking a pure state �xed byan eigenstate of A. The point is however whether the statistical description becomesirrelevant for all observables. To show that this is not the case consider A=P� for a cer-tain one-dimensional projection. If � would be dispersion free then one would have 8� 2H

(Tr �P�)2=Tr �P�2=Tr �P�;


so that Tr �P� is equal to either 0 or 1. Since Tr �P�= h�j��i, it cannot be always equalto 0, otherwise � should be the zero operator, neither it can always be equal to 1, other-wise � should be the identity operator. Therefore there exists at least a 2H such thath j� i = 1 and therefore Tr�(1 ¡ P ) = 0, in particular Tr �P� = 0 8�? . Let us con-sider therefore the vector �= cos��+ sin� , we have

Tr�P� = sin2�+ sin�cos�[h�j� i+ h j��i];

this function is continuous in � and takes the values 0 for � = 0, and 1 for � = �/2, ittherefore passes through intermediate points, leading to Hilbert space vectors for whichthe expectation value is neither 0 nor 1.

Maximal entropy stateIn the general case, especially for large systems, the information on a state does not

correspond to the fact that it has been obtained by performing a projective measure-ment on an impinging state, possibly selecting according to the outcomes. The availableinformation rather has the form of knowledge of mean values of a selection of observ-ables, which we call relevant observables. Note that variances can also be included, con-sidering the mean values of squares of operators. Of course this is in general a restrictedset, if all mean values were known, this would su�ce to �x the state uniquely. Relevantobservables are typically given by conserved quantities or the integral of densities of con-served quantities over suitable macroscopic space regions. Also variances can be takeninto account, considering both an operator and its square. Let

M = fRigi

the set of relevant observables, so that each Ri is a self-adjoint operator, and let

m = frigi

be the set of real numbers corresponding to the assigned mean values. Let us furtherintroduce the convex set of statistical operators compatible with these mean values

M(H) = f�2S(H) jTr�Ri= ri 8Ri2M g:

Among the states in M(H) we want to characterize the state which takes into accountthis information but includes no further bias, namely as discussed above the state whichsatis�es these constraints and has maximum von Neumann entropy. We therefore face amaximization problem with constraints, given by the assigned mean values as well asnormalization. We will see that the solution to this problem indeed is a statistical oper-ator, in particular a strictly positive operator. Given the expression of the von Neumannentropy

S(�) = ¡kBTr�log�

we look for stationary points of the functional

S 0(�) = S(�)+ kB(+1)h1i�¡ kBXi

�ihRii�

= ¡kBTr��log�¡ (+1)+

Xi

�iRi

�;

2 States 33

where the Lagrange parameters are real. Since the statistical operator can always bewritten as � =

P� ��j�ih�j with fj�ig a SONC, we have to consider variations both

with respect to the choice of SONC, which can be implemented via unitary transforma-tions, and with respect to the distribution of the eigenvalues. Let us �rst consider varia-tions with respect to the SONC. Since such unitary transformations are induced by aself-adjoint operator W we can write

�� = i~ [W; �]��

with W =W y and �� the parameter characterizing the in�nitesimal transformation. Wetherefore come to

�S 0(�) = ¡kBTr"��log�¡ (+1)+

Xi

�iRi

�+ ��

#

= ¡kBTr ��log�+

Xi

�iRi

�!;

where we have used the fact that �� is traceless. Further exploiting the explicit expres-sion of �� and the useful relation

Tr[A;B]C = TrA[B;C]

due to cyclic invariance of the trace operation one has

�S 0(�) = ¡kBTr�i~ [W; �]log�

��¡ kBTr

i~ [W; �]

Xi

�iRi

!��

= ¡kBTr Wi~

��;Xi

�iRi

�!��:

Asking the variation to be zero for arbitrary self-adjoint W we can make in particularthe choice

W = i~

��;Xi

�iRi

�;

so that a necessary condition for the functional S 0(�) to be stationary is given by

i

~

��;Xi

�iRi

�= 0;

so that in particular there exist a common spectral decomposition of the commutingself-adjoint operators

Pi �iRi and �. Writing to �x the notationX

i

�iRi =X�

R�j�ih�j;

this implies that the � we are looking for can be written as

� =X�

��j�ih�j;

for suitable weights ��. The commutativity ofP

i �iRi and � is actually also a su�cientcondition for stationarity of the functional. Indeed suppose

� =X

� j ih j;


for a certain SONC fj ig, we then have

S 0(�) = ¡kBX

�

�log� ¡ (+1)+ h j

Xi

�iRij i�

and the latter expression due to the variational theorem in quantum mechanics is sta-tionary provided the j i are eigenvectors of

Pi �iRi. Indeed the functional W ['] =

h'jH'i is stationary if ' is an eigenvector of H. To show this we consider stationarityof W ['] under variations of the state vector which have to preserve the norm. This canbe included through a constraint, so that we have to verify the relation

0 = �(W [']¡�k'k2)= h�'jH'i+ h'jH�'i¡�h�'j'i¡�h'j�'i= 2<(h�'jH'i¡�h�'j'i)= 2<h�'j(H ¡�)'i:

Since this must be true 8�', it follows that one has (H ¡ �)j'i = 0, so that ' is aneigenvector of H . As a result the condition

i~

��;Xi

�iRi

�= 0;

is both necessary and su�cient in order to have stationarity with respect to variation ofthe SONC. We thus have

S 0(�) = ¡kBX�

��flog��¡ (+1)+R�g

and varying with respect to the eigenvalues

�S 0(�) = ¡kBflog��¡+R�g��

and asking this quantity to be zero for all � we obtain

�� = e¡R�

so that the eigenvalues are indeed non-negative, and furthermore strictly positive. Thesolution of the stationary problem is indeed a statistical operator. We have

� =X�

e¡R�j�ih�j

= ee¡P

i�iRi

Normalization implies

e¡ = Tr e¡P

i�iRi

so that de�ning the quantity Z according to standard notation in statistical mechanics

= ¡logTr e¡P

i�iRi

= ¡logZ

the state which makes the functional S 0 stationary is identi�ed to be

w = e¡P

i�iRi

Z:

2 States 35

We now show that among the states in M(H), that is with the assigned mean values, ithas indeed maximum entropy. Suppose �2M(H), exploiting the Klein inequality

S(�) 6 ¡kBTr �logw;

we have

Tr�log� > Tr�logw

= Tr��¡Xi

�iRi¡ logZ�

= Trw�¡Xi

�iRi¡ logZ�

= Trw logw;

since both these operators have the same mean values for the relevant observables. As aconsequence

S(w) = ¡kBTrw logw> ¡kBTr�log�= S(�);

for all �2M(H). We have in particular the expression

S(w) = kBXi

�iri+ kBlogZ:

We have thus identi�ed the state which maximizes the entropy for �xed mean values.This state is of Gibbs form and its expression has now been obtained adopting the max-imum information principle to identify the state. Given its structure in terms of the rele-vant variables and the Lagrange parameters let us consider the equations which implic-itly �x the parameters in terms of the mean values. Starting from

= ¡logX�

e¡R�

we have

@@�i

=

P�@R�@�i

e¡R�P� e

¡R�;

but@R�@�i

= @@�ih�jXj

�jRj j�i

= h�jRij�i;

where always because of the variational theorem one need not take a variation withrespect to the basis, since the j�i are eigenvectors of

Pj �jRj. Finally we have

@@�i

= ¡@logZ@�i

=P

� h�jRij�ie¡R�P

� e¡R�

= TrRi e¡P

j�jRj

Tr e¡P

j�jRj

= hRii= ri:


The parameters appearing in the expression of the statistical operator are thereforeimplicitly �xed by the solution of the equations

= ¡logTr e¡P

i�iRi

@@�i

= ri;

which can have solutions provided the set m = frig provides a set of expectation valuescompatible with quantum mechanical requirements.�

As a last remark note that in order to evaluate the derivatives we have actuallyobtained the formula

@@�

Tr eA(�) = Tr@A(�)@�

eA(�);

which follows due to ciclic invariance of the trace from the following general relationshipkeeping track of the fact that the operators @A(�)/@� and A(�) do generally not com-mute

@@�eA(�) =

Z0

1

du euA(�)@A(�)@�

e(1¡u)A(�):

�

3 Observables

So far we have given the mathematical representative of equivalence classes of basicexperiments with yes-no answer. In terms of these basic building blocks we can now con-struct the general notion of observable with arbitrary outcome space.

Let us start from a discrete setting. Suppose that each outcome is represented by ane�ect Ej corresponding to a yes-no signature in the measurement apparatus, e.g. a click,then given a state � it takes place with probability Tr�Ej, and since only one eventtakes place at a time we have X

j

Tr�Ej 6 1;

and if the inequality is not saturated we can always introduce a further e�ect given by1 ¡

Pj Ej, corresponding to no event in the measurement apparatus given that the

preparation apparatus has �red. Indeed we have seen that the linear sum of e�ectswhich still stays below the identity is also an e�ect, so that they can be observedtogether, being di�erent exclusive outcomes of the same observable. Thus we assumewithout loss of generality that

Pj Tr�Ej = 1. Such an observable is therefore described

by a collection of operators fEjg such that 0 6 Tr�Ej 6 1 andP

j Tr�Ej = 1. Sincethese relations must hold for any state �, they uniquely identify a collection of e�ectssumming up to the identity.

Positive operator-valued measureLet us put this result in a general framework. Consider the outcome space of a given

physical quantity, be it (; F), where F is the �-algebra of measurable outcomes in thesample space . In the case of discrete or numerable outcomes, will consist of a �niteor denumerable set of real numbers. In the general case =Rn for a certain n, and F isthe Borel �-algebra of open sets B(Rn).

3 Observables 37

We de�ne as positive operator-valued measure (POVM) a map

F :F ! E(H)�B(H)

such that

i. 06F (M)61 for any M 2Fii. F (;)= 0, F ()=1

iii. F (Si Mi) =

Pi F (Mi) in the weak topology for any sequence of disjoint sets in

F , i.e. Mi\Mj= ; for i=/ j.

These properties express the fact that a POVM takes values in the set of positive opera-tors, between zero and unit so as to extract probabilities, is �-additive as a measureshould be. The weak topology allows for a natural connection with classical scalar mea-sures.

Indeed given an arbitrary � 2 S(H) and an arbitrary POVM F one immediatelyobtains a classical probability measure according to the formula

��F(M) = Tr�F (M);

as granted by linearity and continuity of the trace operation.We take POVMs as the most general description of observables, and the classical

measure ��F then provides the probability distribution of the observables associated to Fgiven a preparation �, that is the distribution of its possible outcomes once the state is�. Otherwise stated the map F is a POVM provided the map F 3M 7! Tr�F (M) is aprobability measure for any � 2 S(H). To substantiate the fact that positive operator-valued measures do indeed provide the general notion of observable, let us introduce theconvex set Prob() of all classical probability measures on the outcome space (; F).We can show that each map associating to any element � of the convex set of quantumstates S(H) an element of the convex set of classical states Prob(), namely an observ-able compatible with the statistical interpretation of quantum mechanics, can be identi-�ed with a POVM. We formulate this statement as a theorem.

Theorem Given a POVM F the map

�F :S(H) ! Prob()� Tr�F (�)

provides an a�ne map from the convex set of quantum states to the convex set of clas-sical probability distributions over (;F). Viceversa any such a�ne map can be realizedas �=�F for some POVM F .

Proof.Given a positive operator-valued measure F one immediately checks that for any �

the measure ��F(M) =Tr�F (M) belongs to Prob(), and thanks to linearity of the traceoperation the map is a�ne in its dependence on �. To show that the converse holds sup-pose we are given an a�ne map

�:S(H) ! Prob()� �[�]

where each �[�] is a probability distribution over . For �xed M 2 F we can considerthe map

�M:S(H) ! [0; 1]� �M[�]��[�](M)


which since � is a�ne is also a�ne in its dependence on � and takes values in theinterval [0; 1]. As shown before such a map can be represented as an e�ect, let us call itF (M), since it depends on M . It remains to show that the dependence on M is such asto make F (�) a positive operator-valued measure. Indeed being an e�ect we alreadyknow 0 6 F (M) 6 1 8M 2 F . Moreover since �[�] 2 Prob() we immediately haveF (;) = 0 and F () = 1. Let us now consider a sequence of disjoint sets in F , i.e. Mi \Mj= ; for i=/ j. We have, since �[�] is �-additive

Tr�F�[

i

Mi

�= �

SiMi[�]

= �[�]�[

i

Mi

�=Xi

�[�](Mi)

=Xi

�Mi[�]

= Tr�Xi

F (Mi)

and therefore �-additivity in the weak operator topology F (Si Mi)=

Pi F (Mi).

To connect with the standard notion of observable as a self-adjoint operator let usconsider the very special case in which F (M) = F 2(M) 8M 2 F , so that the measureactually takes values in the set of projections. We therefore de�ne as projection-valuedmeasure (PVM) a map

E:F ! P(H)�B(H)

such that

i. E2(M)=E(M), 06E(M)61 for any M 2Fii. E(;)= 0, E()=1

iii. E(Si Mi) =

Pi E(Mi) in the weak topology for any sequence of disjoint sets in

F , i.e. Mi\Mj= ; for i=/ j.

Observables which do correspond to positive operator-valued measures which are in par-ticular projection-valued measures are called sharp observables.

The crucial di�erence is idempotency, which is not necessary in order to obtain astatistical description of the measurement outcomes in terms of a classical probabilitymeasure ��

E. It has however important consequences from the mathematical point ofview. For the case of projection-valued measures we know that there is a one to one cor-respondence with self-adjoint operators. Indeed given a projection-valued measure EA

one can construct a self-adjoint operator A according to

A =ZRxdEA(x);

where we have assumed that A takes values in R. An important point is that allmoments of the probability distribution obtained from EA once �xed a state � can beexpressed directly in terms of this operator, which is therefore of great signi�cance.Indeed �xed an arbitrary state � we have, denoting by Mean�(EA) the �rst moment ofthe classical probability distribution obtained from the projection-valued measure EA

Mean�(EA) =ZRxd��

EA(x)

= TrA�;

3 Observables 39

where ��EA(M)=Tr�EA(M). Thanks to the identity

f(A) =ZRf(x) dEA(x);

so that in particular

A2 =ZRx2 dEA(x);

we can further express the variance as

Var�(EA) = TrA2�¡ (TrA�)2:

Similarly any moment or combination of moments of the statistical distribution can bedirectly expressed in terms of powers of the operator A. This is no more true in the caseof a POVM. The integral

B(1) =

ZRxdF (x)

still generally de�nes an operator, but the latter is not necessarily self-adjoint, and evenif this is the case the operator corresponding to the second moment

B(2) =ZRx2 dF (x)

is generally not the square of the �rst B(2) =/¡B(1)

�2. This opens the complicated

moments problem. As a general fact the focus remains on the positive operator-valuedmeasure itself and cannot be shifted to an operator.

As a �rst example of an observable in the sense of a proper POVM, let us considerthe following. We take H=C2 and consider the following POVM with three outcomes

F (fkg) = 12(�k1+mk��)

with k = 1; 2; 3 and in order for F (fkg) to be an e�ect we further need as shown before�k2R, mk2R3 together with kmkk6�k6 2¡kmkk.�

In order for the triple of operators fF (fkg)gk=1;2;3 to be a POVM we further needXk=1

3

F (fkg) = 1

implyingP

k=13 �k= 2, and

Pk=13 mk= 0, so that the three dimensional vectors mk are

linearly dependent, lying in a plane. A choice compatible with these conditions corre-

sponds to �k=2

3for k= 1; 2; 3 together with m1=

�0; 2

3; 0�, m2=

�+ 1

3p ;¡1

3; 0�, m3=�

¡ 1

3p ; ¡1

3; 0�so that the three vectors are coplanar, lie in the x-y plane and form an

angle of 120 degrees according to (mk/kmkk)�(mj/kmjk) = ¡1

2for j =/ k. The proba-

bility formula if the preparation is described by �= 1

2(1+ r ��) then implies

P (fkg) = Tr�F (fkg)

= Tr�12(1+ r ��)1

2(�k1+mk��)

�= 1

3+ 12r �mk

= 13(1+ r �mk);


where r = (x; y; z) is a vector with norm less or equal to one and mk =mk/kmkk. Inparticular we have P (fkg) 6 2/3, corresponding to the fact that the outcome is nevercertain, whatever the state, as it should be corresponding to a joint measurement of spinalong two distinct directions. In this case we can also completely spell out the map �Fwhich associates to a quantum state a classical probability distribution thanks to thePOVM

�F :S(H) ! Prob()� Tr�F (�):

For the case at hand = f1; 2; 3g, so that an element p2Prob() simply corresponds toa probability vector, and the map explicitly reads

�F : � p=(Tr�F (f1g);Tr�F (f2g);Tr�F (f3g))

with

p =

13(1+ y); 1

3

1+ 3

p

2x¡ 1

2y

!;13

1¡ 3

p

2x¡ 1

2y

!!:

�We thus have in particular kpk 6 1

2p . The image of S(H) through �F has to be a

convex subset of Prob(), a proper subset to comply with the constraints inherent inthe quantum states. In the case at hand Prob() is the set of three components proba-bility vectors. In three dimensional space such vectors lie in the �rst octant, and theconstraint

Pk=1

3 pk = 1 implies the restriction to a plane, so that the points lie in a tri-angle with corners (1,0,0), (0,1,0) and (0,0,1). The image of S(H) through �F is theintersection of this triangle with the sphere kpk6 1

2p .

�

Note the occurrence of a proper POVM in the space H = C2 for the case at hand isimplied by the fact that we have three outcomes. Indeed since the range of a PVM isgiven by orthogonal projections, for a PVM in this space one can only has the two out-comes corresponding to two orthogonal projections P�n =

1

2(1 � n��), so that the pre-

vious probability formula

Tr�F (fkg) = 13(1+ r �mk)

which is bound by 2/3 is to be compared with

Tr�P�n = 12(1� r �n)

which can attain the value one corresponding to a sharp statement.In order to see how an unsharp spin measurement might arise, let us consider the fol-

lowing slightly more re�ned description of the Stern Gerlach experiment. We considerthe Hilbert space H = L2(R3) C2 where both centre of mass and internal degrees offreedom are described. We devise a preparation procedure in two steps as follows. Firstone prepares a pure state of the form

�in = '= (c+v++ c¡v¡);

3 Observables 41

where 2 L2(R3) is a suitable wave packet describing the centre of mass degrees offreedom, while fv+; v¡g denote a basis in C2. The complex weights satisfy jc+j2+ jc¡j2=1 according to normalization. As a second step a inhomogeneous magnetic �eld isapplied, in order to sort the spatial direction of the atoms in the sample according totheir magnetic moment. We describe this transformation through a unitary operation asfollows

�out = U�in

= c+ + v++ c¡ ¡ v¡;

where � denote the de�ected wave packets, which are generally non-orthogonalh +j ¡i=/ 0. These two steps combined together then lead to our preparation proceduredescribed by �out. We now suppose to observe on the overall system the following PVMon L2(R3)C2 with two outcomes

E� = P�1C2;

where P� 2 P(L2(R3)) correspond to projections on the upper and lower half-planesrespectively. The probability to have localization in the upper or lower half-plane thusbecomes

P (�) = TrP�outE�

= TrP�inUyE�U

= h�outjP�1C2�outi= jc+j2h +jP� +i+ jc¡j2h ¡jP� ¡i= jc+j2kP� +k2+ jc¡j2kP� ¡k2

and upon setting

F� = Pv+kP� +k2+Pv¡kP� ¡k2

which identi�es a POVM with two outcomes on C2 one also has

P (�) = TrP'F�:

Thus the projective measurement on the overall system do correspond to a POVM forthe subsystem corresponding to the internal degrees of freedom only. Note that F� is aproper POVM unless kP� +k2 and kP� ¡k2 are exactly equal to 0 or 1. In this case,which corresponds to a sharp localization in the upper/lower plane, the observablebecomes a PVM. Coming back to the Stern Gerlach experiment, the unsharp situationis realized when due to the width of the wave packet, one can obtain counts in the upperplane corresponding to atoms which according to their magnetic moment have beende�ected in the down direction. A sharp observable corresponds to a situation in whichone can perfectly discriminate among the two signals, and the pattern on the screen isgiven by two sharp lines, with no deposition of atoms in between the region with highestdensity. Let us note that we can also write

F� = h jU y (P�1C2)U i:

�


Naimark's dilationThis is actually a general situation when one has a system interacting with some

environment. A projective measurement on the overall system corresponds in general toa POVM giving an e�ective description of the statistics of a certain physical quantity ofthe system only, according to the external noise induced by the interaction. That in thissetting a PVM generally becomes a POVM is immediately seen as follows. Consider aPVM on the system ES(M), a factorized system environment initial state �S �E, anevolution described by a unitary operator U and a measurement on the side of thesystem only, we then have for the distribution of the outcomes

TrSTrEfU(�S �E)U yES(M)1Eg = TrS�STrEf�EU yES(M)1EU g= TrS�SFS(M);

with

FS(M) = TrEf�EU yES(M)1EU g:

According to this expression one can check that indeed FS(M) is an observable in thesense of a POVM whenever ES(M) is a PVM. Actually it can be shown, as follows by atheorem due to Naimark, that any POVM can be seen as obtained by a PVM on alarger Hilbert space. The considered example is therefore in a sense paradigmatic. Notehowever that some, but not all, unsharp observables arise as smeared version of a sharpobservable. The unsharpness brought about by coarse-graining may or may not admitthe kind of ignorance interpretation familiar from classical physics. In general theunsharpness is the re�ection of a genuine quantum indeterminacy.

We now provide a �rst formulation of Naimark's dilation theorem, stating that anyPOVM can be dilated to a PVM in a larger Hilbert space. A more general proof will beprovided later on, exploiting the notion of complete positivity. We restrict ourselves forthe time being to a �nite Hilbert space H = Cn, and consider a POVM with �nite out-come space = f1; :::; Ng. A POVM on such a space can be identi�ed with a map asso-ciating to each k 2 1; :::; N a semipositive de�nite n � n matrix F (fkg) whose greatesteigenvalue lies below one. One further has the normalization condition

Pk=1N F (fkg) =

1n�n. Without loss of generality we take the e�ects F (fkg) to be given by rank oneoperators, so that F (fkg) =j'kih'k j with f'kg 2 Cn, generally not orthogonal amongthemselves. Due to the constraint

Pk=1N j'kih'kj = 1n�n one must have N > n, that is

the number of outcomes have to be higher than the dimensionality of the space. If N =n the f'kg have to provide an orthogonal set, so that the F (fkg) are actually a collec-tion of mutually orthogonal projections, and the POVM is actually already a PVM.Suppose now N > n, which is the case of interest, and consider the N � n matrix givenby

M =

0@ '1��'N

1A;where each element belongs to Cn and can therefore be expressed in terms of n compo-nents f'k1; :::; 'kng. This matrix realizes an isometry thanks to the normalization con-straint M yM =1n�n.

�

3 Observables 43

Its columns can be seen as n orthonormal vectors in a N dimensional space. It ittherefore possible to �nd a further matrix M 0 of dimension N � (N ¡ n) such that thematrix U = (M; M 0), where the �rst n columns come from M and the latter N ¡ ncolumns come from M 0, is a unitary matrix in a Hilbert space of dimension N . Thematrix M 0 is not uniquely identi�ed, since there are di�erent ways to add N ¡ ncolumns of other orthogonal vectors to the matrix M so as to make it unitary. The rowsof U therefore provide N orthonormal vectors which lead to a PVM in CN . Their pro-jection on the original H=Cn recovers the starting POVM.

Note that this minimal construction asks an enlargement of the original Hilbertspace depending on N ¡n, the di�erence between the dimension of the outcome space ofthe POVM and the dimension of the original Hilbert space.��Coexistence

We now want to discuss the problem of considering di�erent observables together,and see the major changes taking place in this respect when moving from PVM toPOVM. We �rst introduce the notion of range of an observable F on the outcome space(;F) as the set

ran(F ) = fF (M);M 2Fg�E(H):We further say that the e�ects E1; E2; :::; En are coexistent if there exists an observableF in the sense of POVM such that

E1; E2; :::; En � ran(F ):

This holds true ifP

j=1n Ej 6 1, since in this case as in previous examples we can con-

struct an observable adding the e�ect 1 ¡P

j=1n Ej. But this is only a su�cient condi-

tion, as we shall see by means of example. In this case the e�ects can be realizedtogether within a POVM, in general the fact that they are coexistent means that theycan be read within a POVM. On the same footing we say that two or more POVM arecoexistent if both their range are contained within the range of a single POVM. Other-wise stated one can �nd a third observable whose statistics contains those of the othertwo.

In order to clarify this notion, let us �rst remain in H = C2, and consider a furtherspin observable, which we can describe as a spin direction observable, realized by the fol-lowing POVM with outcome space (; F) where is the surface of the sphere of radiusone in three dimensions, and F the �-algebra naturally induced by the Borel �-algebrain R3

F sphere(M) = 14�

ZMdn(1+n��);

where n is a unit vector �xing a given direction, and the integral is over the solid angle.The de�ning properties of a POVM can be easily checked.

For any direction m we can further consider the two hemispheres with poles �m, letus call them M�, and consider the spin observable with two outcomes, which provides adiscretization of the previous observable

F hemisphere(�) = 14�

ZM�

dn(1+n��)

= 12

�1� 1

2m��

�= 1

2121+ 1

2

�12(1�m��)

�:


Note that the two outcomes can be seen as a convex combination of a projection andhalf the identity, the former corresponding to a sharp measurement and the latter corre-sponding to some extra noise, which reduces the accuracy. Once again one can checkthat this is a POVM. Now for any unit vector m we have di�erent hemispheres andtherefore di�erent POVMs. All of them are obviously within the range of F sphere, sinceF hemisphere(�) = F sphere(M�), so that they are coexistent. However one can �nd e�ectsin their range whose sum is no more an e�ect, and therefore cannot be in the range ofany observable, so that indeed the condition

Pj=1n Ej 6 1 for coexistence is indeed only

su�cient.It is further worth to indicate the relationship between coexistence and commuta-

tivity, which is the standard notion for joint measurability of observables in the stan-dard framework of quantum mechanics. When we speak of commutativity of two observ-ables we mean the standard notion of commutativity of the e�ects in their range. Giventwo observables we have that coexistence coincides with commutativity if at least one ofthe two is actually a PVM. If both are POVM, commutativity implies coexistence, butnot viceversa. It is just the release of commutativity that signi�cantly enlarges the classof physical quantities that can be considered by means of POVM.

To stress the fact that the basic di�erence between POVM and PVM lies in idempo-tency and commutativity of the operators in the range, let us consider the followingstatement. Let A be an observable with outcome space (; F), then the following state-ments are equivalent

i. A is a PVM

ii. A(M)A(N)=A(M \N) for all M;N 2F

iii. A(M)A(MC)= 0 for all M 2F

where the statement is known as multiplicativity and implies that the range of a PVM isgiven by commuting projections. To show the equivalence of the statements we showthat each one implies the one that follows.

i.)ii. Since M \ N �M and M �M [ N from the properties of PVM we have theordering A(M \N)6A(M)6A(M [N), which implies A(M \N)A(M)=A(M \N) aswell as A(M)A(M [N)=A(M). For a PVM we further have

A(M [N)+A(M \N) = A(M)+A(N)

so that multiplying by A(M) we have

A(M)+A(M \N) = A2(M)+A(M)A(N)

and therefore thanks to idempotency

A(M)A(N) = A(M \N)= A(N \M)= A(N)A(M)

which shows multiplicativity and commutativity.ii.)iii. Immediately follows by taking N =MC.iii.)i. Follows from exploiting once again the properties of being an observable

0 = A(M)A(MC)= A(M)[1¡A(M)]= A(M)¡A2(M)

3 Observables 45

so that we have idempotency and therefore A is a PVM.

CovarianceWe now want to consider a few examples of POVM and PVM concentrating on posi-

tion and momentum, showing in particular how symmetry properties can be a veryimportant guiding principle in the determination of meaningful observables, once weleave the correspondence principle focussed on quantum mechanics as a new mechanicswith respect to the classical one. As we shall see while for the case of either position ormomentum alone POVM are essentially given by a suitable coarse-graining with respectto the usual PVM, if one wants to give statistical predictions for the measurement ofboth position and momentum together the corresponding observable is given by neces-sity in terms of a POVM.

Let us �rst start by introducing the notion of mapping covariant under a given sym-metry group G. As we will show this notion is of great interest in many situations, bothfor the construction of POVM and general dynamical mappings. Consider a measurespace X with a Borel �-algebra of sets B(X ). Such a space is called a G-space if thereexists an action of G on X de�ned as a mapping that sends group elements g 2 G totransformation mappings �g on X , realizing a group omomorphism between G and thegroup of automorphisms of X

�:G ! Aut(X )g �g

so that group composition, inverse operation and identity are preserved

�g�h = �gh 8g; h2G�e = 1X

(�g)¡1 = �g¡1

If furthermore G acts transitively on X , in the sense that any two point of X can bemapped one into the other with �g 0 for a suitable g 02G, then X is called a transitive G-space. Consider for example X = R3, then X is a transitive G-space with respect to thegroup of translations. The elements of the group are 3-dimensional vectors acting in theobvious way on the Borel sets of R3, i.e. �a(M) =M + a for all a 2 R3 and for all M 2B(R3). Consider as well a unitary representation U(g) of the same group G on a Hilbertspace H

j gi=U(g)j i 2H g 2G;

in terms of which one also has a representation of G on a space A(H) of operatorsacting on H

Ag=U y(g)AU(g) A2A(H):

A mapping M de�ned on B(X ) and taking values in A(H) is said to be covariant withrespect to the symmetry group G provided it commutes with the action of the group inthe sense that

U y(g)M(X)U(g)=M(�g¡1(X)) 8X 2B(X ) 8g 2G:

A symmetry transformation on the domain of the mapping is mapped into the sym-metry transformation corresponding to the same group element on the range of the map-ping. The action of G on X commutes with the automorphism group representation ofG on A(H) induced by the unitary representation of G on H.


Considering the case of a POVM on the measure space (; F), supposing the exis-tence of an action �g of the group G on the space , the requirement of covariance canbe written as

U y(g)F (M)U(g)=F (�g¡1(M)) 8M 2F 8g 2G;

and corresponds to commutativity of the following diagram

F !F E(H)�g¡1# #Ug

F !F E(H)

Position observableAccording to the previous results we describe observables as POVM, and look for a

position observable. Rather than relying on the usual correspondence principle withrespect to classical mechanics, we want to give an operational de�nition of the positionobservable describing localization, �xing its behaviour with respect to the action of therelevant symmetry group, which in this case is the isochronous Galilei group, containingtranslations, rotations and boosts that is to say velocity transformations. In order for ameasurement Fx(M) to be a position measurement we have to ask that the probabilitydistribution of the outcomes of Fx(M) performed on a state � and on the correspond-ingly transformed one �g=U(g)�U y(g) should satisfy the relation

Tr �gFx(�g(M)) = Tr �Fx(M) 8�2S(H) 8M 2B(R3) 8g 2 G

where now G denotes the isochronous Galilei group. The group acts in the natural wayon the Borel sets of R3, and the covariance equations that we require for an observableto be interpreted as position observable then become

U y(a)Fx(M)U(a) =Fx (M ¡a) 8a2R3

U y(R)Fx(M)U(R) =Fx(R¡1M) 8R2SO(3)U y(q)Fx(M)U(q) =Fx(M) 8q 2R3;

where we have used the de�nitions U(a)= eia�p, U(q)= e¡iq�x.�

The mapping Fx to be interpreted as a position observable has to transform covari-antly with respect to translations and rotations, and to be invariant under a velocitytransformation. These equations can also be seen as a requirement on the possiblemacroscopic apparata possibly performing such a measurement. The apparatus used totest whether the considered system is localized in the translated region M ¡ a should bein the equivalence class to which the translated apparatus used to test localization in theregion M belongs, and similarly for rotations. Localization measurements should insteadbe una�ected by boost transformations. A solution of these covariance equations, that isto say a POVM complying with the covariance conditions, is now a position observable.

If one looks for such a solution asking moreover that the POVM be in particular aPVM, the solution is uniquely given by the usual spectral decomposition of the positionoperator

Ex(M)= �M(x̂)=ZMd3xjxihxj

where �M denotes the characteristic function of the set M .

3 Observables 47

The basic idea goes as follows. If Ex has to be a PVM, it identi�es a triple of com-muting self-adjoint operators. At the same time according to Stone's theorem the uni-tary representation U(q) of the group of boosts can be uniquely expressed in terms of aPVM EU

x according to

U(q) =ZR3

e¡iq�ydEUx(y):

The latter de�nes a triple of self-adjoint operators in terms of the integrals

x̂j =ZRydEU

xj(y);

and the latter are the generators of boosts. The invariance under boosts of our PVMlocalization observable Ex then implies that the generator of boosts x̂ can be identi�edwith the triple of self-adjoint operators de�ned by Ex. A similar argument leads to theidenti�cation of a PVM momentum observable and the generators of translations,namely the usual momentum observable. Covariance under rotations then warrants thatthese triples indeed transform as vectors.

The �rst moment of the spectral measure therefore gives the usual triple of com-muting position operators

x̂j=ZRdxjxj jxjihxj j;

whose powers coincide with the higher moments of Ex

x̂jn=ZRdxjxj

njxjihxj j:�

In particular for a given state � mean values and variances of the classical proba-bility distribution giving the position distribution can be expressed by means of theoperator x̂

Mean�(Ex)=Tr�x̂ = hx̂ i�Var�(Ex)=Tr�x̂2¡ (Tr�x̂)2= hx̂2i�¡hx̂ i�2:

�The couple (U ; Ex), where U is the unitary representation of the symmetry group,

here the isochronous Galilei group, and Ex a PVM covariant under the action of U iscalled a system of imprimitivity.

More generally a solution of the covariance conditions as a POVM can be obtainedas follows. Let us consider a distribution function h(x) with zero mean such that

h(x)� 0Zd3xh(x)= 1 h(Rx)=h(x);

so that it can be identi�ed with a rotationally invariant probability density. Since it haszero mean the variance is given by

Var(h)=Zd3xx2h(x):

One can then indeed check that the expression

Fhx(M)= (h � �M)(x̂)=

ZMd3y

ZR3d3xh (x¡ y)jxihxj


where � denotes convolution, actually is a POVM complying with covariance, and in factprovides the general solution. Indeed one immediately sees that thanks to positivity of hthe operator Fh

x(M) is positive 8M 2 F . Moreover since the POVM is built in terms ofan integral with an operator density one immediately has Fh

x(;) = 0, Fhx() = 1 as wellas �-additivity.

In the case of the PVM Ex we have in the position representation for a pure state 2H

hy jEx(M) i = (Ex(M) )(y)= �M(y) (y)

while

hy jFhx(M) i = (Fhx(M) )(y)= (h � �M)(y) (y)

=ZMd3xh (y¡x) (y):

In particular

h jEx(M) i =ZMd3y j (y)j2

while

h jFhx(M) i =ZMd3x

ZR3d3yh (x¡ y)j (y)j2:

As a useful exercise let us directly check that Fh indeed obeys the covariance require-ments. We have for arbitrary M 2F

U y(a)Fhx(M)U(a) =ZMd3y

ZR3d3xh (x¡ y)U y(a)jxihxjU(a)

=ZMd3y

ZR3d3xh (x¡ y)jx¡aihx¡aj

=ZMd3y

ZR3d3xh (x+a¡ y)jxihxj

=ZM¡a

d3y

ZR3d3xh (x¡ y)jxihxj

= Fhx(M ¡a):

As far as rotations are concerned we exploit invariance of h under rotations

U y(R)Fhx(M)U(R) =ZMd3y

ZR3d3xh (x¡ y)U y(R)jxihxjU(R)

=ZMd3y

ZR3d3xh (x¡ y)jR¡1xihR¡1xj

=ZMd3y

ZR3d3xh (Rx¡ y)jxihxj

=ZMd3y

ZR3d3xh (x¡R¡1y)jxihxj

=ZR¡1M

d3y

ZR3d3xh (x¡ y)jxihxj:

= Fx(R¡1M)

3 Observables 49

Finally we have invariance under boosts, namely

U y(q)Fx(M)U(q) =ZMd3y

ZR3d3xh (x¡ y)U y(q)jxihxjU(q)

=ZMd3y

ZR3d3xh (x¡ y)eiq�xjxihxje¡iq�x

= Fx(M):

The POVM actually is a smeared version of the usual sharp position observable, theprobability density h(x) which �xes the POVM being understood as the actual, �niteresolution of the registration apparatus. For any state � the �rst moment of the associ-ated probability density can still be expressed as the mean value of the usual positionoperator, since Mean(h)= 0

Mean�(Fhx) =

ZR3

ydTr�Fhx(y)

=ZR3d3y

ZR3d3xyh (x¡ y)Tr�jxihxj

= hx̂ i�:= Mean�(Ex)

�The second moment however di�ers

Var�(Fhx) =

ZR3d3yy2 dTr�Fh

x(y)¡hx̂ i�2

=ZR3d3y

ZR3d3xy2h (x¡ y)Tr�jxihxj ¡ hx̂ i�2

= hx̂2i�¡hx̂ i�2+Var(h)= Var�(Ex)+Var(h):

�The variance is no more expressed only by the mean value of the operator which can

be used to evaluate the �rst moment and by its square. Besides the variance for a sharpposition measurement, a further contribution Var(h) appears, which is state independentand re�ects the �nite resolution of the equivalence class of apparata used for the local-ization measurement. Note that the usual result is recovered in the limit of a sharplypeaked probability density h(x)! �3(x), corresponding to a pure state in the set of clas-sical probability measures. Taking e.g. a distribution of the form

h�(x)=�

12��2

�3

2 e¡1

2

x2

�2!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !�!0�3(x)

one has that in the limit of an in�nite accuracy in the localization measurement of theapparatus exploited the POVM reduces to the standard PVM

Fh�x (M)=

ZMd3y

ZR3d3x

�1

2��2

�3

2 e¡ 1

2�2(x¡y)2jxihxj

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !�!0

ZMd3y

ZR3d3x�3 (x¡ y)jxihxj

=ZMd3xjxihxj:


Analogous results can obviously be obtained for a momentum observable, asking for thecorresponding covariance properties.

Position and momentum observableA more interesting situation appears when considering apparata performing both a

measurement of the spatial location of a particle as well as of its momentum. As it iswell known no observable can be associated to such a measurement in the framework ofstandard textbook quantum mechanics. Let us consider the covariance equations of suchan observable in the more general framework of POVM. A position and momentumobservable should be given by a POVM Fx;p de�ned on B (R3 � R3) satisfying the fol-lowing covariance equations under the action of translations, rotations and boostsrespectively

U y(a)Fx;p (M �N)U(a) =Fx;p (M ¡a�N) 8a2R3

U y(R)Fx;p (M �N)U(R) =Fx;p (R¡1M �R¡1N) 8R2SO(3)U y(q)Fx;p (M �N)U(q) =Fx;p (M �N ¡ q) 8q 2R3:

Such covariance equations, de�ning a position and momentum observable by means ofits operational meaning, do not admit any solution within the set of PVM, while thegeneral solution within the set of POVM is given by

FSx;p (M �N)= 1

(2�~)3

ZMd3x

ZNd3pW (x; p)SW y(x; p)

where S is a trace class operator, positive, with trace equal to one and invariant underrotations

S 2T (H) S � 0 TrS=1 U y(R)SU(R)=S

so that it is in fact a statistical operator, even though it does not have the meaning of astate, while the unitaries

W (x; p) = e¡ i

~ (x�p̂¡x̂�p)

= e¡ i

~x�p̂e+i

~ x̂�pe+

i

2~x�p

are the Weyl operators built in terms of the canonical position and momentum opera-tors, namely the generators of translations and boosts respectively. One can directlycheck as a useful exercise that FS

x;p (M � N) is a POVM. Indeed since it is built by aLebesgue integral with an operator kernel, one immediately has �-additivity, as well asFSx;p (;)= 0. Furthermore normalization can be checked evaluating the matrix elements

hz jFSx;p (R3�R3)z 0i = 1

(2�~)3

ZR3d3x

ZR3d3phz jW (x; p)SW y(x; p)jz 0i

= 1(2�~)3

ZR3d3x

ZR3d3p

�z��e¡ i

~x�p̂e+i

~ x̂�pS e¡ i

~ x̂�pe+i

~x�p̂��z 0�

= 1(2�~)3

ZR3d3x

ZR3d3p

�z ¡x

��e+ i

~ (z¡x)�pS e¡ i

~ (z0¡x)�p

��z 0¡x

�= �3(z¡ z 0)

ZR3d3xhxjS jxi

= �3(z¡ z 0):

3 Observables 51

The covariance of the proposed POVM can be directly checked, exploiting the propertiesof S and working with the matrix elements of the operator expression. The couple (U ;Fx;p), where U is the unitary representation of the symmetry group and Fx;p a POVMcovariant under its action is called system of covariance.

The connection with position and momentum observable as well as the reason whysuch a joint observable can be expressed only in the formalism of POVM, where positionobservables alone are generally given by smeared versions of the usual position observ-able, and similarly for momentum, can be understood looking at the marginal observ-ables. Starting from the POVM one can consider a measure of position irrespective ofthe momentum of the particle, thus coming to the marginal position observable

FSx(M) = FS

x;p (M �R3)

= 1(2� ~)3

ZMd3x

ZR3d3pW (x; p)SW y(x; p)

= 1(2� ~)3

ZMd3x

ZR3d3pe

¡ i

~x�p̂ZR3d3z

ZR3d3z 0e

+i

~ (z¡z0)�pjzihz jS jz 0ihz 0j e+

i

~x�p̂

=ZMd3x

ZR3d3z jz+xihz jS jzihz+xj

=ZMd3y

ZR3d3xjxihx¡ y jS jx¡ yihxj

=ZMd3y

ZR3d3xhSx (x¡ y)jxihxj;

where the function

hSx(x)= hxjS jxi

is a well de�ned probability density invariant under rotations due to the fact that theoperator S has all the properties of a statistical operator invariant under rotations, sothat hxjS jxi would be the position probability density of a system described by thestate S. On similar grounds the marginal momentum observable is given by

FSp(M)=FS

x;p (R3�N)

=ZNd3q

ZR3d3pjpihp¡ q jS jp¡ qihpj

=ZNd3q

ZR3d3phSp (p¡ q)jpihpj

where again the function

hSp(p)= hpjS jpi

is a well de�ned probability density, which corresponds to the momentum probabilitydensity of a system described by the statistical operator S. As it appears the marginalobservables are given by two POVM characterized by a smearing of the standard posi-tion and momentum observables by means of the probability densities hSx(x) andhSp(p) respectively. The latter probability densities as we have seen originate from thesame statistical operator S, so that we also have

Vari(hSx)Vari(hSp) �~24

i=x; y; z:


It is exactly this �nite resolution in the measurement of both position and momentum,stemming from FS

x;p that allows for a joint measurement for position and momentum inquantum mechanics, without violating Heisenberg's uncertainty relations. Two positionand momentum observables FS

x and FSp obtained from con�dence functions hSx(x) and

hSp(p) satisfying the above uncertainty relation are called a Fourier couple. Consideringthe product of the variance of the two marginals FS

x and FSp in a given state � and

restricting for the sake of simplicity to one dimension one has

Var�(FSx)Var�(FSp) = (Var�(Ex)+Var(hSx))(Var�(Ep)+Var(hSp))

= Var�(Ex)Var�(Ep)+Var(hSx)Var(hSp)+Var�(Ex)Var(hSp)+Var�(Ep)Var(hSx);

where due to Heisenberg's uncertainty relation and the Fourier contraints the �rst twoterms at the r.h.s. are both greater than ~2/4. Indeed the variances of hSx and hSp sat-isfy as� shown an uncertainty principle, while Ex and Ep are the usual PVM observ-ables for position and momentum. Exploiting the same constraints the last two terms atthe r.h.s. obey the following inequality

Var�(Ex)Var(hSp)+Var�(Ep)Var(hSx) >~24

�Var�(Ex)Var(hSx)

+ Var(hSx)Var�(Ex)

�> ~2

2;

since the function x + 1/x is bounded from below by 2. As a result we have theinequality

Var�(FSx)Var�(FSp) > ~2:

Thus if one consider the statistics of a measurement of position obtained from the mar-ginal of a given joint position momentum observable, as well as measurement ofmomentum obtained from the marginal of the same joint position momentum observ-able, the product of the variances stay above the lower bound that one has for sharpposition and momentum observables, just because of the added unsharpness necessary inorder to consider a joint measurement.

In order to consider a de�nite example we take S to be a pure state corresponding toa Gaussian of width �

hxj i=�

12��2

�3

4 e¡ 1

4�2x2

;

on which the Weyl operators act as a translation in both position and momentum,leading to

hxjW (x0; p0)j i=�

12��2

�3

4 e¡ 1

4�2(x¡x0)2+

i

~p0�(x¡x0)= hxj x0;p0i:

In particular one has for i=x; y; z

h x(x) =�

12��2

�3

2 e¡ 1

2�2x2

Vari(h x) = �2

3 Observables 53

together with

h p(p) =�2�2

�~2

�3

2 e¡2�2

~2p2

Vari(h p) = ~24�2

so that

Vari(h x)Vari(h p) = ~24:

The POVM now reads

Fx;p (M �N)= 1(2� ~)3

ZMd3x0

ZNd3p0j x0p0ih x0p0j;

and it is worthwhile to verify directly the properties of POVM. Note that this is only aPOVM and not a PVM just because the vectors f x0p0gx0;p02R3 are not orthogonal, butrather provide an overcomplete set. We have in fact

h x0p0j x00p00i = e¡ �2

2~2(p0¡p00)2

e¡ 1

8�2(x0¡x00)2

e¡ i

~

¡p0+p0

0 �2

�(x0¡x00);

and they correspond to coherent states with complex parameter �= 1

2�

�x0+

i

~2�2p0

�Fx;p(;) = 0, as well as �-additivity directly follows from the properties of the inte-

gral. Regarding normalization we have, taking matrix elements

hz jFx;p (R3�R3)z 0i = 1(2�~)3

ZR3d3x0

ZR3d3p0hz j x0p0ih x0p0jz 0i

= 1(2�~)3

�1

2��2

�3

2

ZR3d3x0

ZR3d3p0 e

¡ 1

4�2[(z¡x0)2+(z0¡x0)2]+

i

~p0�(z¡z0)

=�

12��2

�3

2

ZR3d3x0e

¡ 1

4�2[(z¡x0)2+(z0¡x0)2] 1

(2�~)3

ZR3d3p0e

+i

~p0�(z¡z0)

= �3(z¡ z 0);

which corresponds to the statement that the vectors f x0p0g provide an overcompleteset

1 = 1(2� ~)3

ZR3d3x0

ZR3d3p0j x0p0ih x0p0j:

The marginals are given by

Fx(M)=ZMd3x0

ZR3d3x

�1

2��2

�3

2 e¡ 1

2�2(x¡x0)2jxihxj

and

F p(N)=ZNd3p0

ZR3d3p

�2�2

�~2

�3

2 e¡2�2

~2(p¡p0)2jpihpj

for position and momentum respectively. It is now clear that depending on the value of� one can have more or less coarse-grained position and momentum observables. Nolimit on � can however be taken in order to have a sharp observable for both positionand momentum. In the limit � ! 0 one has as before Fx! Ex, but the marginal formomentum would identically vanish, intuitively corresponding to a complete lack ofinformation on momentum, and vice versa.


4 Transformations of states and observables

In order to understand and describe the basic statistical structure of quantummechanics, we have considered as basic statistical experiments those describable as adirect interaction of a single particle prepared by a preparation apparatus and a�ectinga registration apparatus.

[preparationapparatus] [registrationapparatus]:

In this setting the preparation procedure produces a quantum output, namely a state�2S(H), while the registration procedure accepts a quantum input and produces a clas-sical output corresponding to the probability distribution of the considered observable.In general a measurement can be performed in di�erent steps, however the scheme canalways be traced back to this one by including intermediate transformations either onthe side of the preparation or of the registration. Note that if many intermediate stepscan be put into evidence such inclusion can generally be performed in di�erent ways.Think for example of the Stern Gerlach experiment. One prepares a beam of particles,then spatially separates the contributions with di�erent spin components along a givenaxis thanks to an inhomogeneous magnetic �eld, �nally a measurement of position isperformed on a screen. The action of the magnetic �eld, which gives the proper dynam-ical interaction, can be seen as part of the state preparation corresponding to a transfor-mation on the space of states, thus corresponding to the Schrödinger picture

[beampreparation] [magnetic�eld]preparation apparatus

[detection screen]registration apparatus

;

or equivalently it can be seen as part of the registration, determining a transformationon the space of observables and thus corresponding to the Heisenberg picture

[beampreparation]preparation apparatus

[magnetic�eld] [detection screen]registration apparatus

:

Given this situation it immediately appears that it is of interest to characterize the pos-sible transformations of states and of observables, which lead to a description of theintermediate steps in the characterization of a measurement procedure. We will calloperations such transformations, which accept a quantum input and produce a quantumoutput. These transformations of relevant spaces of operators into themselves are obvi-ously of great importance. Each transformation can be seen in a dual way, exchangingthe role of states and observables. They allow to describe not only intermediate steps ina given measurement setting, they describe symmetry transformations, input/outputtransformations over a �nite time, so called one shot transformations, as well as a gen-eral time dependent dynamics. Such transformations are moreover necessary if besides asingle measurement, leading from a quantum state to a classical probability distribution,one wants to describe repeated measurements of the same or of di�erent observables,leading to conditional probabilities. Indeed in order to perform two measurements oneafter the other one needs to know, besides the statistics of the �rst measurement, thestate transformed as a consequence of the �rst measurement, so as to apply to this newstate the second measurement. In such a way one can obtain conditional probabilities.By iterating this procedure one can also think about performing a high number of subse-quent measurements of the same observable, so that in the limit of arbitrarily small timein between these subsequent measurements a description of measurement continuous intime can be obtained, thus recovering in a suitable framework the notion of trajectoryalso in quantum mechanics.

4 Transformations of states and observables 55

We focus on transformations of the space of states S(H)� T (H). Such maps have torespect the convex structure of the space of states, and as already seen such maps canbe uniquely extended to the whole linear space T (H), which then allows to exploituseful duality relations. In a general transformation an ensemble of quantum states willbe possibly sent to a subensemble. Think e.g. of the situation in which in the prepara-tion one takes out of an unpolarized beam particles those which are polarized along agiven axis, thus picking up only a fraction of the beam. Such state transformations thatwe have called operations will thus send elements of S(H) in the space S~(H) = f� 2T (H); � > 0; Tr � 6 1g � T (H) of subcollections or subnormalized statistical operators.Denoting by O such an operation

O: T (H) ! T (H)� O[�]

the natural requirements are: i) linearity, ii) positivity, namely O[�] > 0 for � > 0, andiii) trace non-increasing, namely Tr O[T ] 6 Tr T for all T 2 T (H). The latter conditionre�ects the fact that a state can be sent to a subcollection. It turns out however thatthe requirement of being positivity preserving is not su�cient, since it allows for the fol-lowing unphysical situation. Suppose we have an operation acting on T (H), and we con-sider our system described on H together with another system described on a Hilbertspace Cn with n 2 N. We extend our map on T (H Cn) by letting it act in a trivialway on the other degrees of freedom, namely we consider the map O 1. It mighthappen that the latter map is not positive, so that states are no more sent to states.Such a behaviour, due to the non commutativity of the space of operators, calls for amore stringent requirement on the transformation maps known as complete positivity.

Complete positivityA property of maps on spaces of operators which becomes of relevance when the

algebra of observables is non-commutative is complete positivity. To introduce it let us�rst recall that the space of bounded operators B(H) is the dual of the space of traceclass operators T (H), i.e. B(H)�T 0(H) according to the duality formula

Tr:B(H)�T (H) 7! C

(B; �) TrBy�

with B 2B(H) and �2T (H). Given a map � acting on the states

�: T (H) 7! T (H)

the duality relation allows to introduce the adjoint map acting on the dual space

�0:B(H) 7! B(H);

whose action is de�ned asking

Tr(B�[�]) = Tr(�0[B]�)

for all B 2B(H) and �2T (H). Note that from the relation

Tr(�[T ]) = Tr(�[T ]1)= Tr(T�0[1])


for all T 2 T (H) one immediately has that if � is trace non increasing, then �0[1] is ane�ect, that is to say 0 6 �0[1] 6 1, as can be seen taking as trace class operators onedimensional projections. In particular trace preservation of � corresponds to the factthat �0 is unit preserving, namely �0[1] =1. For example consider the map

�(t)[�] = U(t)�U y(t);

which gives the time evolution in Schrödinger picture. Its adjoint is easily seen to begiven by

�0(t)[X] = U y(t)XU(t);

which gives the time evolution in Heisenberg picture. The notion of dual map allows tointroduce a notion of time evolution for the observables even if the dynamics is no moreunitary.

We now de�ne an important property of these maps, known as complete positivity.We say that a map �0

�0:B(H) 7! B(H)

is completely positive if the following equivalent conditions hold:

I. For any n2N the maps

�01:B(HCn) 7! B(HCn)

send positive operators to positive operators.

II. For any n2N the inequality holdsXi; j=1

n

h ij�0¡BiyBj

�j ji> 0 8f ig�H;8fBig�B(H) :

Note that due to the relation (� 1)0 = �0 1, which can be checked starting from theduality relation, the requirement of complete positivity can be equivalently set on themap � itself rather than its adjoint, that is asking

�1:T (HCn) 7! T (HCn)

to be positive for all n 2N (or up to the dimension of H in the �nite dimensional case).This condition allows to rephrase the condition asking that the map extended in thetrivial way to a tensor product structure remains positive. If � describes e.g. aSchrödinger evolution, the extension of the description of the dynamics to anothersystem which has itself a trivial dynamics and is not coupled to the system of intereststill leads to an evolution preserving positivity. As we shall see below the di�erencebetween positivity and complete positivity can be detected considering entangled states.

To see the equivalence of the two de�nitions recall that HCn�L

i=1n H, so that a

pure state in HCn can be represented as

ji =

0@ 1�� n

1A;


with j 2 H, at the same time an element of B(H Cn) can be identi�ed as a n � nmatrix with operator entries B(H Cn) �Mn(B(H)). A positive operator A in B(H Cn) can be written in the form

A =Xj;k=1

n

BjyBk jejihekj

=Xj;k=1

n

BjyBkEjk ;

where fejg denote the canonical basis of vectors in Cn, and fEjkg the canonical basis ofmatrices in Mn(C). Of course other bases can be considered, e.g. f'jg, and in thematrix elements relative to this basis setting Ejk = j'jih'k j we still have that theseoperators can be represented as the canonical basis of matrices in Mn(C). We cantherefore write for a positive operator A

A =

0B@ B1yB1 ::: B1

yBn��

BnyB1 ::: Bn

yBn

1CA= B yB

with B � (B1; :::; Bn) a vector of operators. Positivity of the map �0 1 on B(H Cn)is then expressed as

hj�01(B yB)ji> 0 8f ig� (HCn); 8fBig�B(H)

which can equivalently be writtenXi; j=1

n

h ij�0¡BiyBj�j ji> 0 8n2N;8f ig�H; 8fBig�B(H):

Examples of completely positive maps are given by unitary transformations

�[�] = U�U y;

directly from the de�nition one hasXi; j=1

n

h ij�0¡BiyBj

�j ji =

Xi; j=1

n

h ijU y¡BiyBj

�U j ji

=Xi; j=1

n

h ijU yBiyUU yBjU j ji

=

Xj=1n

U yBjU j ji

H

2

> 0:

The same holds for the trace of a matrix

tr:Mn(C) 7! C

A Xi=1

n

Aii

and note that this map can be represented as follows

trA =Xi=1

n

eiAeiy


with ei= (0; :::; 1; :::; 0), whose unique nonzero element is in the i-th position. This caseis however trivial, since positivity and complete positivity are equivalent conditions ifthe elements of either the initial or �nal space commute. The notion of complete posi-tivity is therefore only relevant for quantum probability.

Also the partial trace TrE is completely positive, indeed one has, 8f ig � HS,8fBig�B(HSHE) and 8n2NX

i; j=1

n

h ijTrE¡BiyBj

�j ji =

Xi; j=1

n X�

h ijh'�jBiyBj j'�i j ji

=X�

Xj=1n

Bj j'�i j ji

HSHE

2

> 0

Note that according to the de�nition the composition of completely positive maps iscompletely positive, in fact, if both � and � are completely positive, then both � 1and �1 are positive, so that such is

(�1) � (�1) = (� ��)1

and therefore � � � is completely positive. The same goes for linear combinations withpositive coe�cients, so that the set of completely positive maps forms a positive cone.

OperationsWe are now in the position to actually de�ne an operation as a map

O: T (H) ! T (H)T O[T ]

which is linear, completely positive and trace non increasing, i.e. Tr O[T ] 6 Tr T for allT 2 T (H). Note that actually, as we shall see in the examples, one can consider situa-tions in which the initial and �nal Hilbert spaces are not the same. If in particularTr O[T ] = Tr T for all T 2 T (H), the quantum operation is called a channel. Note thatan operation generally sends S(H) into S~(H), so that a statistical operator is sent to asubcollection, while a channel sends S(H) into S(H), so that statistical operators aretransformed in statistical operators. As we have seen if O is trace preserving, than thedual map O 0 preserves the identity and is called unital, i.e. O 0[1] = 1. In general if O istrace decreasing O 0[1] is a positive operator between zero and one, that is an e�ect. Fur-thermore as we have seen the map O is completely positive i� such is O 0, so that com-plete positivity can be equally well be veri�ed on the map or its dual. Note �nally thatthe space of operations or channels is closed under convex combinations.

Let us consider a few examples. For a given unitary operator U 2U(HS), we considerthe map AU[�] = U�U y, which is in particular a state automorphism, that is to say abijection of the space of states S(HS) onto itself, which preserves its convex structure.Such a map is an operations and in particular a channel, a so called unitary channel.Linearity and trace preservation immediately appear, while regarding complete posi-tivity, for any Hilbert space HE, taken T 2 T (HS HE), T > 0 we have, for an arbitrary 2HSHE

h j(AU 1E)[T ] i = h j(U 1E)T (U 1E)y i= h 0jT 0i> 0;


which grants complete positivity. Note that if U is anti unitary, such a map does notde�ne a channel any more.

As a further basic example we consider a basic simple operation built as follows.Consider an operator S 2B(H) and de�ne

OS[T ] = STS y:

Linearity is obvious, while regarding complete positivity we argue as before

h j(OS1E)[T ] i = h j(S 1E)T (S 1E)y i= h 0jT 0i> 0:

For OS to be an operation we further need it to be trace non increasing. This conditionleads to

TrOS[T ] = TrS yST6 TrT ;

for arbitrary trace class operator T . The map OS then is an operation i� S yS is ane�ect, that is S yS 6 1 or equivalently kSk6 1. The adjoint map has the same form withS replaced by its adjoint, namely OS0 =OSy.

Let F be a non zero positive trace class operator F 2 T (H) and de�ne a map on thespace of trace class operators as follows

OF [T ] = TrTF

TrF;

so that each operator is sent to an operator proportional to the state F/(Tr F ). Thedual map is de�ned verifying the identity

TrTOF0 [B] = TrOF [T ]B

for all B 2B(H) and T 2T (H). We thus have

TrTOF0 [B] = TrOF [T ]B

= Tr�TrT

FTrF

B

�= Tr

�TTr(BF )TrF

1

�and therefore

OF0 [B] = Tr(BF )TrF

1:

From the last expression one immediately sees complete positivity. In particular, whilethe map OF in Schrödinger picture sends all the states to the state F/(Tr F ), the dualmap OF0 sends all observables in a multiple of the identity. In particular OF0 contractsall the space on the commutative algebra of operators proportional to the identity. Inthis respect we recall the fact that the notion of complete positivity coincides with thenotion of positivity if either initial of �nal space are commutative.


We now consider a situation in which an operation connects operators acting on dif-ferent Hilbert spaces. Given a �xed statistical operator �E on the Hilbert space HE wede�ne the map

A�E: T (HS) ! T (HSHE)T T �E

often called assignment map, whose complete positivity is immediately seen consideringthat extending it to another ancillary Hilbert space HA, for any positive operator X 2T (HSHA) one has

(A�E1A)X = X �E

which is still a positive operator. This operation is in particular a channel. The dualmap is de�ned as

A�E0 :B(HSHE) ! B(HS)

B A�E0 [B]

satisfying

TrSTA�E0 [B] = TrSEA�E[T ]B

= TrSET �EB= TrSE(T 1E) (1S �E)B= TrS T TrE(1S �E)B

so that we have

A�E0 [B] = TrE(1S �E)B;

which is in particular unital, corresponding to the fact that A�E is a channel.The partial trace is another example of completely positive map, which sends states

to states. It is a mapping

TrE:T (HSHE) ! T (HS)T TrE [T ]

de�ned by the requirement

TrS(TrE [T ]B) = TrSE(T (B 1E))

for all B 2B(HS) and all T 2T (HSHE). The dual map

(TrE)0:B(HS) ! B(HSHE)B (TrE)0[B]

is then obtained from the relation

TrSE(T (TrE)0[B]) = TrS(TrE [T ]B)= TrSE(T (B 1E));

so that

(TrE)0[B] = B 1E:


From the latter expression one immediately reads complete positivity of the map, as wellas trace preservation. The dual map thus amounts to embed the original observable in alarger algebra.�

Operations and measurementWe now dwell on the physical meaning of operations in connection with conditional

measurements. As we have already seen to an operation one uniquely associates ane�ect via E =O 0[1]. Vice versa many distinct operations can lead to the same e�ect, sothat to an e�ect one can associate a whole equivalence class of operations [O]E, whichare E compatible in the sense that O 0[1] = E for all O 2 [O]E. This many to one rela-tionship is associated to the meaning of operation as state transformation correspondingto the measurement of the elementary observable described by the e�ect E. There are ingeneral many di�erent way to measure the same e�ect, which lead to a di�erent statechange, and they correspond to the set of operations compatible with the given e�ect.Let us take the pre-measurement � 2 S(H). The probability associated to the eventdescribed by the e�ect E is then given by

P (E) = ��E

= Tr�E= Tr �O 0[1]= TrO[�];

where O is any operation compatible with the given e�ect. As it appears, the proba-bility for the realization of the e�ect is given by the trace of the subcollection O[�],which provides the statistical weight associated to the event. While as far as the mea-surement of E only is concerned, any operation in the equivalence class [O]E leads tothe same result, things change when we consider to perform subsequently the measure-ment of another e�ect, let us call it F . To perform such a second measurement we needto know not only the statistics of the �rst measurement, but also the transformed state,so that the notion of operation now becomes essential. The post-measurement state con-ditioned on the occurrence of E is given by

�jE = O[�]TrO[�] ;

which is a suitably normalized statistical operator. We can now express the conditionalprobability for the occurrence of the e�ect F in the second measurement given theoccurrence of E in the �rst measurement as

P (F jE) = ��jEF

= Tr �jEF

= TrO[�]FTrO[�]

= TrO[�]FP (E)

;

which also leads to the expression of the joint probability as

P (F ;E) = P (F jE)P (E)= TrO[�]F= Tr �O 0[F ]:


Note that the order in which the measurement are performed is actually relevant.Indeed denoting by OE and OF operations in the equivalence class compatible with thee�ects E and F respectively, we have the relations

P (F ;E) = TrOF [OE[�]]= TrOE0 [OF0 [1]]

to be compared with

P (E;F ) = TrOE[OF [�]]= TrOF0 [OE0 [1]]:

Note further how the second measurement actually depends on how the state has beentransformed through the operation O in performing the �rst measurement. Di�erent ele-ments of the equivalence class [O]E lead to di�erent results for P (F ; E). While thetransformation on the state O[�] corresponds to a Schrödinger picture, the transforma-tion on the e�ect O 0[F ] can be seen as a Heisenberg picture. As already discussed thestatistics of an experiment in which di�erent steps are performed can be divided in dif-ferent ways in a preparation and a registration part. We note further that while ane�ect E uniquely �xes its complementary e�ect E?, the operation O? which determinesthe post-measurement state linked to occurrence of the e�ect E? is still not �xed evenonce a representative O in the equivalence class [O]E has been �xed, since the only con-straint is O?0[1] +O 0[1] = 1, so that once again a whole equivalence class of operations[O]E? leads to the same statistics.

It is important to stress that while O is a linear map, the map which gives the stateconditioned on the outcome

� �jE=O[�]

TrO[�]

is no more linear.

�Indeed considering a convex combination of states �= p1�1+ p2�2 we have

�jE = p1O[�1] + p2O[�2]p1Tr �1E+ p2Tr �2E

= p1Tr �1Ep1Tr �1E+ p2Tr �2E

� O[�1]Tr �1E

+ p2Tr �2Ep1Tr �1E+ p2Tr �2E

� O[�2]Tr �2E

= p(1jE)�1jE+ p(2jE)�2jE;

where p(1jE) is the probability of the �rst case conditioned on the outcome E and simi-larly for p(2jE), so that a di�erent convex combination appears, corresponding to a lackof linearity.

As a last remark let us come back to our example of simple operation OS[T ] =STS y,requiring the condition S yS 6 1 which ensures that OS0 [1] is an e�ect. We immediatelysee that more in general the map O[T ] =

Pk SkTSk

y still gives an operation provided the

e�ects Ek = SkySk are compatible, that is

Pk Sk

ySk still is an e�ect lying below the iden-tity.

Operations and state preparation


We now describe the connection between operations and state preparations. As wealready discussed, a possible way to perform the preparation of a state is to exploitapparata for the sharp measurement of a complete set of commuting observables whichcan be identi�ed with a projection-valued measure. In this setting let us denote by Athis set of observables and a the corresponding eigenvalues, which are non degenerateand correspond to certain eigenstates ua. We thus have the spectral representation(A)i =

Pa aiPa, where the Pa are one-dimensional projections, Pa = juaihuaj. Let us

denote by �in our pre-measurement state. We want to consider an operation Oadescribing the statistics of the measurement of our observable, so that it has to satisfythe constraint

TrOa[�in] = P (a)= Tr �inPa:

It is natural to consider the Ansatz

Oa[�in] = Pa�inPa;

which turns out to be a proper operation obviously compatible with our constraint. Thee�ect associated to each of these operations Oa0 [1] = Pa is actually a projection. Thesechoice of operations lead to certain special features for our state transformation whichare known as repeatability and ideality. Repeatability means that if we repeat the mea-surement on the state selected according to a given outcome, with probability one werecover the same outcome.

�Ideality means that if an outcome occurs with probability one, then the initial state

is not disturbed. More speci�cally, suppose TrOa[�in] = 1 for a certain a, then Pa�inPa=�in. Indeed one has

1 = TrOa[�in]= Tr �inOa0 [1]= Tr �inPa;

so that

Tr(1¡Pa)�in = 0;

but if Q is a projection, Tr Q� = 0 implies Q� = 0. In fact we then have 0 = Tr�Q =Tr( �p

Q)y( �p

Q)= k �p

Qk22, so that since the Hilbert-Schmidt norm is equal to zero onealso has 0 = �Q = Q�. As a result we have (1¡ Pa)�in= �in(1¡ Pa) = 0, and therefore�inPa = Pa�in = �in as desired. Note that these features are related to the fact that theassociated e�ects are actually projections, and the spectrum is discrete. The post-mea-surement state conditioned on the occurrence of the e�ect Pa is given according to theprevious formulae by

�outja = Oa[�in]TrOa[�in]

= Pa�inPaTr �inPa

= juaihuaj;


that is the expected pure state. If no selection is performed, the state which a prioricomes out of the measurement procedure is the mixture

�out =Xa

P (a)�outja

=Xa

P (a) Oa[�in]TrOa[�in]

=Xa

Pa�inPa;

diagonal in the basis common to the complete set of commuting observables.

�

Mathematical characterization of operationsWe now want to provide a general characterization of completely positive maps, both

in view of their structure and of their physical interpretation. This result will be accom-plished by means of a few basic theorems, which we consider now. We �rst mention abasic result on the general structure of completely positive maps between algebras.

Theorem (Stinespring, 1955) A linear map �0: B(HS) ! B(HS) is completely posi-tive i� there exists a Hilbert space HE, a bounded operator V : HS ! HS HE and anormal, unit preserving �-homomorphism �: B(HS)! B(HS HE) such that �0[AS] =V y�(AS)V holds. � can be always realized as �(AS) =AS 1E, so that we have the rep-resentation

�0[AS] = V yAS1EV :

�We recall that according to its de�nition the �-homomorphism � is a map from the

algebra B(HS) to the larger algebra B(HSHE)

�:B(HS) 7! B(HSHE)AS �(AS)

that satis�es

�(ASBS) = �(AS)�(BS)�¡ASy � = �(AS)y

�(1S) = 1S1E:

These properties imply in particular that � is a positive map, and more than this a com-pletely positive map. For a normal map this embedding in the larger algebra can alwayswe written as �(AS) = AS 1E, so that it is equivalent to the map (TrE)0 considered inthe previous examples. Normality is a regularity requirement, implying that convergentmonotone non-decreasing sequences of operators are transformed into convergentmonotone non-decreasing sequences.

��

As a �rst application of Stinespring dilation's theorem we provide a general proof ofthe already discussed Naimark's theorem.


Theorem (Naimark, 1943) Let F be a positive operator-valued measure on the mea-sure space (; F), so that F : F ! B(HS). Then there exists a Hilbert space HE, abounded operator V :HS!HS HE and a projection-valued measure on the same mea-sure space E:F!B(HSHE) such that

F (M) = V yE(M)V :

The proof goes as follows. First we observe that the positive operator-valued measure,de�ned on the measure space (; F), induces in a natural way a map acting on thealgebra C() of continuous functions on , which becomes a Banach �-algebra with thesupremum norm, as follows

�F :C() ! B(HS)

h �F(h)=Zh(x) dF (x);

where the operator �F(h) with matrix elements

h'j�F(h) i =Zh (x)dh'jF (x) i

is indeed bounded, as follows from the inequality

h'j�F(h)'i =Zh (x)dh'jF (x)'i

6 khk1 :

Zdh'jF (x)'i

= khk1k'k2

valid 8'2HS:

�Namely, to each function one associates its integral over the measure space with the

given positive operator-valued measure.This map is obviously positive, and therefore, since the space C() is abelian, it is

also completely positive. We can therefore exploit Stinespring theorem to write it in theform

�F(h) = V y�(h)V ;

where � is a �-homomorphism �:C()!B(HS HE) between algebras. All such homo-morphisms are of the form

�(h) =Zh (x)dE(x)

where E is a projection-valued measure on the measure space (; F) taking values inB(HS HE). The de�ning properties of a �-homomorphism can be directly veri�ed, andin particular one has

h�j�(h f)i =Zh (x)f (x)dh�jE(x)i

= h�j�(h)�(f)i;

just thanks to idempotency of the elements of the projection-valued measure. We havethus associated to the initial positive operator-valued measure F taking values in B(HS)a projection-valued measure E on the same measure space taking values in B(HSHE).


We now exploit Stinespring to obtain a most important structural characterization ofcompletely positive maps. We consider for de�niteness the case of a trace preservingcompletely positive map � with equal initial and �nal space, however the result works aswell in the case of di�erent initial and �nal spaces, as well as considering general tracenon increasing maps, the only relevant feature being complete positivity.

Theorem (Kraus, 1971) Given a completely positive trace preserving map �:T (HS)!T (HS) the following statements are equivalent:

1. � is completely positive according to either of the two equivalent formulations

2. � can be written in Kraus form

�[�] =XK

AK�AKy

where the sum is over a set which is at most denumerable, convergence is in theweak sense and

PK AK

y AK=1

3. � can be expressed in the form

�[�] = TrEfU� �EU yg

for a certain Hilbert space HE, a state �E, which can be taken to be pure, and aunitary operator U on HSHE.

The expressions appearing in 2 and 3 are known as �rst and second representation the-orem for a completely positive map. While the �rst representation is very useful since itgives the generic structure of an arbitrary completely positive map in terms of operatorsin the Hilbert space of interest only, the second representation tells us that any com-pletely positive transformation can be seen as originating from the interaction with anexternal system, taking an initially factorized state between system and environment.Each completely positive dynamics can be obtained taking the partial trace with respectto a unitary evolution in a larger Hilbert space. The non uniqueness of this constructionre�ects itself in the non uniqueness of the Kraus representation.

Let us prove the sequence 1) 3) 2) 1.1) 3 By de�nition � is completely positive i� �0:B(HS)!B(HS) has this property.

We now use Stinespring's theorem adapted to the case in which �0 is a normal map. Inthis case for a completely positive map �0 on the algebra B(HS) there exists a Hilbertspace HE and a bounded operator V :HS!HSHE such that

�0[AS] = V yAS1EV ;

while identity preservation implies V yV =1. For �xed 2HE of norm one we can there-fore de�ne the operator

U(' ) = V' 8'2HS ;

which turns out to be an isometry thanks to the property of V and the fact that is ofnorm one, indeed

hU('0 )jU(' )i = hV'0jV'i= h'0jV yV'i= h'0j'i= h'0 j' i:


One can now extend U to de�ne a unitary operator on the whole HS HE. With thisresults at hand we have the following chain of equalities, for arbitrary BS 2 B(HS), � 2T (HS) and a basis f'jg in HS

TrS�[�]BS = TrS��0[BS]= TrS�V y(BS1E)V=Xj

h'j j�V y(BS1E)V'ji

=Xj

hV�'j j(BS1E)V'ji

=Xj

hU(�'j )j(BS1E)U('j )i

=Xj

h'j j�1EUy(BS1E)U('j )i

= TrSE�1EUy(BS1E)U1SP

= TrSE(BS1E)U(�P )U y

= TrSBSTrEfU�P U yg;

that is our statement, with �E=P given in particular by a pure state.3) 2 Follows directly by considering an orthogonal resolution for �E, namely �E =P� ��j'�ih'�j and using the same orthonormal basis to evaluate the trace

�[�] = TrEfU� �EU yg=X��

¡��

ph'�jU j'�i

��¡

��p

h'�jU j'�i�y

=XK

WK�WKy

with K=�; � a multi-index, and the operators WK de�ned on HS via

WK = ��p

h'�jU j'�i;

which further satisfy XK

WKyWK = 1:

2 ) 1 This can be proved noting that the Kraus representation immediately impliescomplete positivity according to the very de�nition. Given the Kraus representation onehas immediately

�1n[T ] =XK

(WK1C2)T (WK1Cn)y;

so that �1n send positive operators into positive operators for all n.We have thus obtained a fundamental result according to which any operation

admits a representation in the form

O[�] =Xk

Ak�Aky

with the constraintP

k AkyAk 6 1. If the Hilbert space HS of the system has dimension

d, it is always possible to use d2 or less operators in the sum. To show an example of thenon uniqueness of the Kraus representation consider the following completely positivetrace preserving map de�ned on HS=Cd

A: T (HS) ! T (HS)

T Tr (T ) 1d;


which sends all states to the maximally mixed state. We point to two di�erent Krausrepresentation. Consider �rst a SONC in HS given by f'igi=1;:::;d and the operatorsEij=j'iih'j j which provide the canonical basis in Md(C). They satisfyX

i; j=1

d

EijyEij =

Xi; j=1

d

j'jih'ij'iih'j j

= d1;

so that one has the Kraus representation

A[T ] = 1d

Xi; j=1

d

Eij TEijy :

As an alternative consider another basis in Md(C) obtained from the unitary operators

Urs =Xk=0

d¡1

e¡i2�

dsk jk rihk j;

where the symbol denotes the di�erence modulo d, fjkigk=0;:::;d¡1 is a basis. Theseoperators are unitary, i.e. Urs

y Urs=Urs Ursy = 1, as follows from their very de�nition, and

orthogonal according to the Hilbert-Schmidt scalar product, as follows from the identityXk=0

d¡1

ei2�

d(s¡s0)k = d �s;s0;

so that one can normalize the basis by dividing each operator by dp

. We claim that onehas the Kraus representation

A[T ] = 1d2

Xi; j=0

d¡1

Uij TUijy :

To see this relabelling for convenience the operators as Uij/ dp

! U�, since the latterform an orthonormal basis we have

T =X�

U�Tr¡U�yT�:

Given any two states '; we have

TrP P' = h jTr(j ih'j)1'i

but also

TrP P' =X�

Tr¡P U�Tr

¡U�yP'

��=X�

Tr (P U�)Tr¡U�yP'

�= h j

�X�

jU� ihU�'j�'i;

so that we have

Tr(j ih'j)1 =X�

U�j ih'jU�y

and since any operator can be expressed by means of rank one operators �nally

Tr (T )1 =X�

U�TU�y:


To clarify the freedom left in connecting two di�erent Kraus decompositions, we �nallypoint to the fact that two collections of Kraus operators fA1; :::; Ang and fB1; :::; Bmgde�ne the same operation i�

Aj =Xk=1

m

ujkBk

where the complex numbers ujk satisfyP

j=1n ujl

� ujk = �lk. In fact given this hypothesis,for any trace class operator T we haveX

j=1

n

AjTAjy =

Xj=1

n Xk;l=1

m

ujkBkTujl�Bl

y

=Xk=1

m

BkTBky:

Viceversa, suppose that for any trace class operator T the identityXj=1

n

AjTAjy =

Xk=1

m

BkTBky

holds, take T =j ih j for an arbitrary norm one vector, than one has two distinct spec-tral representations of the same operator by means of subnormalized vectors fjAj igand fjBk ig, but then it can be shown that any two such decomposition must satisfyAj =

Pk=1m ujkBk , with again

Pj=1n ujl

� ujk = �lk. Since this relationship must be truefor any unit vector we have the desired connection.

Trace distanceAs we have seen operations have a natural interpretation as transformations on the

space of states corresponding to the measurement of an elementary observable describedby an e�ect. Operations can describe the preparation of a state or mixing of a state insubcollections, as well as conditional state measurements and state preparations. Wenow want to provide a way to put into evidence the measurement character of a certainstate transformation. First we stress the fact that the measurement character is relatedto irreversibility. Indeed one has the result that a completely positive trace preservingmap E : T (HS)!T (HS) admits an inverse i� it is a unitary transformation. This can beshown directly, or referring to Wigner's theorem and the fact that operations describedby anti-unitary operators do not provide a channel, being related to transposition.�

To a completely positive trace preserving transformation one can associate a notionof disturbance, which compares the state after and before the transformation, namely�in and �out=�[�in]. This leads to quantify their di�erence or distance, which intuitivelyis maximal when input and output states are orthogonal. Such a disturbance is notrelated to the measurement character, and also a unitary channel can transform a statein another orthogonal to the given one. Another notion which can be associated to acompletely positive trace preserving map is the noise it introduces in the state, a�ectingits information content. This feature thus in�uences the comparison of two states beforeand after the action of the transformation, namely �in

1;2 and �out1;2 = �

��in1;2�; rather than

the comparison of a given input with the corresponding output. To quantify this mea-surement character, we now introduce a quanti�er of distance or distinguishabilitybetween quantum states. On the space of states S(H) we introduce a distance accordingto the following de�nition

D(�1; �2) = 12k�1¡ �2k1 �1; �22S(H):


One can verify that such a quantity is a proper distance, in particular the triangularinequality follows from the property of the trace norm k�k1 used in the de�nition. Onefurther has

D(�1; �2) = 12Trj�1¡ �2j

= 12

Xi

j�ij;

where f�ig are the eigenvalues of the self-adjoint trace class traceless operator �1 ¡ �2,so that

06D(�1; �2)6 1:

Again, since k�k1 is a norm, one has D = 0 i� �1 = �2, while D = 1 i� �1 and �2 haveorthogonal support. It can be shown that if the two states are in particular pure states,the trace distance reads

D(P'; P ) = 1¡ jh'j ij2q

:

We now show that this distance on the space of states, as it immediately appearslooking at the special situation of pure states, is directly connected to the distinguisha-bility among states. To this aim we show that given a preparation procedure that leadswith equal probability to prepare the states �1 and �2, the highest probability of successin discriminating among the two states is given by

Poptimal =12(1+D(�1; �2));

so that the trace distance provides the bias in favor of a correct identi�cation of thestate. To realize that the optimal strategy actually leads to this result let us �rst provethe identity

D(�1; �2) = maxP

TrP (�1¡ �2);

where the maximum is taken over all possible orthogonal projections. Since �1 ¡ �2 isself-adjoint and traceless we have the decomposition

�1¡ �2 = N+¡N¡;

where N+ and N¡ are positive operators with orthogonal �nite dimensional support andsuch that Tr jN+j=TrN+=Tr jN¡j=TrN¡. We thus have

D(�1; �2) = 12Trj�1¡ �2j

= 12Tr(N++N¡)

= TrN+:

If we now take the projection P+ on the eigenspace of N+, we have

TrP+(�1¡ �2) = TrP+(N+¡N¡)= TrP+N+

= TrN+

= 12Trj�1¡ �2j

= D(�1; �2):


Moreover for any other projection P we have

TrP (�1¡ �2) = TrP (N+¡N¡)6 TrPN+

6 TrN+

= 12Trj�1¡ �2j

= D(�1; �2):

Considering now the law of total probability we express our success probability as fol-lows, in terms of conditional probabilities. We measure a projection P and associate tothe outcome 1 the state �1, and to the outcome 0 the state �2. We then have

Psuccess = p(P =1j�1)p(�1)+ p(P =0j�2)p(�2)= 1

2[TrP�1+Tr(1¡P )�2]

= 12[1+TrP (�1¡ �2)];

since p(�1)= p(�2)=1

2, so that indeed the best strategy is to take P!P+ and leads to

Poptimal =12(1+D(�1; �2)):

As one immediately sees from the fact that a unitary transformation does not changethe eigenvalues, one has that the trace distance among two states is invariant under theaction of a unitary transformation

D(U�1U y; U�2U y) = D(�1; �2):

We have however the important fact that completely positive trace preserving maps arecontractions with respect to the trace distance, that is for any completely positive tracepreserving map � we have

D(�(�1);�(�2)) 6 D(�1; �2):

In fact we have

D(�(�1);�(�2)) = 12Trj�(�1¡ �2)j

= 12Trj�(N+¡N¡)j

6 12Trj�(N+)j+

12Trj�(N¡)j

= 12Tr�(N+)+

12Tr�(N¡)

= 12TrN++

12TrN¡

= 12Trj�1¡ �2j

= D(�1; �2);

where we have exploited the triangular inequality, positivity and trace preservation.Indeed here complete positivity does not play any role. We thus have the result that theaction of a completely positive, or even simply positive, trace preserving map doesindeed reduce the trace distance between states. In particular the action of a trace pre-serving operation, that is a channel, decreases the distance between states and thereforeaccording to the previous discussion their distinguishability.

Instruments


We have seen that to the measurement of an e�ect one can associate, in a way whichis not unique, a transformation on the space of states known as operation whichdescribes a possible state change associated to the observation of the elementary observ-able. Moving from elementary observables, such as e�ects, to the general notion ofobservable as a positive operator-valued measure, we are led to ask about the generalproperties of a state transformation which reproduces the statistics of the given positiveoperator-valued measure. The mathematical object describing this situation is known asinstrument and de�ned as follows.

Given a measure space (;F) we call instrument or operation-valued measure a map

I:F ! O(T (H));

where O(T (H)) denotes the set of operations, such that

i. I(M) is an operation 8M 2Fii. Tr I()[T ]=Tr[T ] 8T 2T (H)iii. I(

Si Mi)=

Pi I(Mi) in the weak topology for any sequence of disjoint sets in F ,

i.e. Mi\Mj= ; for i=/ j.

The �rst condition tells us that I(M) sends states to subcollections for any M 2 F ,while the last two express the notion of normalized measure. An instrument uniquelyidenti�es a positive operator-valued measure as follows. Setting

F (M) = I(M)0[1]

one can directly check that the map

F (�):F ! E(H)M I(M)0[1]

is indeed a positive operator-valued measure. Given a pre-measurement state � 2 S(H)the probability to obtain an outcome in M for the observable F is given by the formula

P (M) = ��F(M)

= Tr �F (M)= Tr � I(M)0[1]= TrI(M)[�];

otherwise stated the probability of a de�nite outcome M is given by the weight of thecorresponding subcollection I(M)[�]. The instrument also provides the new state condi-tioned on the occurrence of an outcome in M , namely

�(M) = I(M)[�]TrI(M)[�]

= I(M)[�]��F(M)

= I(M)[�]P (M)

:

Note that as we noticed discussing e�ects, the map associating to the pre-measurementstate a new state conditioned on the measurement outcome � ! �(M) is not linear.Moreover it is not additive in its dependence on the outcome M . If now we consider thespecial case in which M becomes the whole space , we have P () = 1, so that thetransformed state becomes

�() = I()[�];


which is called the a priori state. We recall that I() is a quantum channel. This is thestate that we can a priori obtain as output of the measurement, given the instrumentand the pre-measurement state only, namely if we do not look at the outcomes andtherefore do not select the state according to the measurement outcomes. If we considerthe other extreme situation in which rather than the whole space we select as outcomean in�nitesimal region dw around the point w 2 we are lead to consider the so called aposteriori state

�(w) = I(dw)[�]TrI(dw)[�] ;

which is the state we can associate to the system if we perform a measurement andobtain an outcome w. Note that while �(M) is a function de�ned on F , the a posterioristate is a function de�ned on . It is a random variable depending on the outcomes dis-tributed according to the probability distribution P (w) = Tr I(w)[�]. The a priori stateis then the expectation value of the a posteriori state with this probability measure

I()[�] =Z�(w)P (dw);

and in general the subcollection corresponding to a given outcome is given by

I(M)[�] =ZM�(w)P (dw):

Given an instrument, which provides both the statistics of a positive operator-valuedmeasure and the transformed state, we are able to consider subsequent measurementsleading to conditional probabilities. Indeed considering another positive operator-valuedmeasure G on the outcome space (¡; G), given the pre-measurement � and an outcomeM 2 F , the conditional probability to obtain in a subsequent measurement an outcomeN 2 G for the positive operator-valued measure G is given by

P (N jM) = ��(M)G (N)

= TrG(N)�(M)

= TrG(N) I(M)[�]TrI(M)[�]

= TrG(N)I(M)[�]P (M)

:

The joint probability distribution is then given by

P (N;M) = P (N jM)P (M)= TrG(N) I(M)[�]= Tr � I(M)0[G(N)]:

If we now consider an instrument K compatible with the positive operator-valued mea-sure G in the sense that

Tr �G(N) = TrK(N)[�] 8N 2 G ; 8�2S(H);

so that it correctly reproduces the statistics of the measurement, implying in particular

G(N) = K(N)0[1]

we can write the joint probability distribution as

P (N;M) = TrK(N)0[1]I(M)[�]= TrK(N)[I(M)[�]]= Tr � I(M)0[K(N)0[1]]:


The last two expressions, corresponding to a Schrödinger and Heisenberg picture respec-tively, actually lead to a probability distribution on the rectangles M �N 2� ¡, fromwhich one can construct a well de�ned probability on F G. We have thus built a newinstrument, de�ned on the measure space ( � ¡; F G), given by the composition ofthe two original instruments K � I, which describes a joint measurement. We have thusput multiple subsequent measurements on the same formal footing as a single measure-ment, thus opening the way to speak about measurements continuous in time.�

Similarly to the case of operations, while an instrument as we have seen uniquelydetermines a positive operator-valued measure, the inverse connection is many to one, inthat many instruments, leading to di�erent state transformations, provide the same sta-tistics of the outcomes and therefore the same positive operator-valued measure. Toeach positive operator-valued measure F corresponds a whole equivalence class of instru-ments [I]F which are compatible with it. As usual the meaning of the appearance ofthese equivalence classes [I]F is the fact that the statistics of the same observable can beobtained a�ecting the state in di�erent ways, corresponding to di�erent registration pro-cedures. These di�erent registration procedures do however a�ect as we have seen subse-quent measurements.

We now provide some examples of instruments, originating from a given positiveoperator-valued measure or in particular projection-valued measure. Let us �rst considera conditional state preparation or random state generation. Given a positive operator-valued measure F with a �nite outcome space = f1; :::; Ng, and a set of statisticaloperators f�kgk=1;:::;N the formula

I(fkg)[�] = Tr(F (fkg)�)�k

actually de�nes an instrument, given that we de�ne as usual I(M) =P

k2M I(fkg) tocomply with additivity. To the pre-measurement state we thus associate di�erent �xedstates �k according to the probability distribution Tr(F (fkg)�). If all the �k are equalto � we have the so called trivial instrument, and the dual map is given by

I(fkg)0[B] = Tr(B�)F (fkg):

In the Schrödinger picture all states are sent to a �xed set of states, in the Heisenbergpicture all observables are sent to the positive operators F (fkg).

Von Neumann instrumentsAnother example of major relevance of instrument is the so called von Neumann-

Lüders instrument, obtained as follows. Consider an observable in the sense of a self-adjoint operator A with discrete spectrum, not necessarily non degenerate. We haveaccording to the spectral theorem

A =Xk

akPk

and denoting by EA the uniquely associated projection-valued measure we have

EA(fkg) = Pk:

An instrument compatible with this projection-valued measure can be immediately con-structed de�ning

M(fkg)[�] = EA(fkg)�EA(fkg)= Pk�Pk


and setting to comply with �-additivity

M(M)[�] =X

fkjak2MgEA(fkg)�EA(fkg):

Such an instrument is called a von Neumann instrument in the absence of degeneracy,so that the Pk are one-dimensional projections, or more generally a von Neumann-Lüders instrument. The conditional state reads

�(M) = M(M)[�]TrM(M)[�]

=

Pfk jak2Mg

Pk�PkPfkjak2Mg

TrPk�:

The a posteriori state is given by

�(fkg) = Pk�PkTrPk�

so that in particular in the absence of degeneracy �(fkg) is the pure state Pk = jukihukjindependently of the purity of the pre-measurement state �, while for a Lüders instru-ment �(fkg) is pure i� � is pure. The a priori state is given by

�() =Xk

Pk�Pk;

diagonal in the basis determined by the original observable. Such an instrument has twoimportant features, which actually single it out from all other instruments, namelyrepeatability and ideality. Repeatability corresponds to the fact that the state singledout according to a certain outcome, leads to the same outcome if the instrument isapplied once again and can be expressed as follows

M(fjg)[M(fkg)[�]] = �jkM(fkg)[�]:

Otherwise stated the conditional state relative to a certain outcome gives with proba-bility one the same outcome in a subsequent measurement. Denoting as usual by F ( � )the positive operator-valued measure uniquely associated to the instrument we have

Tr �(M)F (M) = 1:

Ideality corresponds to a minimality in the perturbation of the state in the measurementof the considered observable and corresponds to the fact that

TrM(fkg)[�] = 1 ) M(fkg)[�] = �

that is if the outcome is certain the state is not modi�ed. As we have shown before, thiscan be demonstrated just relying on the speci�c properties of orthogonal projections.Otherwise stated, the state after the action of the instrument is diagonal in the basis ofthe projective observable corresponding to the instrument. If the outcome for thisobservable are certain, the state is already diagonal in this basis, and it does not changeunder the action of the instrument.

It is an important fact that any instrument which satis�es these two properties corre-sponding to repeatability and ideality has to be of the von Neumann-Lüders form, whichimplies in particular the pure point spectrum of the associated observable. Observableswith a continuous spectrum therefore do not admit, as is well known, an ideal, that isnon perturbing, measurement. In this sense pure and continuous spectrum indeed dohave crucially di�erent features.


Also for the general case of a positive operator-valued measure F (�) there is a simpleway to associate to it an instrument compatible with the statistics of the observable.Still restricting to a discrete outcome space we set

I(M)[�] =Xk2M

F (fkg)p

� F (fkg)p

;

which can be checked to be a proper instrument, moreover compatible with F (�) as fol-lows from

TrI(M)[�] = TrXk2M

F (fkg)p

� F (fkg)p

= TrXk2M

F (fkg)�

= TrF (M)�:

Since one of the important advancement in introducing instruments is the possibility toperform on an equal footing single and repeated measurements, it is interesting to seewhat happens when we consider the subsequent application of two von Neumann-Lüdersinstruments. We already know that their composition can be described as single instru-ment on an outcome space given by the cartesian product of the outcome spaces. Thequestion is whether the class of von Neumann-Lüders instruments is stable under such acomposition law. Let us therefore consider two von Neumann-Lüders instruments MA

and MB, related to two projective measurements corresponding to the observables

A =Xk

akPk

B =Xj

bjQj

respectively. If we apply in sequence the two instruments, for the subcollection obtainedstarting from the pre-measurement state � and selecting an outcome in M in the �rstinstance and an outcome in N in the second we have

MB(N)[MA(M)[�]] =X

�j��bj2N

Xfk jak2Mg

QjPk�PkQj

= MB �MA(N �M)[�];

where in the last line we have stressed the fact that the subsequent action of the twoinstruments equals the action of a single instrument obtained composing the originaltwo. The overall instrument is however no more of the von Neumann-Lüders type,unless the two observables commute, i.e. [Pk; Qj] = 0 8j; k so that [A;B] = 0. Indeed theoperator QjPk is positive but not idempotent in the general case. This fact is best seennoting that the uniquely associated observable is no more a projection-valued measure

(MB �MA)0(N �M)[1] = MA0(M)�MB 0(N)[1]

�=

X�j��bj2N

Xfk jak2Mg

PkQjPk;

since PkQjPk 2 E(H), but PkQjPk 2 P(H), unless A and B commute. The joint proba-bility for an outcome in M for the observable A and an outcome in N for the observableB is given by

P (N;M) = TrMB(N)[MA(M)[�]]=

X�j��bj2N

Xfk jak2Mg

TrQjPk�PkQj ;


where the last expression is known as Wigner's formula.We now provide the generic expression of an instrument which encompasses the situ-

ation in which the outcome space is not necessarily �nite dimensional or discrete. Let be a measurable space, � a postive �nite measure on it, and suppose that a measurablefunction taking operator values is given

V : ! B(H);

such that the constraint holds Zd�(x)V y(x)V (x) = 1:

Then one can de�ne an instrument as follows

I(M)[�] =ZMd�(x)V (x)�V y(x):

The associated positive operator-valued measure is given by

F (M) = I(M)0[1]

=ZMd�(x)V y(x)V (x);

which turns out to be well de�ned according to the previous constraint. The statistics ofthe measurement is given by

P (M) = TrZMd�(x)V y(x)V (x)�;

the a priori state reads

I()[�] =Zd�(x)V (x)�V y(x);

while the a posteriori state is

�(x) = V (x)�V y(x)Tr �V y(x)V (x)

:

It is clear that this example includes the previously considered cases by asking the mea-sure to have a pure discrete support.

This formalism allows us to introduce an example of instrument leading to the jointmeasurement of position and momentum considered before. Given a Gaussian wavepacket x0;p0 centered in x0; p0

hxjW (x0; p0)j i=�

12��2

�3

4 e¡ 1

4�2(x¡x0)2+ i

~p0�(x¡x0)= hxj x0;p0i:

where W (x0; p0) denote as before the Weyl operators, we can consider the instrument

Ix;p (M �N)[�] = 1(2�~)3

ZMd3x0

ZNd3p0j x0p0ih x0p0j� x0p0ih x0p0j;

which turns out to be well de�ned and associated to the joint position and momentumobservable

Ix;p (M �N)0[1] = 1(2�~)3

ZMd3x0

ZNd3p0j x0p0ih x0p0j

= Fx;p (M �N):

�


To recover the expression of an instrument associated to the general form of a jointposition and momentum observable we consider

Ix;p (M �N)[�] = 1(2�~)3

ZMd3x0

ZNd3p0 W (x0; p0) S

pW (x0; p0)y� W (x0;

p0) Sp

W (x0; p0)y;

which has all the desired properties and lead to the positive operator-valued measure

Fx;p (M �N) = 1(2�~)3

ZMd3x0

ZNd3p0W (x0; p0)SW y(x0; p0):

Measurement modelsUp to now we have considered two levels of description of the measurement, the sta-

tistics of the outcomes and the related transformation of the states, showing that thetwo descriptions are generally given by positive operator-valued measures and instru-ments respectively. As already stressed while a positive operator-valued measure onlyprovides the statistics of the outcomes, an instrument associates to a given pre-measure-ment state the subcollection obtained performing a selection according to a certain out-come. To any instrument one uniquely associates a positive operator-valued measurewhose statistics is compatible with it, while the inverse relation is many to one. Indeed awhole equivalence class of instruments share the same statistics, though a�ecting thestate in di�erent ways. We now make a further step in the description of the measure-ment, considering a third level of description, which we call measurement model, inwhich the measurement is described as an indirect measurement on a probe or measure-ment apparatus, coupled via a unitary interaction to the considered system. We thusprovide a dynamical description of the measurement, obtained by letting the systeminteract with a quantum measurement apparatus. Denoting by M a measurementscheme, once again to a given instrument it corresponds a whole equivalence class ofmeasurement schemes compatible with it, in symbols

F! [I] I! [M]:

As usual this construction comes about considering a suitable dilation. As we discussedin connection to operations, we have two basic representation theorems for operations,and therefore in particular channels. The �rst points to the so called Kraus representa-tion in terms of a denumerable set of Kraus operators, the second refers to a dilation ina larger Hilbert space. The second representation as we shall soon see directly leads toassociate to an instrument a measurement scheme. Regarding the �rst representation,which is often of great convenience, this is not in general readily available for an instru-ment. Indeed an instrument is an operation-valued measure, so that for any �xed out-come one has an operation, and therefore a Kraus representation, with a discrete set ofKraus operators. However if the outcome space is not discrete, there is no general recipefor the structural characterization of an instrument. On the contrary, if the outcomespace is �nite or denumerable, one obtains a general representation of the instrumentsimilar to a Kraus representation by suitably summing up the Kraus representations cor-responding to the single outcomes.

Let us now come to a general dilation of an instrument which allows to connect it toa measurement scheme.

Theorem (Ozawa, 1984) Let I be an instrument on the outcome space (; F), thenthere exist a Hilbert space K, a projection-valued measure E on K over the same out-come space, a statistical operator � 2 T (K) and a unitary operator on H K such thatthe representation holds

I(M)[�] = TrKfU(��)U y(1HE(M))g 8M 2F 8�2T (H):


We note that the right hand side properly de�nes an instrument for any choice ofHilbert space K, unitary operator U , state � and observable E, which can also be takento be a positive operator-valued measure. In this model of indirect measurement wehave obtained the states transformed according to the given instrument, and thereforealso the statistics of the associated observable, by a dynamical description of the interac-tion between system and apparatus. Let us prove this statement for the special case inwhich the Hilbert space H of the system has �nite dimension n, together with the out-come space. Given the �nite dimension of the outcome space for any �xed outcome wehave the Kraus representation of the corresponding operation

I(fkg)[�] =Xj2Ik

Aj�Ajy;

where the index j runs over a �nite set Ik with at most n2 elements, corresponding tothe cardinality of the representation of I(fkg). Summing up all these representations wehave a representation for the channel I() determined by the given instrument in termsof a �nite set, say N =

Pk Ik of Kraus operators, so that we have

I()[�] =Xj=1

N

Aj�Ajy:

We now consider a dilation of the channel I(), whose existence is warranted by Stine-spring's theorem. For a certain Hilbert space K, whose dimension can always be takento be N , and a state in K, which in particular can be taken to be pure, say P� = j�ih� j,we therefore have the representation

I()[�] = TrKfU(�P�)U yg:

What is still missing in this description, is the pointer basis, let us introduce it as fol-lows. First we observe that any basis, say f'igi=1;:::;N , in K induces a set of Kraus oper-ators by evaluating the trace

I()[�] =Xj=1

N

Bj�Bjy;

where the operators fBjg are de�ned as Bj = h'j jU �i, so that their matrix elementsread

h�jBji = h� 'j jU ( �)i 8�;2H:

Now as we know the two Kraus representation, which in particular have the samedimension, have to be related by a square N � N matrix [ujk], whose elements satisfyP

j=1N ujk

� ujl= �kl as follows

Bk =Xj=1

N

Ajujk

Aj =Xk=1

N

Bkujk� :

Exploiting this matrix we can therefore introduce a new orthonormal system in K asfollows

'i0 =

Xk=1

N

uij'j ;


and given this basis an associated projection-valued measure as follows

E 0(fjg) = j'j0 ih'j0 j:

We can now consider the following expression of indirect measurement

TrKfU(�P�)U y(1HE 0(fjg))g;

and build in terms of it the instrument we started with. We have in fact

TrKfU(�P�)U y(1HE 0(fjg))g = h'j0 jU�i�(h'j0 jU�i)y

=Xr;s=1

N

ujr� h'rjU�i� uksh� j'si

=Xr=1

N

ujr� Br �

Xs=1

N

ujsBsy

= Aj�Ajy;

for each of the N possible values of j. The desired representation of the instrument isobtained by suitably collecting the indexes in groups of cardinality Ik, recalling thatP

j2Ik E0(fjg) still is a projection.

As we have seen a measurement model M can be identi�ed with a quadruple (K; U ;�; G), where K can be seen as the Hilbert space of the apparatus, U is a unitary oper-ator on H K coupling system and apparatus, � a statistical operator for the appa-ratus, which can be taken to be pure, G an observable for the apparatus, which can betaken to be projection-valued. As we have already seen, given a positive operator-valuedmeasure F , the latter is compatible with an instrument I provided

TrF (M)� = TrI(M)[�] 8M 2F 8�2T (H):

On the same footing an instrument I is said to be compatible with a measurementmodel M� (K; U ; �;G) if

I(M)[�] = TrKfU(��)U y(1HG(M))g 8M 2F 8�2T (H):

It is therefore natural to say that a positive operator-valued measure F is compatiblewith a measurement model M� (K; U ; �;G) if

TrF (M)� = TrHTrKfU(��)U y(1HG(M))g 8M 2F 8�2T (H);

where the last equality is also known as probability reproducibility condition, as well asNaimark's dilation of the initial observable.

We now consider an example of measurement model leading to a von Neumanninstrument, and therefore to a sharp observable. Consider on the Hilbert space H asharp observable with discrete and non degenerate spectrum E(fkg) = j'kih'k j, wheref'kg is a basis in H. Take as apparatus space K one with the same dimensionality of Hand consider here a basis f ig which has the same cardinality and allows to constructthe projective observable for the apparatus F (fig) = j iih ij. Given that we initializethe apparatus in the state P 1= j 1ih 1j, for a measurement model we still only have tospecify the unitary interaction. This is obtained by considering a unitary extension ofthe isometry

V ('k 1) = 'k k 8k:


Suppose now the system starts in a state given by a coherent superposition of eigen-states of the considered observable E

' =Xk

ck'k;

where the complex coe�cients ck satisfyP

k jck j2 = 1. For such a case statistics of the

outcomes of the observable E are given by the Born probabilities

P (fkg) = jckj2:

We now put together our measurement model showing that it is compatible with thisstatistics for the outcomes and that it leads to an instrument which is of the von Neu-mann form. To this aim we have to evaluate the trace, or the partial trace with respectto K only, of the operator

V (P'P 1)V y(1HF (fkg)):

We evaluate �rst

V (P'P 1)V y = V (j'ih'j j 1ih 1j)V y

= jV (' 1)ihV (' 1)j=Xj;k

cjck� j'jih'kj j jih kj;

so that the instrument determined by the considered measurement model reads

I(fkg)[P'] = TrKfV (P'P 1)V y(1HF (fkg))g= jck j2j'kih'k j= Tr(P'E(fkg))E(fkg)= PkP'Pk;

that is indeed a von Neumann instrument, while the statistics �xed by the measurementmodel leads to

TrHTrKfV (P'P 1)V y(1HF (fkg))g = jckj2

= Tr(P'E(fkg));

indeed compatible with the projective measurement considered as starting point.


Part II

Open quantum systems

``Natürlich ist man nicht so einfältig zu denken, dass solchermaÿen zuerraten sei, wie es auf der Welt wirklich zugeht.''

(Erwin Schrödinger)

5 Open quantum systems as composite systemsA ubiquitous situation in quantum mechanics is given by composite systems, i.e. a situa-tion in which the overall set of degrees of freedom considered, the physical object onetries to describe or observe, can be naturally divided in two parts. In this case theoverall system is described on a Hilbert space with a tensor product structure. Notehowever that given the overall Hilbert space its expression as a tensor product of smallerHilbert spaces is generally not unique. Think for example of the hydrogen atom. Theoverall system includes the centre of mass degrees of freedom of proton and electron, sothat H=Lproton

2 (R3)Lelectron2 (R3). The most convenient tensor product representationof the overall space is however given by H=Lcm

2 (R3)Lrelative2 (R3). This has the advan-tage that given an overall state factorized between centre of mass and relative degrees offreedom, it remains factorized, these two kinds of degrees of freedom are not coupled bythe interaction. We thus concentrate on the internal degrees of freedom and obtain thelevel spectrum of the hydrogen atom. This is why in a sense we can neglect that theoverall system is composite, since for a natural choice of degrees of freedom the latterare decoupled. On the same footing when describing an electron we should generallyconsider a composite structure given by centre of mass and spin degrees of freedom, butif the latter are not coupled by some interaction, we can neglect the bipartite structure.In these cases the Hamiltonian does not contain interaction terms so that we can ignore,for special initial preparation and dynamics, the composite structure.

More importantly a bipartite structure may arise because the system we want toconsider does interact with some external �environment� (often called reservoir or bath ifone wants to stress that it is characterized by a system with very many degrees offreedom at thermal equilibrium), which modi�es its dynamics. If the system we areinterested in lives in a Hilbert space HS, we say that it is closed if its dynamics can bedescribed by a unitary evolution, which thanks to Stone's theorem identi�es a self-adjoint Hamiltonian HS. This dynamics is then reversible. Then the closed dynamics isgiven by the Schrödinger equation �xed by HS, or the Liouville von Neumann equationto cope with mixed states. This is however an often idealized situation, since it wouldcorrespond to a perfect shielding of our system from the rest of the universe. In generalif interaction with an external environment cannot be neglected so that the overallHamiltonian reads

H = HS1E+1SHE+HI

we are naturally led to consider together with the system S also a system E, to bedescribed on the tensor product space HS HE. In this case we say that the system isopen, and its dynamics will generally not be given by a Schrödinger equation. This alsoimplies that new dynamical e�ects such as dissipation and decoherence can appear, notdescribed by a unitary evolution. Typical examples of open systems are given by a two-level atom coupled to a laser �eld or a nuclear spin coupled to a magnetic �eld.

Given a state on the overall space � 2 S(HS HE), with � = �y, � > 0, TrSE� = 1we can obtain the marginal states for system S and environment E through the partialtrace operations TrS and TrE according to

TrS�= �E TrE�= �S ;

5 Open quantum systems as composite systems 85

with �S 2 S(HS) and �E 2 S(HE) respectively. We have already encountered the partialtrace as an example of completely positive trace preserving map among two di�erent ini-tial and �nal spaces. The partial trace TrE is the unique linear mapping

TrE: T (HSHE) ! T (HS)T TrET

satisfying

TrSE[T (AS1E)] = TrS(TrET )AS 8AS 2B(HS); 8T 2T (HSHE):

This constraint implies in particular for factorized operators T =TSTE 2T (HSHE)

TrSf[TrE(TSTE)]ASg = TrSE[TSTE(AS1E)]= TrSE[TSASTE]= TrSTSASTrETE ;

where the latter equality follows from the standard de�nition of trace over a givenHilbert space. Since this relation is valid for any AS it follows that for factorized statesthe partial trace reads

TrE(TSTE) = (TrETE)TS:

Exploiting this result and the relation

T =Xj;k

Xm;n

j jih mjj'kih'njh j 'k jT m 'ni;

which expresses in a unique way the operator T in a operator basis obtained from theSONC f ig and f'ng in HS and HE respectively, together with the identity

TrE j jih mjj'kih'nj = j jih mj�k;n

as obtained above one has

TrET =Xj;k;m

j jih mjh j 'kjT m 'ki

=Xk

h'kjT'ki;

which provides a direct way to evaluate the partial trace and de�nes it uniquely since wehave expressed its action on a basis of operators. The partial trace is thus obtained bysumming over the diagonal matrix element of the operator with respect to a basis inHE.

Coming back to the action of the partial trace on a statistical operator � de�ned onthe total space, one immediately checks that �S =TrE� still is a statistical operator, andas shown it is �xed by the fact that it provides the correct statistics for any observableon the system only. Indeed for an observable of the form AS1E we have

hAi� = TrSE�AS1E

= TrS�SAS ;

so that �S describes the statistics of any measurement on S only and in this sense canbe taken as state of the reduced system S. Note that �S is mixed even if � is pure, apure � leads to a pure �S i� it is the tensor product of two pure states. On the contrary�S can be pure even for mixed � (think of the trivial case �= j ih jS �E).Reduced dynamics


Given the fact that our system is in interaction with an external environment, thequestion is whether we can eliminate the degrees of freedom of the environment toobtain e�ective closed equations of motion, let us say master equations, for the systemonly. This turns out to be feasible under the following working hypotheses:

i) factorized initial state

�(t0) = �S(t0) �E ;

an hypothesis which essentially cannot be released to obtain a physically well-de�nedclosed e�ective dynamics;

ii) overall unitary dynamics

�(t) = U(t; t0)�(t0)=U(t; t0)�(t0)U y(t; t0);

hypothesis which could be released allowing e.g. for a semigroup dynamics for the wholesystem.

For �xed �E we then have the commutative diagram, now assuming t0=0

ρS(0) ⊗ ρE

TrE

��

U(t) // ρ(t)

TrE

��ρS(0)

A

OO

Φ(t) // ρS(t)

where A�E is the so-called assignment map already encountered as an example ofcompletely positive trace preserving map

A�E: T (HS) 7! T (HSHE)�S �S �E

which furthermore and most importantly satis�es the compatibility condition

TrE �A�E = 1T (HS):

As a result for �xed �E the assignment

�S(0) 7! �S(t)= TrE �U(t) �A�E

= TrEfU(t)�S(0) �EU y(t)g= �(t)�S(0)

is a linear map which preserves hermiticity and trace, further sending positive operatorsto positive operators, therefore sending states to states: let us call it quantum dynamicalmap. It is in particular a completely positive map being a composition of completelypositive maps. The properties of this map can be better checked considering a orthog-onal decomposition for �E

�E =X�

�� j'�ih'� j

so that �� > 0,P

� �� = 1 and f'�g SONC in HE, and using the same basis in HE toevaluate the partial trace we have

�(t)� =XK

WK(t)�WKy (t)


with K a multi-index, which immediately appears to be a Kraus representation of thecompletely positive trace preserving map. The operators WK(t) acting on HS arede�ned via

WK(t) = ��p

h'�jU(t)j'�i;

and further satisfy XK

WKy (t)WK(t) = 1:

The latter equality grants trace preservation.

Bipartite states and entanglementConsidering states on a bipartite Hilbert space leads to a new feature, called entan-

glement (originally Verschränkung). Let us introduce its de�nition. We start with purestates on the total system, where the characterization can be tamed.

Schmidt decomposition

Let 2HS HE be a pure state describing the overall system, then one has the fol-lowing so called �Schmidt decomposition�

ji =Xi=1

D

�ip

j�iSi j�iEi

with D = min fdimHS ; dimHEg. For simplicity we work assuming dimHS = dimHE.Here

��iSand

��iEare orthonormal bases in HS and HE respectively, called Schmidt

bases. The coe�cients �ip

are non-negative numbers called Schmidt coe�cients, suchthat

Pi=1D �i= 1. The number of non-zero coe�cients, the Schmidt number, is uniquely

determined (while this is not true for the bases and the value of the single coe�cients).

Proof. We can always write

ji =Xi; j=1

D

�ij j'iSi j jEi

with�'iS

and� iE

orthonormal basis in HS and HE respectively. Consider thesquare matrix

(A)ij = �ij:

The singular value decomposition theorem states that any square matrix can be broughtin the form

A = U�V ;

where � is diagonal with non-negative entries and U , V are unitary. We thus have

ji =Xi; j

Xk;l

uik�kl�kvkj j'iSi j jEi

=Xk

�kXi

uik j'iSi|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }j�kSi

Xj

vkj j jEi||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }j�kEi

�

�


If dimHS = dimHE, the decomposition still holds considering block matrices, andonly at most D coe�cients can be di�erent from zero.�Pure entangled states

If the Schmidt number is equal to 1 the pure state is called factorized. This happensi� the state is of the form

ji = j'Si j Ei;

that is to say a product state. Note that a pure state in a bipartite system is separablei� its marginals are pure. For such a product state one has

hASAEi = hASi'ShAEi E

so that the statistical outcomes of product observables are uncorrelated.If the Schmidt number is bigger than 1 the pure state is called entangled.If the Schmidt number is equal to its maximum value D and all Schmidt coe�cients

have the same modulus, that is the state is of the form

ji = 1D

pXi=1

D

j�iSi j�iEi;

the state is said to be maximally entangled.In a given Hilbert space one can consider a basis of maximally entangled states, also

called Bell basis. Consider HS =HE =C2, and a basis in this space f +; ¡g, with �e.g. eigenvectors of �z. Out of it one can consider the standard basis of factorized stateson C2C2, given by f + +; ¡ +; + ¡; ¡ ¡g, which are eigenvectors of�z�z. Taking linear combinations of these vectors we can consider in particular a basisof common eigenvectors of the commuting observables �z �z and �x �x. Indeedexploiting the relation

[AB;C D] = [A;C]BD+CA [B;D];

and the usual relation for the Pauli matrices

�i�j = �ij+ i"ijk�k

we have

[�x�x; �z�z] = [�x; �z]�x�z+�z�x [�x; �z]= 2�y�y¡ 2�y�y= 0:

The common eigenvectors of these operators provide a Bell basis f e+; o+; e¡; o¡gaccording to the de�nition

e� = 1

2p ( + +� ¡ ¡)

o� = 1

2p ( + ¡� ¡ +);

where the subscripts mean even or odd parity. The subscripts e/o correspond to theeigenvalues �1 for �z �z, the superscripts � to the eigenvalues for �x �x. One cancheck that a state diagonal in this Bell basis in matrix form has a typical X expression,that is to say

� = pe+P e

++ pe¡P e

¡+ po+P o

++ po¡P o

¡;


where the positive weights sum up to one in the computational basis reads

� = 12

0BBB@pe++ pe

¡ 0 0 pe+¡ pe¡

0 po++ po

¡ po+¡ po¡ 0

0 po+¡ po¡ po

++ po¡ 0

pe+¡ pe¡ 0 0 pe

++ pe¡

1CCCA:Note that for weights all equal to 1/4 one obtains the state proportional to the identity,which can be expressed as a product of states proportional to the identity in each space.�

Let us consider the explicit calculation for the case �=P e+. We have

P e+ = 1

2(P +P ++P ¡P ¡+j +ih ¡j j +ih ¡j+j ¡ih +j j ¡ih +j);

and considering the matrix representation obtained starting from the basis of eigenvec-tors of �z, we have e.g.

P + =�1 00 0

�and therefore

P e+ = 1

2

8>><>>:0BB@

1 0 0 00 0 0 00 0 0 00 0 0 0

1CCA+0BB@

0 0 0 00 0 0 00 0 0 00 0 0 1

1CCA+0BB@

0 0 0 10 0 0 00 0 0 00 0 0 0

1CCA+0BB@

0 0 0 00 0 0 00 0 0 01 0 0 0

1CCA9>>=>>;:

If one thinks of two spin one-half systems o+ corresponds to the �01 triplet state, e

�

are a linear combination of the other two triplet states, while o¡ corresponds to a �0

0

singlet state.It is further instructive to consider the expression of the marginal states associated

to a total pure state. Exploiting the Schmidt decomposition we have

�S = TrE jihj=Xi=1

D

�ij'iSih'iS j

�E = TrS jihj=Xi=1

D

�ij iEih iE j:

It follows in particular that both �S and �E are pure states if and only if the state isfactorized. Both �S and �E are proportional to the identity if and only if the state ismaximally entangled.

In particular �S and �E have the same eigenvalues with the same multiplicity (apartfrom eigenvalue zero), and therefore the same von Neumann entropy, given by

S(�S) = ¡kBTrS�Slog�S=¡kBXi=1

D

�ilog�i

S(�E) = ¡kBTrE�Elog�E=¡kBXi=1

D

�ilog�i:

For a factorized pure state the von Neumann entropy of the marginals is equal to zero,while for a maximally entangled state it takes the maximum value kB log D. Thus max-imal correlations in the total pure state implies that the marginals describe a trivial sta-tistics.


Mixed entangled statesMore generally one can consider mixed states on the bipartite system. In this case

the characterization is much more di�cult, we can however distinguish three kind ofstates.

A state �2T (HSHE) is said to be factorized if it is given by the tensor product oftwo states

� = �S �E:

In this case the statistical outcomes for a product observable is simply the product ofthe outcomes

hASAEi� = hASi�ShAEi�E:

A state � 2 T (HS HE) is said to be separable or classically correlated if it can bewritten as a convex combination of product states

� =Xi

pi �Si �Ei

where pi is a probability distribution, so that pi> 0 andP

i pi=1. Classically correlatedstates correspond to a classical mixture of preparation procedures, and the statisticaloutcomes for a product observable are classically correlated

hASAEi� =Xi

pi hASi�Si hAEi�Ei :

A state �2T (HSHE) is said to be entangled otherwise.The characterization of entangled states is a di�cult issue, and requires the knowl-

edge of the notion of complete positivity.Note that the linear combination of entangled states is not necessarily entangled, e.g.

trivially as observed above 1

4

Pi=14 Pi=

1

212 1

212, which is separable. Also note that if

either �S or �E is a pure state, that is just one of the eigenvalues is di�erent from zero,then � is factorized. In fact if � is pure this immediately follows from the Schmidtdecomposition, which must have Schmidt number equal to one. Supposing � to bemixed, we have, taking e.g. �S to be pure and considering an orthogonal resolution of�=

Pk pkj'kih'k j, where pk=/ 1 for all k since the state is mixed

�S =Xk

pkTrE(j'kih'kj)

=Xk

pk�Sk

which implies, being �S pure, that actually �Sk does not depend on k, in particular it has

to be pure. But then according to the previous argument the 'k have to be factorized as'k= S �Ek and the state � is of the form

� = j Sih S jXk

pkj�Ek ih�Ek j

= j Sih S j �E:

Puri�cationFrom the Schmidt decomposition we also learn that any state can be obtained, actu-

ally in many ways, as partial trace from a pure state on a larger Hilbert space. Indeedgiven �S 2S(HS) we may consider the spectral decomposition

�S =Xi

�ij'iih'ij


and consider another Hilbert space HE (with dimension at least dimHS) and a basisf ig in HE. The pure state on HSHE

ji =Xi=1

�ip

j'iSi j iEi

then has the required marginal. Note that for every unitary operator UE on HE thestate 1 UE ji is another puri�cation of �S. The fact that a puri�cation always existsand is highly non unique, is re�ected in the fact that the state in the ancillary Hilbertspace of a dilation can always be taken to be pure, as well as in the non uniqueness ofthe dilation.

Positive maps and entanglement detectionOne can infer about entanglement or non-entanglement of a state by acting on it

with maps having known positivity or complete positivity properties. If � is a positivemap, and � a separable state, then (� 1)� is positive, even if � is not CP. Indeed if �is a separable state it is of the form �=

Pi pi �S

i �Ei . We then have that for positive �

the operator (�1)�=P

i pi�(�i)S �Ei is positive being the tensor product of positive

operators.As a consequence the fact that (� 1)� 6 0 for some positive but not completely

positive map �, so that the transformed operator has negative eigenvalues, is a su�cientcondition for entanglement of the state �. Indeed a theorem by the Horodecki familystates that a given state � is separable if and only if (� 1)� is positive for all positivemaps �. If the map � is positive, but not completely positive, the condition (� 1)�>0 is a necessary condition for separability of �, while (� 1)� 6 0 is a su�cient condi-tion for entanglement of �.

An example of a positive but not completely positive map is the transposition. Notethat in order to actually de�ne the transposed map one has to �x a basis. Denoting byT the transposition, we have that T is positive, but T 1 is not. To show this weexplicitly exhibit a state which is transformed into a non positive operator. From theabove discussion, the state should not be separable, we take the projector on one ele-ment of the previously considered Bell basis, which in the so called computational basiswe write as

j e+i = 12

p (j11i+ j00i)

Proof. We have

j e+ih e+j =12(�+�¡�+�¡+�¡�+�¡�++�+�++�¡�¡)

and therefore

(T 1)j e+ih e+j =12(T (�+�¡)�+�¡+ T (�¡�+) �¡�++T (�+) �++T (�¡)

�¡)

which in the computational basis reads

(T 1)j e+ih e+j =12

0BB@1 0 0 00 0 1 00 1 0 00 0 0 1

1CCA


with eigenvaluesn1

2;1

2;1

2;¡1

2

o�

Note that the insurgence of negative eigenvalues together with trace preservation isequivalent to growth of the trace norm of �. The trace norm of a self-adjoint trace classoperator is given by the sum of the modulus of its eigenvalues

kAk1 =Xi

jaij:

For a statistical operator one has k�k1=Tr �=P

i pi=1. In our case one has

j e+ih e+j =12

0BB@1 0 0 10 0 0 00 0 0 01 0 0 1

1CCAwith eigenvalues f1; 0; 0; 0g so that

kj e+ih e+jk1 = 1

but

k(T 1)j e+ih e+jk1 = 2:

More generally, consider the tensor product Cn Cn, and restrict the attention to purestates

� = jihj;

where according to the Schmidt decomposition we can write

ji =Xi=1

n

�ij�i(1)i j�i

(2)i

with f�ig denotes a basis on either copy of the Hilbert space. Introducing the basis ofoperators

Ejk = j�jih�kj

we thus have

� =Xi; j=1

n

�i�jEij Eij

and according to the action of the transposition T (Eij)=Eji

(T 1)� =Xi; j=1

n

�i�jT (Eij)Eij

=Xi; j=1

n

�i�jEjiEij

=Xi; j=1

n

�i�j

�j�j(1)i j�i

(2)i��h�i

(1)jh�j(2)j�


and the latter operator has negative eigenvalues as soon as the number of non zero �j isgreater or equal to two, so that according to the Schmidt decomposition the state � isentangled. Indeed suppose �s=/ 0, �r=/ 0, so that they are strictly positive. The state

12

p�j�r(1)i j�s

(2)i¡j�s(1)i j�r

(2)i�

is eigenstate of the operator (T 1)� with eigenvalue ¡�s�r. Thus the map T 1,known as partial transposition, detects all pure entangled states.

Parametrization of quantum mapsWe have seen that for completely positive maps important representation theorems

exist, pointing either to a Kraus representation or to a dilation to a unitary transforma-tion in a larger Hilbert space. Despite the relevance of these results, they apply only tocompletely positive maps, and involve a certain arbitrariness. Indeed as we have seenthe Kraus operators as well as their cardinality are not uniquely �xed. At the same timethe dilation can be obtained in many di�erent ways. It is therefore convenient to obtainother unambiguous representation for maps on space of operators, including transforma-tions which are not necessarily completely positive. This turns out to be feasible in ageneral way in �nite dimension. We will provide three related ways to perform such atask. First we will see such a map as an operator in a Hilbert space and consider itsmatrix representation in a suitable basis. We will then consider di�erent bases of mapsand express the given map in terms of these bases, leading to the so called �-matrix rep-resentation. At last we will obtain Choi's theorem on the characterization of completelypositive maps, pointing to the connection between Choi's matrix and the �-matrix rep-resentation.

We therefore consider a Hilbert space of �nite dimension H = Cn, so that thebounded operators on this space can be identi�ed with n � n matrices with complexentries B(Cn)=Mn(C). The space B(Cn) is not only a Banach space, but thanks to theconstraint on the dimensions it is also a Hilbert space, with scalar product given by theHilbert-Schmidt expression. Each map acting on operators on Cn can thus be seen as alinear operator in a Hilbert space, and for a �xed basis can be expressed accordingly asa matrix. In particular, this map is a self-adjoint operator if

h�(�); !iHS = h�;�(!)iHS 8�; !2B(Cn)

and it is a positive operator if

h!;�(!)iHS > 0 8! 2B(Cn);

where the Hilbert-Schmidt scalar product takes the form

h�; !iHS = Tr�y!:

Note that in general for a map � acting on T (H) one also has the notion of dual map �0

acting on B(H), determined according to the duality form. In the case of a �nite dimen-sional Hilbert space H the two notions of dual and adjoint map coincide. To obtain amatrix representation for � we �rst consider a basis in the Hilbert space B(Cn), whoseelements are therefore operators rather than �vectors�, we denote it as f��g�=1;:::;n2,orthonormal with respect to the Hilbert-Schmidt scalar product

h��; ��iHS = Tr��y��

= ��:


Each operator ! 2 B(Cn) can be identi�ed in this basis as a vector with components

!� = h��; !iHS = Tr��y!�. We thus have, corresponding to the expression A

�� i =Pn;m

��'nih'njA'mih'mj i�(!) =

X�;�=1

n2

h��;�(��)iHSh�� ; !iHS��

=X�;�=1

n2

Tr��y�(��)

�Tr��y!��

=X�;�=1

n2

��!� ��;

where we have de�ned the matrix of coe�cients associated to the map in this basis as

�� = Tr��y�(��)

�:

�

Given this representation of the map, the action of the map on an operator is trans-lated to a multiplication between a matrix and a vector, and the composition of maps istranslated to the product of matrices. One has in fact

(�(!)) = Tr¡� y�(!)

�=X�=1

n2

� �!�;

and also

(� �¡)�� = Tr��y(� �¡)(��)

�= Tr

��y�(¡(��))

�=

X�0;� 0=1

n2

Trh��y��0� 0Tr

�� 0y ¡(��)

��0i

=X

�0;� 0=1

n2

��0� 0¡� 0�Tr��y��0

�=

X� 0=1

n2

�� 0¡� 0�:

Hermiticity and positivity of � as linear operator on the Hilbert space re�ect themselvesin hermiticity and positivity of the matrix ��. For example one has

h!;�(!)iHS = Tr(!y�(!))

=X�;�=1

n2

Tr(��!y)Tr��y�(��)

�Tr��y!�

=X�;�=1

n2

!��!�

> 0 8! 2B(Cn);

so that the linear operator � is positive i� such is the matrix ��. Note that this prop-erty does not correspond to the notion of positivity or complete positivity of � as amap.


Canonical basis and Bloch basis representationWe now point to two particularly relevant examples of basis in B(Cn). The �rst is

the so called canonical basis, given by matrices with only one non zero entry equal toone. Given an orthonormal system in Cn, say f'igi=1;:::n, we have the operator basisfEijgi; j=1;:::n with Eij = j'iih'j j, often simply denoted jiihj j. In C2 this basis becomesfP+; �¡; �+; P¡g= fj+ih+j; j¡ih+j; j+ih¡j; j¡ih¡jg.

The other example is the so called Bloch basis, in which the �rst element of the basisis proportional to the identity, and the others are taken to be hermitian and traceless,namely �0= 1, �i= �i

y, Tr �i=0 for i=1; :::; n2¡ 1 together with Tr �i�j= n �ij. In such acase the action of a map has a geometric interpretation as they can be read as a�netransformations. We can in fact write

� = 1n(1+ � � r);

where r 2Rn2¡1 and is given by r=Tr �� .�

Putting for convenience

t =�1n;rn

�we also have

� = 1n

Xj=0

n2¡1

�jTr ��j

=Xj=0

n2¡1�j tj:

The action of the map now reads, with ��!�jk

�(�) = 1n

Xj;k=0

n2¡1

�jk�jTr ��k

=Xj=0

n2¡1

�j tj0

with

tj0 =

Xk=0

n2¡1

�jktk:

Assuming further the map to be trace preserving we have the constraint t0 = t00 = 1/n.

This implies in particular �0k= �0k so that one has

t0 ! t00 =�00t0= t0

tj ! tj0 =

Xk=1

n2¡1

�jktk+�j0t0 j=/ 0:

Putting �j0 = (b)j and Tij = �ij for i; j = 1; :::; n2 ¡ 1, the transformation associated tothe trace preserving map � can be described as an a�ne transformation in the para-meter space

r ! r 0=T r+ b


ruled by the matrix

� =�1 0b T

�:

Let us now take a slightly di�erent point of view, and look at �, rather than a linearoperator in the Hilbert space B(Cn), as a linear map from B(Cn) into itself, � 2L(B(Cn); B(Cn)). Such a space is also a Hilbert space of dimension n2 � n2, which canbe seen as a direct sum �k=1n2 B(Cn), and in it we have a scalar product constructed fromthe one in B(Cn), namely for any basis f� g =1;:::;n2

h�;¡i =X =1

n2

h�(� );¡(� )iHS

=X =1

n2

Tr(�(� )y¡(� )):

In this space of maps one can introduce two relevant basis of maps, orthonormalaccording to this scalar product. Let us �rst de�ne a basis E�� as follows

E��(!) = ��Tr��y!�;

orthonormal as follows from

hE�� ; E�0� 0i =X =1

n2

hE��(� ); E�0� 0(� )iHS

=X =1

n2

Tr�h��Tr

��y�

�iy��0Tr

�� 0y �

��= �� 0�� 0:

This basis is strictly related to the matrix representation we have considered before. Letus indeed expand the generic map � in this basis, we have

�(!) =X�;�=1

n2

hE�� ;�iE��(!)

=X =1

n2 X�;�=1

n2

hE��(� );�(� )iHSE��(!)

=X =1

n2 X�;�=1

n2

Trh��Tr

��y�

��y�(� )

iE��(!)

=X�;�=1

n2

Tr��y�(��)

�E��(!)

=X�;�=1

n2

��E��(!);

so that the matrix associated to � in this basis of maps corresponding to �� = hE�� ;�iis the same encountered before considering it as a linear operator, and the same consid-erations apply. Let us now consider another possible operator basis F�� de�ned as

F��(!) = ��! ��y;


orthonormal as follows from

hF�� ; F�0� 0i =X =1

n2

hF��(� ); F� 0� 0(� )iHS

=X =1

n2

Tr��

�� y�y��0� �� 0

y�

=X =1

n2

Tr��

y ��y��0� �� 0

y 0�

= Tr��Tr

¡��y��0

�� 0y 0�

= �� 0�� 0;

where we have exploited the identityX =1

n2

� y � � = 1Tr �;

which can be veri�ed exploiting invariance with respect to the choice of basis and usingthe canonical basis. Let us consider the expression of the matrix of coe�cientsexpressing a generic map � in this basis

�(!) =X�;�=1

n2

hF�� ;�iF��(!)

=X =1

n2 X�;�=1

n2

hF��(� );�(� )iHSF��(!)

=X�;�=1

n2 X =1

n2

Trh��

y�y�(� )

iF��(!)

=X�;�=1

n2 X =1

n2

Tr��

y ��y�(� )

�F��(!)

=X�;�=1

n2

��0 F��(!);

where we have de�ned as

��0 =

X =1

n2

Tr��

y ��y�(� )

�the matrix representation of the map in the basis F��, that is ��0 = hF�� ; �i. Thematrix ��0 provides most important information on the properties of � as a map.Indeed from

(�(!))y =

0@ X�;�=1

n2

��0 ��! ��

y

1Ay

=X�;�=1

n2

��0 � ��!

y ��y

=X�;�=1

n2

��0� ��!

y ��;


one has that � is a hermiticity preserving map, that is

(�(!))y = �(!y)

i� ��0 =��0�, that is the matrix ��0 is hermitian. Most importantly one has the result

that � is completely positive i� the matrix ��0 is non negative. To see this suppose �rst��0 is non negative, then one has

��0 = (UDU y)��

=X =1

n2

� u� u� � ;

where U is a unitary matrix and D a diagonal matrix with entries the positive eigen-values of ��0 . We thus obtain the Kraus decomposition

�(!) =X =1

n2

�

0@X�=1

n2

u� ��

1A!0@X�=1

n2

u� � ��

y

1A=X =1

n2

� �~ !�~ y;

which is called canonical decomposition, since the Kraus operators f�~ g are orthogonal,and is furthermore unique if the eigenvalues are non degenerate. This decomposition,which is a Kraus representation just because � > 0, warrants directly complete posi-tivity of the map, not only positivity. Viceversa, suppose to have a completely positivemap �, then it admits a Kraus decomposition of the form

�(!) =X =1

n2

A !A y ;

and expressing the A in the basis f��g we have

A =X�

c ��;

and therefore

�(!) =X =1

n20@X�=1

n2

c ��

1A!0@X�=1

n2

c ��

1Ay

=X�;�=1

n20@X =1

n2

c �c ��

1A��! ��y;which is of the form

�(!) =X�;�=1

n2

��0 ��! ��y;

with the matrix of coe�cients given by

��0 =X =1

n2

c �c ��


and therefore positive since

X�;�=1

n2

v��0 v�� =

X =1

n20@X�=1

n2

c � v�

1A0@X�=1

n2

c � v�

1A�> 0:

In the canonical basis the matrices �� and ��0 are related by a simple but crucialexchange of indexes as follows. Let us take ��!jiihj j and ��!jkihlj.�

We have, suitably inserting completenesses and exploiting the linearity of the map

�(�) =Xi;k=1

n

jiihij�(�)jkihk j

=Xi;k=1

n

jiihij�

0@ Xj;l=1

n

jjihj j�jlihlj

1Ajkihk j=

Xi; j;k ;l=1

n

hij�(jj ihl j)jki jiihj j�jlihk j

=X�;�=1

n2

��0 ��

y;

leading to the identi�cation

��0 = hij�(jj ihl j)jki

= �ij ;kl0 :

In a similar way, aiming to express the map in the other basis of maps we have

�(�) =Xi; j=1

n

jiihj j hij�(�)jji

=Xi; j=1

n

jiihj jTr(jj ihij�(�))

=Xi; j=1

n

jiihj jTr(jj ihij�(Xk;l=1

n

jkihl jTr[jlihk j�]))

=X

i; j;k ;l=1

n

Tr[jj ihij�(jkihl j)] jiihj jTr(jlihk j�)

=X�;�=1

n2

�� Tr��y��;

leading to the identi�cation

�� = hij�(jkihl j)jj i= �ik ;jl

0


so that in this basis the matrices are related by a suitable exchange of indexes. Notehowever their basically di�erent meaning with reference to properties of the linear oper-ator or map �. As we shall see for a suitable choice of basis the matrix ��0 is alsoknown as Choi matrix.�

To stress the di�erence between positivity as linear operator, namely

h!;�(!)iHS > 0 8! 2B(Cn);

and positivity as a map, that is to say

�(!) > 0 8! 2B(Cn); !> 0;

we point to two simple examples.The map

�(!) = �z!�zy

is obviously completely positive, but it is not a positive linear operator, in fact

h�x;�(�x)iHS = ¡2:

On the other side the linear operator

�(!) = �zTr!�zy

is positive since

h!;�(!)iHS = Tr (!y�z)Tr¡!�z

y �> 0;

but

�(j+ih+j) = �z

= j+ih+j¡ j¡ih¡j;

so that the map is not positive, and therefore in particular it cannot be completely posi-tive.

As a last remark we point to the n4 � n4 matrix which connects the two operatorbases

�� = Tr��y�(��)

�= Tr

24��y X ;�=1

n2

� �0 � �� y

35=

X ;�=1

n2

Tr��y � ��

y�� 0=

X ;�=1

n2

M��; �� 0

where we have de�ned M��; �=Tr��y � ��

�.

Choi-Jamiolkowski isomorphism


A further important characterization of completely positive maps is given by a the-orem due to Choi, which connects complete positivity of a map with positivity of a cer-tain matrix, which will turn out to be related to the matrix ��0 introduced in connec-tion with the so called �-matrix representation of the map, for a suitable choice of basis.In the proof of the theorem we will use the following operator on CnCn

1nX = 1

n

Xj;k=1

n

Ejk Ejk

= 1n

Xj;k=1

n

jej ejihek ekj;

where as before fejg denote the canonical basis of vectors in Cn, and fEjkg the canon-ical basis of matrices in Mn(C). This operator is in particular the projection on themaximally entangled state

= 1n

pXj=1

n

jej eji;

so that it is a state. As a consequence X is a positive operator.�

We have the following important resultTheorem (Choi, 1972) Consider the map �:Mn(C) ! Md(C), then the following

statements are equivalent:

i. � is completely positive

ii. � is n-positive, that is �1Cn is a positive map

iii. For any orthonormal basis fejg in Cn the n d�n d square matrix (known as Choimatrix of �)

�� =

0@ �(je1ihe1j) ::: �(je1ihenj)��

�(jenihe1j) ::: �(jenihenj)

1Ais positive.

Let us prove the statement.i.)ii. Follows from the de�nitionii.)iii. Follows observing that

�� = �1Cn[X]

and as shown above X is a positive operator.iii.)i. A constructive proof which shows the insurgence of the Kraus operators goes

as follows. We have the following way to express the Choi matrix

�� = �1Cn[X]

=Xk;q=1

n

�[jekiheqj] jekiheqj

=Xk;q=1

n

�[Ekq]Ekq

and under the hypothesis that �� is a positive matrix we can diagonalize it, and usingnon normalized eigenvectors obtain the representation

�� =Xj=1

nd

jvjihvj j;


with vj 2CdCn, so that with fejg basis in Cn and vjk vectors in Cd we can write

vj =Xk=1

n

vjk ek

and de�ne an operator which relates the canonical basis in Cn to the vectors vjk

Vj:Cn ! Cd

ek vjk

and �nally obtain

�� =Xj=1

nd Xk;q=1

n

jvjk ekihvjq eqj

=Xj=1

nd Xk;q=1

n

jvjkihvjq j jekiheq j

=Xj=1

nd Xk;q=1

n

Vj jekiheq jVjy jekiheq j

=Xk;q=1

n Xj=1

nd

VjEkqVjyEkq:

Comparing with the previous representation and noting that the Ekq are linearly inde-pendent this implies X

j=1

nd

VjEkqVjy = �[Ekq] 8Ekq ;

but the fEkqg provide a basis in Mn(C) so that the representation holds for any oper-ator, we thus have obtained a Kraus representation for the map

�[T ] =Xj=1

nd

VjTVjy;

which as shown implies its complete positivity.This result by Choi shows in particular that for a map de�ned on a space of opera-

tors on a Hilbert space of dimension n, complete positivity is equivalent to n-positivity.In order to see the explicit connection between the Choi matrix and the matrix ��

0

introduced in connection with the so called �-matrix representation of the map, we con-sider its de�ning expression

��0 =X =1

n2

Tr��

y ��y�(� )

�and make the choice of basis ��! jekiheq j, so that, for �! i; j and �! k; l we haveindeed

�ij ;kl0 =X

n;m=1

n

Tr[jekiheljemihenjejiheij�(jenihemj)]

= heij�(jejihelj)jeki= hei ej j��jek eli

= hei ej jXm;q=1

n

�[Emq]Emq jek eli;


that is the i; k matrix element of the d� d block �(jejihelj).Relying on this construction one can also build a correspondence, known as Choi-

Jamiolkowski isomorphism, between the elements of the vector space of linear maps � 2L(Mn(C);Md(C))

�:Mn(C) ! Md(C)

and the linear operators ��2B(CdCn) on the Hilbert space CdCn

��:CdCn ! CdCn

according to

J :L(Mn(C);Md(C)) B(CdCn)� ��1[P];

where P is the projection on a maximally entangled state, which we previously denotedas (1/n)X.�

This correspondence is actually an isomorphism, in fact we can introduce the inversemapping de�ned as

J ¡1:B(CdCn) L(Mn(C);Md(C))� ��[Y ]�nTrCnf(1CdY T)�g;

with Y generic operator in Mn(C), and the trace is on the second factor of the tensorproduct, thus leading to an operator in Md(C). In this correspondence completely posi-tive maps are sent to positive operators, and in particular completely positive trace pre-serving maps, that is channels, are sent to states, in a suitably higher dimensional space.Linearity of both maps is evident by construction, so that we only have to check thatindeed they are one the inverse of the other. The operator J ¡1 � J act as the identityon the space of maps, indeed we apply it to a generic map �, we have 8 Y 2Mn(C)

(J ¡1 � J )�[Y ] = nTrCnf(1CdY T)�1Cn[P]g

= nTrCn

8<:(1CdY T) 1n

Xk;q=1

n

�[Ekq]Ekq

9=;=

Xk;q=1

n

�[Ekq]TrCnfY TEkqg

=Xk;q=1

n

�[Ekq]heq jY T jeki

=Xk;q=1

n

�[Ekq]hekjY jeqi

= �

24 Xk;q=1

n

hek jY jeqiEkq

35= �

24 Xk;q=1

n

jekihekjY jeqiheq j

35= �[Y ];


where we have used the fact that the trace only acts on the second factor and linearityof the map �. Since the action of (J ¡1 � J )� and � coincide on the generic operator,indeed J ¡1 � J acts as the identity. We now consider the operator J � J ¡1 and verifythat it acts as identity on the space of operators on Cd Cn. We have, using the pre-vious notation

(J �J ¡1)� = (J ¡1�)1Cn[P]

= 1n

Xk;q=1

n

(J ¡1�)[Ekq]Ekq

= 1n

Xk;q=1

n

nTrCn

�¡1CdEkqT

��Ekq

=Xk;q=1

n

hekj�jeqiEkq

=Xk;q=1

n

jekihekj�jeqiheq j

= �;

where we recall that hek j�jeqi 2Md(C), so that again indeed J � J ¡1 acts as the iden-tity.�

6 Dynamics of open quantum systems

Quantum dynamical semigroupsMoving from closed to open systems, we consider quantum dynamical maps which

are not necessarily given by unitary transformations, but we still ask for positivity andtrace preservation, mandatory for the probabilistic interpretation. We further keep com-plete positivity.

The reversible dynamics of a closed system described by a group of transformationsnow is generally replaced by an irreversible dynamics, which can at most satisfy a semi-group composition law.

An explicit general characterization of quantum dynamical maps is known only forspecial cases. An important and general class is obtained just assuming a semigroupcomposition law for the quantum dynamical map as a function of the time argument

�(t+ s) = �(t) ��(s) 8t; s> 0

where each �(t) is a completely positive trace preserving map. This is called the (time-homogeneous) Markovian case. A one-parameter group of unitary operators according toStone's theorem is described by its generator given by a self-adjoint operator. Likewisefor a semigroup of contraction operators a generator characterized by the Hille-Yoshidatheorem exists such that

�(t) = etL;

with L given by

L = limt!0+

�(t)¡1t

:

6 Dynamics of open quantum systems 105

Setting �(t) = �(t)�(0) one has the so called master equation for the time evolution ofthe statistical operator

ddt�(t) = L�(t);

which for the case of �(t) actually given by a unitary evolution gives back the Liouvillevon Neumann equation

ddt�(t) = ¡i[H(t); �(t)];

with H(t) a self-adjoint operator.If for any t the map is completely positive then the collection of these maps is called

a quantum dynamical semigroup.A quantum dynamical semigroup is therefore a collection of maps �(t) such that

�(t+ s) = �(t) ��(s) 8t; s> 0�(t) is completely positive tracepreserving 8t> 0

�(t)!1 for t! 0�(t) is continuous 8t> 0

Physical conditions allowing for semigroup dynamics are typically given by

�E� �R

i.e. environment correlation time (decay time of correlation function) much shorter thanrelaxation time of reduced system.

This separation of time scales (so called Markov condition) together with weak cou-pling (so called Born approximation), which also justi�es the choice of a factorized ini-tial state, typically allows for a description of the dynamics in terms of a quantumdynamical semigroup.

Structure of the generator

The characterization of the structure of the generators of quantum dynamical semi-group is given by the famous Gorini Kossakowski Sudarshan Lindblad theorem. Its �nitedimensional version reads:

Theorem (Gorini, Kossakowski and Sudarshan, 1976; Lindblad, 1976) Let dimHS=n, a linear operator L: T (HS) 7! T (HS) is the generator of a quantum dynamical semi-group, that is to say a one-parameter continuous semigroup of completely positive tracepreserving maps �(t)= etL i� it is of the form

L[�] = ¡i[H; �] +Xk=1

n2¡1 k

�Lk�Lk

y¡ 12�LkyLk; �

�= ¡i[H; �] +

Xk=1

n2¡1

k��Lk; �Lk

y �+�Lk�; Lky �with k> 0; H =H y, Lk2B(HS).


The result extends to in�nite dimensional Hilbert spaces provided one asks for normcontinuity of �(t). Most importantly it is a necessary and su�cient condition. We givean idea of the proof, pointing to extensions of the su�cient condition to account formore general situations.

The key ingredients are: the Hille-Yoshida theorem for the existence of a generator,the Kraus representation for a completely positive map, the Lindblad expression of thegenerator ensuring trace and hermiticity preservation.

Necessary conditionIf �(t) is completely positive trace preserving, according to the Kraus representation

at any time it can be written as �(t)[�] =P

i Ai(t)�Aiy(t) with

Pi Ai

y(t)Ai(t) = 1.Writing the Ai(t) in terms of a basis fFigi=0;1;:::;n2¡1 of operators orthonormal withrespect to the Hilbert-Schmidt scalar product, so that hFi; FjiHS= TrHSFi

yFj = �ij, set-

ting F0=1

np , so that the Fj are traceless for j =/ 0, one has Ai(t) =

Pk=0n2¡1 vik(t)Fk and

therefore

�(t)[�] =Xi; j=0

n2¡1

cij(t)Fi�Fjy

with cij(t) =P

k=0n2¡1 vik(t)vjk� (t) a positive matrix. We know that the generator exists

and is given by

limt!0+

�(t)¡1t

[�] = L[�]:

Relying on the existence of the limit and imposing trace preservation one comes to thedesired result.�

More speci�cally we have

L[�] = limt!0+

�(t)¡1

t[�]

= limt!0+

8>><>>:1n

c00(t)¡nt|||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }a00

� + 1n

pXk=1

n2¡1ck0(t)t

Fk�|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }F

+ 1n

pXk=1

n2¡1ck0� (t)t

�Fky +

Xk;l=1

n2¡1ckl(t)t||||||||||||||||{z}}}}}}}}}}}}}}}}akl

Fk�Fly

9>>=>>;= a00

n�+F�+ �F y+

Xk;l=1

n2¡1

aklFk�Fly

which can also be written

L[�] = ¡i�F y¡F2i

; �

�+ 12

na00n

1+F +F y; �o+Xk;l=1

n2¡1

aklFk�Fly:

We now ask for trace preservation which implies

a00n

1+F +F y = ¡Xk;l=1

n2¡1

aklFlyFk


so that de�ning the self-adjoint operator

H = F y¡F2i

one obtains the following form for the generator

L[�] = ¡i[H; �]+Xk;l=1

n2¡1

akl

�Fk�Fl

y¡ 12�FlyFk; �

�;

where indeed H =H y, and akl is a positive matrix known as Kossakowski matrix, as fol-lows from positivity of ckl(t) and the relation akl= limt!0+ckl(t)/t.

Diagonalization of the positive matrix (A)kl= akl according to

A = U�U y

leads to an explicit Lindblad form with Lk=P

i uikFi. In Schrödinger picture

L[�] = ¡i[H; �] +Xk=1

n2¡1 k

�Lk�Lk

y¡ 12�LkyLk; �

�= ¡i[H; �] + 1

2

Xk=1

n2¡1

k��Lk�; Lk

y �+�Lk; �Lky �corresponding to the Heisenberg picture

L0[X] = +i[H;X] +Xk=1

n2¡1

k

�LkyXLk¡

12�LkyLk; X

�;

which can be equivalently written in so called standard form by means of the completelypositive map

[X ] =Xk=1

n2¡1 kLk

yXLk

as follows

L0[X] = +i[H;X] +[X]¡ 12f[1]; Xg:

The fLkg are typically called Lindblad operators.Note that trace preservation in Schrödinger picture corresponds to identity preserva-

tion in Heisenberg picture. Note furthermore that the generator does not uniquely �xthe operators H and Lk. Indeed the expression of L is invariant under the followingtransformations.

1. kp

Lk! k0

pLk0 =P

j ukj jp

Lj with U =(uij) unitary matrix

2. Lk!Lk0 =Lk+ ak together with H!H 0=H + b+ 1

2i

Pj j¡aj�Lj ¡ ajLj

y�, whereak2Cand b2R

the latter transformation shows that (in �nite dimension) H and Lk can be taken to betraceless.

Su�cient condition I


We want to see that a generator in Lindblad form leads to a completely positivemap. The semigroup composition law follows from the exponential representation �(t) =etL. Preservation of hermiticity and trace is immediately checked. To convince oneself ofcomplete positivity of the time evolution the following representation for in�nitesimal d tmight be inspiring

�(t+ d t) =

0@1 ¡ i~H dt ¡ 1

2

Xk=1

n2¡1

kLkyLkdt

1A�(t)0@1 ¡ i

~H dt ¡

12

Xk=1

n2¡1

kLkyLkdt

1Ay+Xk

kLk�Lkydt+O((d t)2):

We show complete positivity pointing to a perturbation expansion of the solution. Letus put L=LR+LJ, sum of a relaxing and jump part, according to

LR[�] = ¡i[H; �]¡ 12

(Xk

kLkyLk; �

)= ¡i

¡He��¡ �He�

y �LJ[�] =

Xk

kLk�Lky

with LJ a completely positive map. We know that

ddt�(t) = L�(t)

and therefore due to �(t)=�(t)�(0) also

ddt�(t) = L�(t) �(0)=1:

Denoting by R(t) the solution of the relaxing part

ddtR(t) = LRR(t) R(0)=1

given by

R(t)[�] = eLRt[�]= e¡iHe� t�e+iHe�y t

one has the representation, known as Duhamel's formula

�(t) = R(t)+Z0

t

d�R(t¡ � )LJ�(� )

= eLRt+Z0

t

d�eLR(t¡�)LJ�(�)

= R(t)+ (R ?LJ�)(t):

Indeed with this de�nition one has

�(t) = �(t)�(0)

= e¡iHe� t�(0)e+iHe�y t+

Z0

t

d�e¡iHe�(t¡�)Xk

kLk�(� )Lkye+iHe�

y (t¡�)

and one can directly verify that the master equation holds since the time derivative ofthe r.h.s. reads

¡iHe��(t)+ i�(t)He�y +

Xk

kLk�(t)Lky:


This equation is of the form G=G0+G0VG and is solved by the Dyson series

�(t) = R(t)+ (R ?LJR)(t)+ (R ?LJR ?LJR)(t)+ :::

which is a completely positive map by construction, because such are R(t) and LJ.Apparently it is enough to ask R(t) and LJR(t) to be completely positive, but theinverse of R(t) is also completely positive, so that this requirement is not weaker. Thesolution of the master equation can thus be explicitly written as a jump expansion asfollows

�(t) = �(t)�(0)

= R(t)�(0)+Xn=1

1 Z0

t

dtn:::Z0

t2

dt1R(t¡ tn)LJR(tn¡ tn¡1):::LJR(t1)�(0):

Su�cient condition IIThe proof still works if we allow the operators appearing in the Lindblad structure as

well as the decay rates k to become time dependent, provided the latter always staypositive. This situation corresponds to a time-inhomogeneous Markovian case. Startingfrom

ddt�(t) = L(t)�(t)

with �(t) � �(t; 0) and the initial condition �(t; t) = 1, we can still consider a relaxingpart LR(t) and a jump part LJ(t), where the latter still is a completely positive mapthanks to the positivity of the k(t). As done before starting from the solution of thetime-local master equation

ddtR(t; s) = LR(t)R(t; s)

where R(t)�R(t; 0), R(t; t)=1 and t> s> 0, given by

R(t; s) = T~ exp�Z

s

t

d�LR(� )�

where T~ denotes the chronological time ordering. This is a completely positive map sat-isfying the two-time composition law

R(t; �) �R(� ; s) = R(t; s) 8t> � > s:

As a result we can still write a Dyson expansion for the time evolution map, whose verystructure ensures complete positivity of the time evolution. One has

�(t) = �(t)�(0)= R(t)�(0)

+Xn=1

1 Z0

t

dtn:::Z0

t2

dt1R(t; tn)LJ(tn)R(tn; tn¡1):::R(t2; t1)LJ(t1)R(t1)�(0);

and therefore a time evolution map characterized by two time indexes, satisfying thecomposition law

�(t; �) ��(� ; s) = �(t; s) 8t> � > s

where each of the maps �(t; s) is completely positive and can be written as

�(t; s) = T~ exp�Z

s

t

d�L(� )�:


This kind of time local master equations arise e.g. in the time-convolutionless projectionoperator technique.

Projection operatorsGiven the general structure of the master equation for the reduced system operator

which grants complete positivity of the dynamics, one might wonder whether and howsuch master equations do arise from a microscopic description of the underlyingdynamics. To point to this connection we consider a general technique, devised by Naka-jima and Zwanzig (1958 and 1960 respectively), in order to obtain a representation forthe exact equations of motion for the overall statistical operator, splitting it according toa projection operator in a so called relevant and irrelevant part. The basic idea comesfrom non equilibrium statistical mechanics: we have a complex system and try to obtaina manageable dynamics by eliminating degrees of freedom by means of some projectionoperator, thus considering the dynamics or relevant variables only, to be described interms of e�ective master equations.

A projection operator is a map which sends states into states

P: � 7!P�

so that it has to be linear, positive, trace preserving and idempotent P2= P. The latterrequirement corresponding to the de�ning property of a projection. Having in mindthat the total space has a bipartite structure H=HS HE we consider projection oper-ators of the form

P = 1S�

with � a completely positive, trace preserving idempotent map on HE. This choicegrants the crucial property

�S=TrE�=TrEP�

so that knowledge of the dynamics of the relevant part is enough to recover the reducedstate �S. It further implies that separable (product) states are sent to separable (pro-duct) states, so that the space of separable (product) states is invariant under P, namelyno arti�cial entanglement is introduced by means of the chosen projection. Note thatindeed in the preparation procedure only classical correlations are usually introduced.For such a projection operator one has the following representation theorem.

Theorem (Breuer, 2007) A projection operator with the above mentioned propertiescan be written as

P� =Xi

TrEfAi�gBi

with fAig; fBig linearly independent sets of observables on HE with the properties

1. TrEfAiBjg= �ij2.P

i TrEfBigAi=1E

3.P

i AiT Bi> 0

To prove the result, instead of resorting to the Kraus representation for a completelypositive map, we �rst observe that a linear and idempotent map ¡ on a �nite dimen-sional Hilbert space can always be represented in the form

¡j'i =Xk

jfkihekj'i;


with fekg and ffkg two sets of linearly independent vectors, satisfying hekjfji= �k;j.�

In our setting the Hilbert space is given by the space of Hilbert-Schmidt operatorsover the Hilbert space HE which has dimension n. We thus have the representation

�[X] =Xk=1

n2

BkTrE¡AkyX�;

with fAkg and fBkg two sets of linearly independent operators, satisfying TrE¡AkyBj

�=

�k;j. Further asking hermiticity of �, we have that these operators have to be self-adjoint, so that they can be taken as observables. Trace preservation requires thevalidity of

TrE�[X] = TrE

8<:Xk=1

n2

BkTrE(AkX)

9=;= TrE

8<:24Xk=1

n2

TrE(Bk)Ak

35X9=;

= TrEX

for all operators X, thus leading to the requirementXk=1

n2

TrE(Bk)Ak = 1E:

We now have to put conditions on these operators in order to have complete positivityof the map �. To this end we study positivity of the associated Choi's matrix. TheChoi's matrix associated to the map is given by

(1�)[X] = (1�)

24 Xj;k=1

n

Ejk Ejk

35=

Xj;k=1

n

Ejk �[Ejk]

=Xj;k=1

n Xi=1

n

Ejk BiTrE(AiEjk)

=Xj;k=1

n Xi=1

n

Ejk BihekjAijeji

=Xj;k=1

n Xi=1

n

Ejk Bihej��AiT��eki

=Xi=1

n Xj;k=1

n

jekihek��AiT��ejihej j Bi

=Xi=1

n

AiT Bi;

so that complete positivity is granted provided the obtained operator is positiveXi=1

n2

AiT Bi > 0:


�We have thus recovered the three constraints. Viceversa, given sets of observables

fAig; fBig linearly independent satisfying the above constraints, one can checks that theoperator

P� =Xi

TrEfAi�gBi

provides a well de�ned projection with the required properties. Note that in general dif-ferent sets of observables can represent the same mapping, e.g. we can take

Ai0 =

Xk

uikAk

Bi0 =

Xk

vikBk;

where the real non singular matrices U and V satisfy UTV =1.

Standard projectionThe standard projection operator onto a factorized state is obtained for the choice

A=1E B= �E

with �E a �xed environmental state

P� = TrE� �E= �S �E:

Note that at the r.h.s. we do not have the product of the marginals of �, but rather theproduct of the �rst marginal with a �xed environmental state.

Correlated projectionA correlated projection operator is obtained considering an orthogonal decomposition

of unity in HE according to �i=�iy, �i�j= �ij�i,

Pi �i=1E, and de�ning

Ai=�i Bi= �Ei = �i�E�i

TrEf�i�Eg

with �Ei is the collection of statistical operators obtained from a �xed environmental

state �E upon the action of a map which implements a von Neumann instrument. Inthis case we have the expression

P� =Xi

TrEf�i�g�i�E�i

TrEf�i�Eg:

Projected equations of motionIn order to obtain equations of motion for the reduced statistical operator only we

start from the overall unitary dynamics. For the total system one has the Liouville vonNeumann equation with

H = HS+HE|||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }H0

+�V

and therefore in interaction picture with V (t)= eiH0tV e¡iH0t

ddt�(t) = ¡i�[V (t); �(t)]�L(t)�(t);


where we have de�ned the Liouvillian operator L(t) as

L(t)[ � ] = ¡i�[V (t); �]:

To simplify the notation we omit the index I in �(t) and V (t), though they are indeedoperators in interaction picture.

We now use P and the complementary projection operator 1¡Q to write the statis-tical operator in terms of a relevant and irrelevant part according to � = P� + Q�. Thetwo contributions obey the equations

ddtP�(t) = PL(t)P�(t)+PL(t)Q�(t)

ddtQ�(t) = QL(t)P�(t)+QL(t)Q�(t):

By analogy with the case of functions, in which a di�erential equation of the form

ddt�(t) = �(t)(�(t)+ (t))

has the solution

�(t) = eR0td��(�)

�(0)+Z0

t

d� eR�td��(�)

�(�) (�);

the second equation can be solved introducing the operator

G~Q(t; t1) = T~ exp�Z

t1

t

dt2QL(t2)�;

where T~ denotes chronological time ordering, solution of the homogeneous equation

ddtG~Q(t; t1) = QL(t)G~Q(t; t1)

corresponding to the initial condition

G~Q(t; t) = 1:

The solution of the equation for the irrelevant part therefore reads

Q�(t) = G~Q(t; 0)Q�(0)+Z0

t

dt1G~Q(t; t1)QL(t1)P�(t1)

and substituting in the �rst equation one obtains

ddtP�(t) = PL(t)P�(t)+PL(t)G~Q(t; 0)Q�(0)+

Z0

t

dt1PL(t)G~Q(t; t1)QL(t1)P�(t1):

Before going on we consider a couple of simplifying assumptions of general validity.First we suppose that the inhomogeneous term vanishes, so that

Q�(0) = 0;

which means that the initial state is an eigenoperator of the projection on the relevantpart. In the case of the standard projection this corresponds to a factorized initial state,which grants in particular, as already discussed, the existence of the reduced dynamicsat the level of the Hilbert space of the system only. The other assumption is that

PL(t1):::L(t2n+1)P = 0;


which corresponds to

TrEV (t1):::V (t2n+1)�E = 0

and is typically satis�ed in applications. To see the connection between the two equiva-lent conditions let us consider the simplest case, with a single action of the LiouvillianL(t). We have, considering a standard projection

PL(t)P� = TrE(L(t)P�) �E= ¡i�TrE([V (t); �S �E]) �E= ¡i�TrE(V (t)�E)�S �E+ i��STrE(�EV (t)) �E ;

where we have denoted as usual �S = TrE� and TrE(V (t)�E) is an operator on HS. Sothat the term vanishes whenever TrE(V (t)�E) = 0. The evolution equation for the rele-vant part thus reads

ddtP�(t) =

Z0

t

dt1PL(t)G~Q(t; t1)L(t1)P�(t1);

where in the integral we have replaced Q = 1 ¡ P with the identity because of the pre-vious condition. We now want to obtain perturbation expansions of this equation,allowing for approximate solutions, recalling that TrEP�(t) = �S(t). This equation isvalid for di�erent choices of P and leads to equations for di�erent relevant states, whichall lead to the same exact equations for �S, though rearranged in a non perturbativeway.

Introducing the integral kernel

KNZ(t; t1) = PL(t)G~Q(t; t1)L(t1)P

where NZ stands for Nakajima-Zwanzig one has

ddtP�(t) =

Z0

t

dt1KNZ(t; t1)P�(t1):

Note that if the integral kernel only depends on the time di�erence of the two argumentsand we consider the standard projection the exact equations of motion for the reducedstatistical operator �S(t)=�(t)�S(0) can be written as a convolution integral

ddt�S(t) = (KNZ? �S)(t)= (KNZ?�)(t)�S(0)

which in Laplace transform leads to the formal relationship

K̂NZ(u) = u1¡ �̂¡1(u):

Nakajima-Zwanzig master equationOne can now consider a perturbation expansion of the memory kernel

KNZ(t; t1) = PL(t)G~Q(t; t1)L(t1)P

relying on the natural expansion of G~Q(t; t1), where we recall that each Liouvillian L(t)brings with itself a factor �

G~Q(t; t1) = T~ exp�Z

t1

t

dt2QL(t2)�

= 1+Zt1

t

dt2QL(t2)+Zt1

t

dt2

Zt1

t2dt3QL(t2)QL(t3)+ :::


so that we can write KNZ=KNZ(2) +KNZ

(4) + ::: with

KNZ(2)(t; t1) = PL(t)L(t1)P

KNZ(4)(t; t1) =

Zt1

t

dt2

Zt1

t2

dt3[PL(t)L(t2)L(t3)L(t1)P ¡PL(t)L(t2)PL(t3)L(t1)P];

where the order of the perturbation corresponds to the order in powers of �. Theexpression up to second order obtained replacing G~Q(t; t1) with the identity operatorreads

ddtP�(t) =

Z0

t

dt1PL(t)L(t1)P�(t1):

= ¡Z0

t

dt1P[V (t); [V (t1);P�(t1)]]

To obtain the master equation for the reduced state �S(t) we now have to specify thechoice of projection, following the above introduced representation so that

P� =Xi

TrEfAi�gBi:

Let us �rst consider the case of a standard projection, corresponding to A! 1E, B!�E, we obtain

ddt�S(t) �E = ¡

Z0

t

dt1TrEf[V (t); [V (t1); �S(t1) �E]]g �E ;

so that upon taking the partial trace with respect to the environment of the Nakajima-Zwanzig perturbation expansion we obtain the following equation for �S(t) only

ddt�S(t) = ¡

Z0

t

dt1TrEf[V (t); [V (t1); �S(t1) �E]]g

which is sometimes called generalized master equation.If we consider instead a correlated projection where Ai!�i, Bi! �E

i =�i/TrE�i weobtain

ddt

Xi

TrEf�i�(t)g�i

TrE�i= ¡

Z0

t

dt1Xk

TrE

8<:�k24V (t); "

V (t1);

Xj

TrEf�j�(t1)g�j

TrE�j

#359=; �kTrE�k

:

From this expression de�ning the sub-collections

wi(t) = TrEf�i�(t)g;

which are trace class operators on HS with trace less or equal to one we come to

ddtwj(t) = ¡

Z0

t

dt1TrE

(�j

"V (t);

"V (t1);

Xi

wi(t1)�i

TrE�i

##):

According to the relationship

�S(t) =Xj

wj(t)


we �nally have

ddt�S(t) = ¡

Z0

t

dt1TrE

("V (t);

"V (t1);

Xi

wi(t1)�i

TrE�i

##):

Note however that in accordance to the fact that the initial state is not necessarily fac-torized, but is only an eigenoperator of the correlated projection, there generally is noclosed dynamics at the level of the reduced system only. Indeed to solve the last equa-tion one does not simply need to know �S(0), the knowledge of all sub-collections attime zero wi(0) is required. The initial state �S(0) only �xes

Pi wi(0), and not the

single contributions.

Weak coupling master equationAs we have seen, up to second order and with the so called standard choice for the

projection operator the Nakajima-Zwanzig projection operator technique leads to themaster equation

ddt�S(t) = ¡

Z0

t

dt1TrEf[V (t); [V (t1); �S(t1) �E]]g:

The approximation involving the restriction of the perturbation expansion to secondorder is also called Born approximation, in analogy with the Born approximation inscattering theory. Upon performing the change of variables t1= t¡ � in the integral onehas

ddt�S(t) = ¡

Z0

t

d�TrEf[V (t); [V (t¡ �); �S(t¡ � ) �E]]g;

and we can further simplify this expression extending the integral to the whole real lineand replacing �S(t ¡ � ) t �S(t). The latter approximation is known as Markov approxi-mation. The basic justi�cation behind these approximations is a separation of timescales between a typical time scale of the environment �E which characterizes a decaytime of its correlation functions, and the time scale �R of the dynamics of the reducedsystem. If we have

�E � �R;

then the actual contribution to the integral only refers to a short time interval, thus jus-tifying the extension of the integral to the whole real line, as well as the fact that thesmall retardation in the time argument of the statistical operator of the reduced systemis neglected. Thus in the so called Born-Markov approximation the master equationreads

ddt�S(t) = ¡

Z0

1d�TrEf[V (t); [V (t¡ � ); �S(t) �E]]g:

We now proceed to further elaborate this expression in order to obtain the so calledweak-coupling master equation.�

As a �rst step we notice that the interaction term can be generally written in theform

V =X�

A�B�;


where due to self-adjointness of V the operators fA�g and fB�g, acting on the Hilbertspace of system and environment respectively, can also be taken self-adjoint. In order tosort out di�erent contributions to the dynamics according to their time dependence,which will allow to later perform a so called secular approximation, it is convenient todecompose the interaction term into eigenoperators of the free system Hamiltonian HS.Supposing HS to have a pure point spectrum, and denoting by Ei its eigenvalues and�(fEig) the projections on the corresponding eigenspace, so that

HS =XEi

Ei�(fEig)

we can introduce the following decomposition for the system operators appearing in theinteraction term

A� = 1A�1

= X

Ei

�(fEig)!A�

XEi

�(fEig)!

=X!

XEj¡Ei=!

�(fEig)A��(fEjg)

�X!

A�(!)

=X!

A�y(!);

where the real quantities ! do correspond to all possible energy di�erences among dis-tinct eigenvalues of the system. Note that the �(fEig) are not necessarily one dimen-sional. In such a way we have de�ned a collection of operators

A�(!) =X

Ej¡Ei=!�(fEig)A��(fEjg);

which together with their adjoints

A�y(!) = A�(¡!)

are indeed eigenoperators of the free system Hamiltonian corresponding to the eigen-values �!

[HS ; A�(!)] = ¡!A�(!)�HS ; A�

y(!)�= +!A�

y(!);

where the index � can now be seen as a degeneracy index for the eigenoperators corre-sponding to the eigenvalue �!. Exploiting these commutation relations we also immedi-ately have �

HS ; A�y(!)A�(!)

�= 0:

These operators also have a very simple expression in interaction picture, since we have

eiHStA�(!)e¡iHSt = e¡i!tA�(!)

which leads to the following expression for the interaction term in interaction picture,where we omit the standard lower index I

V (t) =X!

X�

e¡i!tA�(!)B�(t)

=X!

X�

e+i!tA�y(!)B�

y(t):


�Let us notice that the condition

PL(t)P = 0

which we used to come to the second order Nakajima-Zwanzig master equation nowtranslates into

TrEB�(t)�E = hB�(t)i�E= 0:

If we now insert the obtained expressions for the interaction term in the master equationwe obtain

ddt�S(t) = ¡

Z0

1d�TrEf¡V (t¡ �)�S(t) �EV (t)+V (t)V (t¡ � )�S(t) �E+h:c:g

=Z0

1d�X!;! 0

�;�

TrE�e¡i!(t¡�)A�(!) B�(t ¡ �)(�S(t) �E)e+i!

0tA�y(! 0)

B�y(t) ¡ e+i!

0tA�y(! 0) B�

y(t)e¡i!(t¡�)A�(!) B�(t ¡ �)�S(t) �E +h:c:

=X!;! 0

�;�

e+i(!0¡!)t

Z0

1d�ei!�TrE

�B�y(t)B�(t ¡ �)�E

��A�(!)�S(t)A�

y(! 0) ¡

A�y(! 0)A�(!)�S(t)

+h:c:;

�where we have used the cyclic property of the trace and the fact that operators on

system and environment only simply commute. In view of the obtained expression it isnow natural to de�ne the following C-number function of the energy di�erences of thesystem !

¡��(!) =Z0

1d�ei!�TrE

�B�y(t)B�(t¡ � )�E

�=Z0

1d�ei!� hB�

y(t)B�(t¡ � )i�E:

One can immediately check that provided

[HE ; �E] = 0;

so that the state of the environment is a function of its free Hamiltonian, the functions¡��(!) are actually independent of the time t and only depend on the time di�erence inthe argument of the two interaction picture operators. We can thus write

¡��(!) =Z0

1d�ei!� hB�

y(�)B�(0)i�E:

As anticipated by introducing the eigenoperators of HS we have put into evidence anexplicit time dependence through the oscillating functions e+i(!

0¡!)t. We now perform aso called secular approximation, consisting in keeping only terms with ! = ! 0, whichprovide contributions which cumulate in time, while the terms with !=/ ! 0 do not give asigni�cant contribution to the time evolution if they oscillate very rapidly. This approxi-mation thus requires that

�S � �R;


where �S is a typical time scale of the free system evolution, which can be estimated as�S t ~/j! ¡ ! 0j, while �R is the typical time scale of the dynamics of the system coupledto the environment, roughly of the order of the inverse of the coupling constantsappearing in the master equation.�

Under this hypothesis the secular approximation applies. We now consider the split-ting

¡��(!) = 12 ��(!)+ i S��(!)

where

��(!) = ¡��(!)+¡�� (!)

and

S��(!) = 12i(¡��(!)¡¡�� (!))

are hermitian matrices for any !, that is ��(!) = �� (!) and S��(!) = S��

� (!). Rear-ranging terms according to

ddt�S(t) =

X!

X�;�

¡��(!)�A�(!)�S(t)A�

y(!)¡A�y(!)A�(!)�S(t)

+h:c:

=X!

X�;�

(¡��(!) + ¡�� (!))A�(!)�S(t)A�

y(!) ¡X!

X�;�

�12 ��(!) +

i S��(!)�A�y(!)A�(!)�S(t)

¡X!

X�;�

�12 ��(!)¡ i S��(!)

��S(t)A�

y(!)A�(!);

we �nally have

ddt�S(t) = ¡i

"X!

X�;�

S��(!)A�y(!)A�(!); �S(t)

#+

X!

X�;�

��(!)�A�(!)�S(t)A�

y(!)¡ 12�A�y(!)A�(!); �S(t)

�:

It is now crucial to observe that the !-dependent matrix ��(!) can be written as

��(!) =Z¡1

1d�ei!� hB�

y(� )B�(0)i�E;

which warrants its positivity. To see this let us �rst introduce the notion of positive de�-nite function. A function f(t) on the real line is said to be positive de�nite if for arbi-trary t1; :::; tn and 8n2N

� Xi; j=1

n

vi�f(ti¡ tj)vj > 0;


where fvig 2Cn, that is to say the n� n matrices aij = f(ti¡ tj) are positive for any n.One can check that the Fourier transform of a positive function is positive de�nite.Viceversa according to Bochner's theorem the Fourier transform of a positive de�nitefunction is positive. In our case we have, setting B =

P� B�w�X

i; j=1

n

vi�hB y(ti¡ tj)B(0)i�Evj =

Xi; j=1

n

vi�TrEfB y(ti¡ tj)B(0)�Egvj

=Xi; j=1

n

vi�TrE

�eiHEtiB ye¡iHEtieiHEtjBe¡iHEtj�E

vj

= TrE

8<: Xi=1

n

eiHEtiBe¡iHEtivi

!y0@Xj=1

n

eiHEtjBe¡iHEtjvj

1A�E9=;

= TrEfX yX�Eg;

so that f(�) = hB y(� )B(0)i�E is positive de�nite, and therefore thanks to Bochner's the-orem its Fourier transform is positive. Given the actual expression of B we have there-fore actually shown that the matrix ��(!) is positive. De�ning the Hamiltonian contri-bution

HLS =X!

X�;�

S��(!)A�y(!)A�(!);

where LS stands for Lamb shift, and the dissipator

D[�] =X!

X�;�

��(!)�A�(!)�A�

y(!)¡ 12�A�y(!)A�(!); �

�we �nally have the master equation

ddt�S(t) = ¡i[HLS; �S(t)] +D[�S(t)];

which thanks to positivity of the matrix ��(!) is indeed of Lindblad form and thereforeprovides a proper generator of quantum dynamical semigroup.��

We further note that thanks to the fact that the operators A�(!) are eigenoperatorsof the system free Hamiltonian the dissipator D is covariant with respect to the unitarytransformation generated by HS. As a result in going from the present equation in inter-action picture to the master equation in Schrödinger picture one only has to add a com-mutator term with HS.

We now introduce other properties that the master equation might have under fur-ther hypothesis.

If the environment is in a canonical equilibrium state, so that its two-point correla-tion function obeys the so called �-KMS property

hB�y(t)B�(0)ith = hB�(0)B�

y(t+ i�)ith;

where KMS stands for Kubo, Martin and Schwinger, which is easily veri�ed if the envi-ronment is a thermal reservoir described by the canonical equilibrium distribution, thenthe following thermal state for the system

�th = e¡�HS

TrSe¡�HS


is a stationary solution of the master equation. Indeed using the �-KMS condition onehas

��(¡!) =Z¡1

1d�e¡i!� hB�

y(�)B�(0)ith

= e¡�! ��(!):

�Exploiting further

�thA�(!) = e+�!A�(!)�th�thA�

y(!) = e¡�!A�y(!)�th;

�together with [�th; HLS] = 0 as follows from the already shown property [HS ; HLS] = 0

one immediately obtains, using again A�y(!)=A�(¡!)

D[�th] =X!

X�;�

��(!)�A�(!)�thA�

y(!)¡ 12�A�y(!)A�(!); �th

�=X!

X�;�

��(!)�e¡�!A�(!)A�

y(!)�th ¡ 12A�y(!)A�(!)�th ¡

12A�y(!)A�(!)�th

�=X!

X�;�

� ��(¡!)e�!A�

y(!)A�(!)�th ¡ 12 ��(!)A�

y(!)A�(!)�th ¡

12 ��(!)A�

y(!)A�(!)�th

�=X!

X�;�

[ ��(¡!)e�!¡ ��(!)]A�y(!)A�(!)�th

= 0:

�A semigroup is said to be relaxing, if the stationary state is unique and all initial

states converge to it. A su�cient condition to ensure this fact is that ��(!) > 0, andthat any operator which commutes with the Hamiltonian and all fA�(!)g and

�A�y(!)

is proportional to the identity operator, that is

�HS ; HLS; A�(!); A�

y(!)�;!0 = c1.

If the pure point spectrum of HS is non degenerate, so that all the �(fEig) are one-dimensional projections, say

HS =XEn

Enj'nih'nj

with the fEng distinct eigenvalues, then the equations for populations and coherencesget decoupled. That is to say the evolution equations for the diagonal matrix elementsof the statistical operator are closed and do not involve o�-diagonal matrix elements. Toshow this one starts from the identity

h'mjA�(!)'ni =Xk;l

El¡Ek=!

h'mj'kih'kjA�'lih'lj'ni

= �En¡Em;! h'mjA�'ni


so that

h'nj[HLS; �S(t)]'ni = 0;

and as far as the dissipative part is concerned, recalling that the A� are self-adjoint, onehas

h'njD[�S(t)]'ni =X!

X�;�

��(!)�h'njA�(!)�S(t)A�

y(!)'ni ¡

12h'nj

�A�y(!)A�(!); �S(t)

'ni

�=X!

X�;�

Xr;s

��(!)

24 h'njA�(!)'ri|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }�Er¡En;!

h'r j�S(t)'sih'sjA�y(!)'ni|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }

�Es¡En;!

¡

12h'njA�

y(!)'rih'r jA�(!)'sih'sj�S(t)'ni ¡12h'nj�S(t)'rih'r jA�

y(!)'sih'sjA�(!)'ni

35=Xr

8<: X�;�

��(Er ¡

En)h'njA�'ri(h'njA�'ri)�!h'r j�S(t)'ri ¡

X�;�

��(En ¡

Er)h'r jA�'ni(h'r jA�'ni)�!h'nj�S(t)'ni

9=;;where we have used �Er¡En;!�Es¡En;! = �r;s�Er¡En;! thanks to non degeneracy of thespectrum.

Upon de�ning the transition rates

W (njr) =X�;�

��(Er¡En)h'njA�'rih'r jA�y'ni;

which are actually positive, the evolution equation for the diagonal matrix elements inthe energy eigenbasis

h'nj�S(t)'ni = Pn(t)

therefore readsddtPn(t) =

Xr

[W (njr)Pr(t)¡W (r jn)Pn(t)]:

This equation is also known as Pauli master equation. As it appears the equation for thepopulations has the form of the master equation for a classical Markovian process, withtransition rates W (njr). If the state of the environment is of the canonical form theserates obey the so called detailed balance condition

W (njr)e¡�Er = W (r jn)e¡�En

as follows from ��(¡!)= e¡�! ��(!) and therefore

W (njr) =X�;�

��(En¡Er)h'rjA�'nih'njA�y'rie¡�(En¡Er)

= W (r jn)e¡�(En¡Er):


The detailed balance condition ensures the canonical stationary solution for the classicalprocess Pn(t), so that the equilibrium populations are distributed according to theBoltzmann law.

Quantum optical master equationStarting from the general expression of the weak coupling master equation, we now

consider a de�nite model of environment, later also specifying a particular system.We suppose the environment to be the quantized electromagnetic �eld. Taking the

�eld to be con�ned in a box of volume V and assuming periodic boundary conditions,apart from an in�nite C-number contribution related to normal ordering the free electro-magnetic Hamiltonian can be written

HE =Xk

X�=1;2

~!kb�y(k)b�(k);

where the sum is over the two polarization states for the photon � and the wave vectork, while the frequencies are �xed by the dispersion relation !k = cjkj. The creation andannihilation operators are assumed to satisfy the canonical commutation relations, sothat we have a Bosonic environment�

b�(k); b�0y (k 0)

�= ��;� 0�k;k0

[b�(k); b�0(k 0)] = 0�b�y(k); b�0

y (k 0)�= 0:

We denote by e�(k) the unit polarization vectors, in the plane orthogonal to the wavevector, chosen so as to form Cartesian axes

k �e�(k) = 0e�(k) �e�0(k) = ��;�0:

Since the components of the polarization vectors can be seen as their direction cosinealong the coordinate axes one has the relation

(e1(k))i(e1(k))j+(e2(k))i(e2(k))j+�kjkj

�i

�kjkj

�j

= �ij

or equivalently X�=1;2

(e�(k))i(e�(k))j = �ij ¡kikjjkj2 ;

which identi�es the linear operator

Pij = �ij ¡kikjjkj2

as projection of a vector on its transverse component with respect to the direction k.�

We consider the system to be matter, e.g. an atom, coupled to the �eld in dipoleapproximation, so that the interaction term reads

V = ¡X

i=1;2;3

DiEi

= ¡D �E ;


so that the previous sum over � now runs over a Cartesian index. The triple of opera-tors Di are the components of the dipole operator for the system, while Ei are the threeCartesian components of the quantized electric �eld. One can show that the dipoleinteraction term V =¡D �E arises from the standard interaction term V =¡e(p �A)/mfor a massive particle coupled to the electromagnetic potential by performing a suitabletime dependent gauge transformation and working in the dipole approximation. In inter-action picture the coupling term to be inserted in the master equation can be written

V (t) = ¡X!

e¡i!tD(!) �E(t)

= ¡X!

e+i!tDy(!) �E(t);

where the sum over � is replaced by the sum over a Cartesian index and the electric�eld can be written

E(t) = iXk

X�=1;2

~!k2"0V

re�(k)

�b�(k)e¡i!kt¡ b�

y(k)e+i!kt�:

�The condition

PL(t)P = 0

now therefore reads

TrEE(t)�E = hE(t)i�E= 0;

which holds if the electromagnetic �eld is in a thermal state or in the vacuum.According to the expression of the coupling the matrix of ! dependent coe�cients in themaster equation now reads

¡��(!) ! ¡ij(!)=1~2

Z0

1d�ei!� hEi(t)Ej(t¡ � )i�E

and therefore

¡ij(!) = 1~2Xk;k0

X�=1;2

X�0=1;2

~!k2"0V

r~!k 02"0V

r(e�(k))i(e�0(k 0))j

Z0

1d�ei!�

��hb�(k)b�0

y (k0)i�Ee¡i!ktei!k 0(t¡�) + hb�y(k)b�0(k0)i�Ee+i!kte¡i!k 0(t¡�) ¡

hb�(k)b�0(k 0)i�Ee¡i!kte¡i!k0(t¡�)¡hb�y(k)b�0

y (k 0)i�Ee+i!kte+i!k 0(t¡�):

We now assume the electromagnetic �eld to be in a thermal state at inverse temperature�, identifying �P1 with the vacuum state. We can thus write

�E = e¡�HE

TrEe¡�HE

=Y�;k

(1¡ e¡�~!k)e¡�~!kb�y(k)b�(k);

and exploit hE(t)i�E=0 as well as

hb�(k)b�0y (k 0)i�E = ��;�0�k;k0(1+N�(!k))

hb�y(k)b�0(k 0)i�E = ��;�0�k;k0N�(!k)

hb�(k)b�0(k 0)i�E = 0hb�y(k)b�0

y (k 0)i�E = 0;


where N�(!k) is the Planck distribution

N�(!k) = 1e�~!k¡ 1

:

The expression of the matrix ¡ij(!) thus simpli�es to

¡ij(!) = 1~2Xk

~!k2"0V

X�=1;2

(e�(k))i(e�(k))j

�(1 + N�(!k))

Z0

1d�e¡i(!k¡!)� +

N�(!k)Z0

1d�e+i(!k+!)�

�:

We now consider the continuum limit, so as to consider the electromagnetic �eld in freespace rather than in a �nite volume. Recalling the relevant dispersion relation and usingthe formula

1V

Xk

!Z

d3k(2�)3

= 1(2�)3c3

Z0

1d!k!k

2

Zd;

further noting that the sum over polarizations only involves the polarization vectors,recalling the previous identity for this sum and using the following result for the integralover the solid angle Z

d��ij ¡

kikjjkj2

�= 8

3��ij

we come to

¡ij(!) = �ij

Z0

1d!k !k

31~12"0

83�

1(2�)3c3

�(1 + N�(!k))

Z0

1d�e¡i(!k¡!)� +

N�(!k)Z0

1d�e+i(!k+!)�

�:

We now evaluate the integral over � exploiting the so called Sokhotski Plemelj formula

lim"!0

Z¡1

1dx

f(x)x¡ y� i" = PV

Z¡1

1dx

f(x)x¡ y � i�f(y);

valid for a function f(z) holomorphic and bounded in a strip 0< Im z < �. We thus have

lim"!0

Z0

1d!f(!)

Z0

1d�e¡i(!¡¡i")� = ¡i lim

"!0

Z0

1d!

f(!)!¡¡ i"

= �f()¡ iPVZ0

1d!

f(!)!¡

and therefore

¡ij(!) = �ij

Z0

1d!k !k

31~12"0

83�

1(2�)3c3

f(1 + N�(!k))��(!k ¡ !) + N�(!k)��(!k +

!)g

+i�ijPVZ0

1d!k!k

31~12"0

83�

1(2�)3c3

�1+N�(!k)!¡!k

+N�(!k)!+!k

�= �ij

14�"0

124!3

3~c3(1 + N�(!)) + i�ij1

4�"02

3�~c3PVZ0

1d!k !k

3

�1+N�(!k)!¡!k

+

N�(!k)!+!k

�:

�


According to the previous notation we rewrite this expression as

¡ij(!) = �ij

�12 (!)+ i S(!)

�identifying the contributions

(!) = 14�"0

4!3

3~c3(1+N�(!))

and

S(!) = 14�"0

23�~c3PV

Z0

1d!k!k

3

�1+N�(!k)!¡!k

+N�(!k)!+!k

�:

�In the latter expression one distinguishes a Lamb shift contribution independent on

the photon number, and a Stark shift contribution proportional to N�(!k). Furthernoting that

N�(¡!k) = ¡(1+N�(!k))

�and inserting all the obtained contributions in the expression for the weak coupling

master equation we �nally haveddt�S(t) = ¡ i~

�X!

~S(!)Dy(!) �D(!); �S(t)�+D[�S(t)]

with

D[�S(t)] =X!>0

14�"0

4!3

3~c3(1+N�(!))�D(!)�S(t)Dy(!)¡ 1

2fDy(!) �D(!); �S(t)g

�+X!>0

14�"0

4!3

3~c3N�(!)�Dy(!)�S(t)D(!)¡ 1

2fD(!) �Dy(!); �S(t)g

�;

where a sum over Cartesian indexes is either implicit or expressed through the dot scalarproduct. The upper and lower terms at the r.h.s. are connected to the transfer of aquantum from the system to the bath or from the bath to the system respectively. Tosee this note that if the vector 'n is an eigenvector of the system Hamiltonian HS corre-sponding to the eigenvalue En

HS'n = En'n;

then the vector Ai(!)'n still is an eigenvector corresponding to the lower eigenvalueEn¡!, as follows from the commutation relations between HS and Ai(!)

HSAi(!)'n = Ai(!)(HS¡!1)'n= (En¡!)Ai(!)'n:

Thus the action of the Lindblad operators appearing in the upper line at the r.h.s.remove an excitation from the system. Note that the rate corresponding to this eventapart from an overall factor is proportional to (1 + N�(!)), so that this contribution isnon vanishing even if the electromagnetic �eld is in the vacuum. The term in the lowerline at the r.h.s. on the contrary describes the inverse process and vanishes in the limitof the vacuum �eld, which cannot transfer excitations to the system. Note that given atypical matrix element for the dipole operator of the system the reference rate appearingin the master equation reads as

0 = 14�"0

4!3

3~c3 jdj2:


To assess the validity of the weak coupling master equation this rate 0t 1/�R has to becompared with a typical frequency of the electromagnetic �eld !0t 1/�E.Bloch equations

We now consider as speci�c example of matter interacting with the electromagnetic�eld a two-level atom stimulated by a laser described as a classical �eld. This system isoften called driven damped two-level system, and the equations describing it are knownas optical Bloch equations. They are of the same form as the equations introduced byBloch for the dynamics of a spin coupled to a driving magnetic �eld in the presence ofan environment.

For the case at hand the eigenoperators of the free Hamiltonian

H0 = 12!0�z

where !0 is the frequency of the two-level system, are easily seen to correspond to the�+ and �¡ operators, so that we have

[HS ; ��] = �!0��:

Assuming no permanent dipole moment, so that h�jV j �i = 0, where +; ¡ denote thetwo level of the atoms singled out by the relevant transition, and introducing the dipolemoment

d = h¡je r̂ j+i;

where r̂ now denotes the position operator for the centre of mass of the two-level systemwe can write

V = ¡�¡d �E+h:c:

The system operators appearing in the master equation are now therefore given by

D(!0) = d�¡

Dy(!0) = d��+:

We thus end up for the dissipative part with

D[�S(t)] = 0(1+N�(!0))��¡�S(t)�+¡

12f�+�¡; �S(t)g

�+ 0N�(!0)

��+�S(t)�¡¡

12f�¡�+; �S(t)g

�:

We now couple the atom to a classical monochromatic electromagnetic �eld in the elec-tric dipole and rotating wave approximation. The driving electric �eld can be written as

E = i "!

2"0V

r[Ae¡i!t¡A�e+i!t]

with A the classical amplitude and " the polarization. With no permanent dipolemoment in the rotating wave approximation one has

V =

id �" !

2"0V

rA�

!�e¡i!t�++

id � " !

2"0V

rA�

!||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }

¡

2

ei!t�¡

and the quantity , called Rabi frequency, can be taken real for a suitable choice ofphase in the de�nition of the states j�i. It is essentially the matrix element of ¡r̂ � E.Including the classical driving �eld the Hamiltonian term is of the form

H(t) = 12!0�z¡

2(e¡i!t�++ei!t�¡)


�where is called Rabi frequency. Leaving out the index S of the statistical operator

for the system we thus have the master equation

ddt� = ¡i[H(t); �]+D�

with

D� = 0(1+N�(!0))||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} } #

��¡��+¡

12f�+�¡; �g

�

+ 0N�(!0)|||||||||||||||||||||||||||||||||||||||||||||||||| |{z}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} } "

��+��¡¡

12f�¡�+; �g

�where 0 is called spontaneous emission rate, N�(!0) as de�ned above is the meannumber of photons in the electromagnetic �eld at inverse temperature � at the resonantfrequency !0.

The �rst term describes induced and spontaneous emission, the second inducedabsorption. In this case we have only two Lindblad operators

1L1 = 0(1+N�(!0))p

�¡

2L2 = 0N�(!0)p

�+:

This master equation can describe dynamics of populations and coherences, spectrum ofemitted radiation, statistics of photon detection. The atom is driven by the laser andcan decay in the electromagnetic �eld.

The time dependence in the Hamiltonian can be removed by a unitary transforma-tion of the form

�I(t) = ei!

2�zt�e¡i

!

2�zt

= U(!t)�U(!t)y;

where we have put

U(!t) = e+i!

2�zt;

which provides a representation of the group of the rotation around the z axis. If != !0that is to say the system is on resonance, this would correspond to a I-picture withrespect to the free Hamiltonian. In general the quantity � = !0 ¡ ! is called detuning.Here it has to be interpreted as moving to a time dependent rotating frame around the�z direction. Note that the Schrödinger picture is recovered via

� = e¡i!

2�zt�Ie

+i!

2�zt

=

(�I)11 (�I)10e¡i!t

(�I)01ei!t (�I)00

!:

One is left with

ddt�I = i

�12!�z; �I

�+ei

!

2�zt d

dt�e¡i

!

2�zt

= LI�I

The expression of LI can be obtained by explicit calculation, but also noting the covari-ance of the dissipative part D. Due to

U(!t)��U(!t)y = e�i!t��


one can check that the dissipative part D is covariant under U(!t), therefore

U(!t)D�U(!t)y = D�I:

The same trivially holds for the free evolution term. It only remains to evaluate

i2U(!t)[e¡i!t�++ei!t�¡; �]U(!t)y = i

2[�++�¡; �I]

= i2[�x; �I]:

We now want to study the equation

ddt�I = ¡i

�12��z¡

2�x; �I

�+D�I:

Recall that any statistical operator in C2 can be represented as

�(t) = 12(1+ h�(t)i ��)

= 12

1+ h�z(t)i h�x(t)i¡ ih�y(t)i

h�x(t)i+ ih�y(t)i 1¡h�z(t)i

!with

h�(t)i = Tr �(t)� ;

it is thus expressed through certain time dependent coe�cients on the basis 1

2p f1;�g.

We have thatddt�I = LI�I

becomes

h�_ (t)i �� = LI1+ h�(t)i � LI�:

Setting = 0(1+ 2N�(!0))= "+ # we obtain

LI1 = ¡ 0�zLI�x = ¡

2�x+��y

LI�y = ¡��x¡ 2�y¡�z

LI�z = �y¡ �z

and therefore exploiting the previous results on the representation of linear maps compo-nentwise

h�_x(t)i = ¡ 2h�x(t)i¡�h�y(t)i

h�_y(t)i = �h�x(t)i¡ 2h�y(t)i+h�z(t)i

h�_z(t)i = ¡h�y(t)i¡ h�z(t)i¡ 0

= ¡h�y(t)i¡ ( "+ #)�h�+�¡(t)i¡ h�¡�+(t)i¡

"¡ # "+ #

�:

�It is not a trivial fact that we have closed equations of motion for the mean values or

�rst moment. This is due to the fact that we are dealing with the generators of thesu(2) Lie algebra.


To better follow the behavior of populations and coherences we can write the equa-tions in the form

ddt12(1+ h�z(t)i) = +i

2(h�+(t)i¡ h�¡(t)i)¡

2h�z(t)i¡

02

ddth�+(t)i =

�+i�¡

2

�h�+(t)i+ i

2h�z(t)i

or equivalently in the notation

�(0) =��11(0) �10(0)�01(0) �00(0)

�one has the equations

ddt(�I)11(t) = ¡i[(�I)10(t)¡ (�I)01(t)]¡ [(�I)11(t)¡ (�I)00(t)]¡

02

ddt(�I)10(t) =

�+i�¡

2

�(�I)10(t)+ i

2[(�I)11(t)¡ (�I)00(t)]:

These equations are known as optical Bloch equations. Here we note that coherences(that is to say o�-diagonal matrix elements of the statistical operator) and populations(that is to say diagonal matrix elements of the statistical operator) are coupled onlythrough the driving �eld. For the dissipative part we recover the decoupling betweenpopulations and coherences discussed when deriving the general form of the masterequation.�

In the absence of environment, i.e. = 0 and on resonance, i.e. � = 0 we have theRabi oscillations described by

h�x(t)i = h�x(0)ih�y(t)i = h�y(0)icos(t)¡h�z(0)isin(t)h�z(t)i = h�y(0)isin(t)+ h�z(0)icos(t)

caused by interaction with a laser with intensity proportional to 2. In this case purestates remain pure. For �> 0 one still has oscillations with higher frequency 2+�2

p.

�Otherwise stated the solution reads

h�(t)i =�1 00 R(t)

�h�(0)i

and formally describes a precession around the axis of the driving �eld.We notice that if

�I(t) = 12(1+ h�(t)i ��)

with h�(t)i solution of the equations given above, by the relation between �I and � wehave

�(t) = 12(1+ h� 0(t)i ��)

with

h� 0(t)i =�R(!t) 00 1

�h�(t)i

= (h�x(0)icos(!t)¡h�y(0)isin(!t); h�x(0)isin(t)+ h�y(0)icos(t); h�z(0)i):

To handle the equations for h�_ (t)i it is convenient to write them in matrix form as fol-lows

h�_ (t)i = b¡Ah�(t)i


�with

A =

0BBB@ 2

� 0

¡� 2¡

0

1CCCAb =

0@ 00¡ 0

1Aand detA> 0 for > 0. The matrix A is normal, i.e. [A; Ay] = 0, since A Ay = AyA, sothat it admits a spectral decomposition.�

The solution reads

h�(t)i = e¡Ath�(0)i+Z0

t

ds e¡A(t¡s)b:

For = 0 the non zero eigenvalues of A are purely imaginary and given by �i 2+�2p

,corresponding to a unitary dynamics. For detA=/ 0 the solution can be written

h�(t)i = e¡Ath�(0)i+ 1¡ e¡AtA

b:

The stationary state corresponds to h�ieq

h�ieq = A¡1b

= ¡ 0

1( 2+22+4�2)

0B@ ¡4�2

2+4�2

1CA:From now on we assume �=0, i.e. we are on resonance. We thus have in particular

h�zieq = ¡ 0

2

2+22

h�+ieq = ¡i 2+22

:

In the absence of driving the stationary state has only one non zero component

h�zieq = ¡ 0

= ¡ 11+2N�(!0)

= ¡Tanh��!02

��

so that the stationary population in the upper level is given by

�11eq = 1

2(1+ h�zieq)

=N�(!0)

1+ 2N�(!0)

= e¡�!02

2Ch��!02

�:


The stationary state is therefore of the canonical form, as expected thanks to thedetailed balance condition obeyed by the coe�cients in front of the di�erent channels inthe master equation

" = #e¡�!0:

We thus have

�eq = e¡�H0

Tr e¡�H0

= e¡�!0�z2

Tr e¡�!0�z2

= 1

2Ch��!02

�0@ e¡�

!02 0

0 e+�!02

1A:In the considered case of zero detuning the eigenvalues of A become

�i =� 2;34 � i�

�with

� = 2¡� 4

�2

r;

while for �=/ 0 one should solve an algebraic equation of the third order. Note that thereal part of the eigenvalues is strictly positive. As we shall see the dynamics is relaxingin the sense that the stationary state is reached with elapsing time for any initial condi-tion. To see this consider the quantity h�(t)i ¡ h�ieq which obeys the homogeneousequation

ddt(h�(t)i¡ h�ieq) = ¡A(h�(t)i¡ h�ieq)

predicting convergence to the stationary state for t!1 due to Re�i> 0. One has there-fore h�ieq = h�(1)i. This is just a special case of the general result stated previously,according to which the dynamics is relaxing if the commutant of Hamiltonian and Lind-blad operators reduces to multiple of the identities, indeed in this case one has f�z; �+;�¡g0= c1.

Before considering the solution of the equations for the case of zero detuning, let uspoint to the connection between the optical Bloch equation and the standard Blochequations introduced by Bloch in 1946 to describe the behavior of nuclear spins. Theequations for h�(t)i can be recast in the following form

h�_x;y(t)i = ¡[h�(t)i�H ]x;y¡h�x;y(t)i

T2

h�_z(t)i = ¡[h�(t)i�H ]z¡h�z(t)iT1

¡ 0

= ¡[h�(t)i�H ]z¡h�z(t)i¡ h�z(1)i

T1

upon the identi�cation 1/T1 = , and 1/T2 = /2, together with H = (; 0; ¡�). Theseequations are known as (magnetic) Bloch equations. In the nuclear magnetic resonance(NMR) terminology T1 is called longitudinal relaxation time (spin-lattice relaxationtime), while T2 is the transverse relaxation time (spin-spin relaxation time). Here T2 =2T1, while in general complete positivity requires 2T1 > T2, which is the typical experi-mental case.


�Recalling that for a spin 1/2 the magnetic moment is proportional to � one can

write

hM(t)i = ¡M(t)�H ¡R[M(t)¡M(1)]

where the relaxation matrix R reads

R =

0BBBBBB@1T2

0 0

0 1T2

0

0 0 1T1

1CCCCCCA:

Let us now consider the general solution of the equations for �= 0. To this aim and forlater reference it is useful to change basis and consider instead of h�(t)i ¡ h�ieq= (�x¡h�xieq; �x ¡ h�xieq; �z ¡ h�zieq) the vector with components � = (�+ ¡ h�+ieq; �¡ ¡h�¡ieq; �z ¡ h�zieq). In the corresponding equations the matrix A has to be replaced byanother matrix G of the form

G =

0BBB@+i�¡

20 i

2

0 ¡i�¡ 2¡i

2i ¡i ¡

1CCCA;so that

h�_ (t)i = ¡Gh�(t)i:

The solution of our system of equations, recalling that we take zero detuning � = 0,reads

h�z(t)i = e¡3

4 t�cos (�t)¡

4�sin(�t)

�h�z(0)i

+i�e¡

3

4 tsin(�t)(h�+(0)i¡ h�¡(0)i)

and

h�+(t)i = 12e¡

1

2 t(h�+(0)i+ h�¡(0)i)

+12e¡

3

4 t�cos (�t)+

4�sin(�t)

�(h�+(0)i¡ h�¡(0)i)

+i 2�

e¡3

4 tsin(�t)h�z(0)i;

together with h�¡(t)i= h�+(t)i�We now consider the solution of these equations for the case of the atom initially in

the ground state, so that �(0) = j¡ih¡j, and for a bath at T = 0 i.e. � P1, so that = 0. The initial conditions for the homogeneous equations become

h�+(0)i = h�+(0)i¡ h�+ieq= ¡h�+ieq= i

0 02+22


and

h�z(0)i = h�z(0)i¡ h�zieq= ¡1¡h�zieq

= ¡1+ 02

02+22

= ¡2 2

02+22

:

For the population in the upper level, if it is initially not occupied one then has

Pexcited(t) = 12(1+ h�z(t)i)

= 12(1+ h�zieq+ h�z(t)i)

= 2

02+22

+12e¡

3

4 0t��

cos (�t)¡ 04�

sin(�t)�¡22 02+22

+ i�sin(�t) 2i 0

02+22

�= 2

02+22

�1¡ e¡

3

4 0t�cos (�t)+ 3

4 0�sin(�t)

��For >

04

one has exponentially damped oscillations, for < 04

the system is over-damped. The stationary value of the upper population is

Peq = 2

02+22

;

that is approximately 1/2 for the case of strong driving. In a similar way one obtains forthe coherences

h�+(t)i = h�+ieq+ h�+(t)i

= ¡i 0 02+22

�1¡ e¡

3

4 0t�cos (�t)+

� 04�¡ 2

0�

�sin(�t)

��:

In the case of strong driving, i.e. � 04, one has

Pexcited(t) '12

h1¡ e¡

3

4 0tcos(t)

ih�+(t)i ' ¡ i

2e¡

3

4 0tsin(t)

so that for long times Pexcited(t) ' 1/2. As an application of these equations we studythe �uorescence spectrum of the emitted radiation in the stationary state. According tothe Wiener-Khintchine theorem the spectrum of the emitted radiation is proportional tothe Fourier transform of the autocorrelation function of the �eld generated by the atom.The latter can be shown to be proportional to the �¡ operator, so that we essentiallyneed to know the correlation function h�+(�)�¡(0)ieq. The expression of multi-time cor-relation functions of this form is generally not warranted by knowledge of the timeevolved statistical operator for the system �S(t), nor by knowledge of the time evolutionmap �(t). Under certain conditions they can however be shown to obey the same evolu-tion equations as the mean values. This results goes under the name quantum regressiontheorem. Let us start providing a statement of the Wiener-Khintchine theorem.

Wiener-Khintchine theorem


Let us consider a stochastic process X(t), taking values in C. We say that such aprocess is wide-sense or second-order stationary if the one and two dimensional distribu-tions only depend on the time di�erence, so that, denoting by E the expectation value,we have

E[X(t)] = E[X(0)] 8t2R

E[X(t+ � )X�(t)] = E[X(� )X�(0)] 8t; � 2R

� GX(�);

where we de�ne GX(�) as autocorrelation function of the process. More generally astrictly stationary stochastic process is such that all �nite dimensional distributions onlydepend on the di�erence of the time arguments. Suppose that GX(�) is absolutely inte-grable, so that it admits Fourier transform, which we suppose to be also absolutely inte-grable, so that the inverse Fourier transform is well de�ned. We consider the truncatedFourier transform of the process, which we de�ne as

X̂T(!) =Z¡T

2

+T

2

dt e¡i!tX(t);

so that we do not necessarily ask X(t) to admit a Fourier transform. We further de�neas truncated power spectral density ST(!) the expectation value of the random variable1

T

��X̂T(!)��2

ST(!) = 1TE��X̂T(!)

��2�:�

The Wiener-Khintchine theorem states that the limit for large T of this quantity

SX(!) = limT!1

ST(!);

which we call the spectral density of the process, exists and is given by the Fouriertransform of the autocorrelation function of the process

SX(!) =Z¡1

+1dt e¡i!tGX(t):

The power spectral density of the process is therefore given by the Fourier transform ofthe autocorrelation function, and viceversa the autocorrelation function can be obtainedas the inverse Fourier transform of the power spectrum

GX(t) =Z¡1

+1d! e+i!tSX(!):

The functions GX(t) and SX(!) therefore constitute a Fourier couple. The result can bedescribed stating that the autocorrelation function of a wide-sense stationary randomprocess has a spectral decomposition given by the power spectrum of the process. It isimportant to stress that for such a stationary stochastic process the autocorrelationfunction is positive de�nite, in fact

Xi; j=1

n

viGX(ti¡ tj)vj� = E

24 Xi=1

n

viX(ti)

!0@Xj=1

n

vjX(tj)

1A�35> 0;


for arbitrary t1; :::; tn, fvig � C and 8n 2 N. Thanks to Bochner's theorem this meansthat indeed its Fourier transform is a positive function, and therefore a well-de�nedspectral density. Moreover note that the value at � = 0 of the autocorrelation function,which is by construction positive, provides the integrated spectral density

GX(0) =Z¡1

+1d!SX(!):

It remains to be proven the existence of the limit. We have

1TE��X̂T(!)

��2� = 1TE

"Z¡T

2

+T

2

dtZ¡T

2

+T

2

d� e¡i!(t¡�)X(t)X�(�)

#

= 1T

Z¡T

2

+T

2dtZ¡T

2

+T

2d� e¡i!(t¡�)E[X(t)X�(�)]

= 1T

Z¡T

2

+T

2dtZ¡T

2

+T

2d� e¡i!(t¡�)GX(t¡ � );

where we have used the stationarity property of the process. We exploit the resultZ¡T

2

+T

2

dtZ¡T

2

+T

2

d� f(t¡ � ) =Z¡T

T

d� (T ¡ j� j)f(� );

to obtain the expression

1TE��X̂T(!)

��2� =Z¡T

T

d��1¡ j� j

T

�GX(� )e¡i!�

=Z¡1

+1d�GX;T (� )e¡i!� ;

where we have de�ned the function

GX;T (�) =

8<:�1¡ j� j

T

�GX(�) j� j<T

0 j� j>T:

We now use Lebesgue dominated convergence theorem with the identi�cation

fn ! GX;T (�)e¡i!� ;

converging to the function

f ! GX(�)e¡i!� ;

and the sequence is dominated by the function

g ! jGX(�)j;

which is by hypothesis absolutely integrable. Since jGX;T (�)e¡i!� j 6 jGX(�)j, we canexchange limit and integral, thus indeed obtaining

SX(!) = limT!1

1TE��X̂T(!)

��2�= lim

T!1

Z¡1

+1d�GX;T (�)e¡i!�

=Z¡1

+1d� lim

T!1GX;T (� )e¡i!�

=Z¡1

+1d�GX(� )e¡i!�:


�

Multi-time correlation functionsSuppose we have a quantum reduced dynamics described by a collection of com-

pletely positive maps of the form

�(t; s) = T~ exp�Z

s

t

d�L(� )�;

where L(� ) is a generator in Lindblad form, so that they satisfy the equation

ddt�(t; s) = L(t)�(t; s)

with initial condition �(t; t)=1, and obey the composition law

�(t; �) ��(� ; s) = �(t; s) 8t> � > s:

We now consider two generic system operators AS ; BS 2 B(HS), and consider the two-time correlation function

hAS(t+ �)BS(t)i � TrSE eiH(t+�)ASe¡iH(t+�)eiHtBSe¡iHt�SE(0);

which we de�ne as the expectation value with respect to the initial state of the overallsystem of the product of the two system operators, in the Heisenberg picture withrespect to the full Hamiltonian. Exploiting the ciclic property of the trace and the factthat the operators act in the system Hilbert space only we have

hAS(t+ � )BS(t)i = TrSE eiH (t+�)ASe¡iH(t+�)eiHtBSe¡iHt�SE(0)= TrSASTrEfe¡iH� [BS (e¡iHt�SE(0)e+iHt)]e+iH� g= TrSASTrEfe¡iH� [BS �SE(t)]e+iH� g= TrSASTrEXt(� ):

Here the operator Xt(� ) 2 T (HS HE), if the initial condition is factorized, thus war-ranting the existence of a dynamical map for the reduced system, reads

Xt(�) = e¡iH� [BS(e¡iHt�S(0) �Ee+iHt)]e+iH� :

It therefore obeys the Liouville-von Neumann equation

dd�Xt(� ) = ¡i[H;Xt(�)];

with initial condition Xt(0) = BS(e¡iHt�S(0) �Ee+iHt). Under the same physicalapproximations under which the reduced system state �S(t) = TrE�SE(t) obeys a timedependent Lindblad dynamics as given by the map �(t; 0), we can assume that thedynamics of the operator TrEXt(�) in its � dependence is given by the same map. Thisimplies in particular that we approximate �SE(t)' �S(t) �E, in keeping with the Born-Markov approximation. We thus take

TrEXt(�) = �(t+ � ; t)fTrEXt(0)g= �(t+ � ; t)fBSTrE e¡iHt�S(0) �Ee+iHtg= �(t+ � ; t)fBS�(t; 0)f�S(0)gg= �(t+ � ; t)fBS�S(t)g;


where as stressed by the curly brackets any superoperator acts on all operators to itsright.�

We thus obtain the important relation

hAS(t+ � )BS(t)i = TrSAS�(t+ � ; t)fBS�(t; 0)f�S(0)gg;

which has the very important feature that it provides the expression of a multi-time cor-relation function in terms of the dynamical map �(t; s) which provides the time evolu-tion of the mean values.�

On a similar footing one can consider the correlation function hBS(t)AS(t + �)i andobtain the result

hBS(t)AS(t+ �)i = TrSAS�(t+ � ; t)f�(t; 0)f�S(0)gBSg;

where the curly brackets delimit the action of the superoperators. We have indeed

hBS(t)AS(t+ �)i = TrSE eiHtBSe¡iHteiH (t+�)ASe¡iH(t+�)�SE(0)= TrSASTrEfe¡iH� [(e¡iHt�SE(0)e+iHt)BS]e+iH� g;

and considering similarly as before the quantity

Y (� ; t) = e¡iH� [(e¡iHt�S(0) �Ee+iHt)BS]e+iH� ;

in the same approximations

hBS(t)AS(t+ �)i = TrSASTrEY (� ; t)= TrSAS�(t+ � ; t)fTrEY (0; t)g= TrSAS�(t+ � ; t)f�(t; 0)f�S(0)gBSg= TrSAS�(t+ � ; t)f�S(t)BSg:

This procedure can be extended to consider arbitrary multi-time correlation functions ofthe form

hA1(s1):::Am(sm)Bn(tn):::B1(t1)i � TrSEfBn(tn):::B1(t1)�SEA1(s1):::Am(sm)g;

with time ordering tn> ::: > t1> 0, sm> ::: > s1> 0, where all operators at the r.h.s. aresystem operators considered in Heisenberg picture with respect to the full Hamiltonian.For a suitable choice of times the expression can be used to describe the repeated actionof a completely positive map, and therefore an instrument. As already stressed the cru-cial fact here is that one can express multi-time correlation functions by means of thesame operator which provides the mean values.

In some cases these relationships can be used to obtain the expression of multi-timecorrelation functions as follows. Suppose that for the considered system a set of systemoperators fAig�B(H) can be considered such that their mean values obey closed homo-geneous evolution equations, as it was the case for the previously considered triple �, sothat

ddthAj(t)i =

Xk

GjkhAk(t)i;

= TrS�X

k

GjkAk

��S(t);


with a suitable matrix of coe�cients Gjk. We then have for any initial system stateddthAj(t)i = d

dtTrSAj�S(t)

= ddt

TrSAj�(t)f�S(0)g= TrSAjL�(t)f�S(0)g= TrS (L0fAjg)�(t)f�S(0)g= TrS (L0fAjg)�S(t);

so that we obtain the operator relation

L0fAjg =Xk

GjkAk:

If we combine this result with the previous regression relation for the multi-time correla-tion function we have for its time dependence, considering a time homogeneous evolutionequation

dd�hAj(t+ �)C(t)i = d

d�TrSAj�(t+ � ; t)fC�(t; 0)f�S(0)gg

= TrSAjL�(t+ � ; t)fC�(t; 0)f�S(0)gg= TrSL0fAjg�(t+ � ; t)fC�(t; 0)f�S(0)gg= TrS

Xk

GjkAk�(t+ � ; t)fC�(t; 0)f�S(0)gg

=Xk

GjkhAk(t+ �)C(t)i:

As a result the higher order correlation functions have the same time dependence, sothat they can be obtained once the time evolution for the mean values is known. Thispowerful result is known as quantum regression theorem.�Resonance �uorescence

Having a proper handle on two-time correlation functions for our model of a two-level atom interacting in dipole approximation with the electromagnetic �eld, we canconsider relevant physical expression such as the �uorescence spectrum and statistics ofthe emitted photons, which depend on multi-time correlation function of second andfourth order respectively. We will focus on the spectrum of the emitted radiation in thestationary state �eq introduced before. The expression of the positive frequency compo-nent of the retarded electromagnetic �eld radiated by the atomic dipole is given by

E(+)(t;x) = !02

c2r[(n�d)�n]�¡

�t¡ r

c

�;

where x=n r so that r= jxj.�

The spectral density radiated per unit solid angle by the dipole is then given by theexpression

dId

(!) = c r2

4�

Z¡1

+1 d�2�

ei!� hE(¡)(t;x)E(+)(t+ � ;x)ieq

= I0(x)S(!);

where we have introduced the quantities

I0(x) = !04

4�2c3j(n�d)�nj2

S(!) = 12�

Z¡1

+1d� e¡i!� h�+(�)�¡(0)ieq:


To evaluate the stationary correlation function h�+(�)�¡(0)ieq we use the quantumregression theorem applied to the present model. The expression of the correlation func-tion in our case is explicitly given by

h�+(� )�¡(0)ieq = TrSf�+e�Lf�¡�eqgg;

where L is the superoperator in Schrödinger picture. To exploit the quantum regressiontheorem we consider the previously introduced operators

� = (�+¡h�+ieq; �¡¡h�¡ieq; �z¡h�zieq)= �¡h�ieq

which obey the homogeneous equations

h�_ (t)i = ¡Gh�(t)i:

The quantum regression theorem then tell us that multi-time correlation functions of theform h�(t+ � )C(t)i obey the same equation of motions as the mean values. In our case,considering the stationary state for the system we have

h�(� )C(0)ieq = h(�(�)¡h�ieq)C(0)ieq= h�(�)C(0)ieq¡h�ieqhC(0)ieq� (�(�); C(0)):

The correlation function de�ned in the last line obeys

dd�(�(�); C(0)) = ¡G(�(�); C(0)):

To obtain the solution of this equation we simply have to consider the previouslyobtained results for h�z(t)i and h�+(t)i and consider the corresponding initial condi-tions (�(0); C(0)). Since we are interested in the quantity h�+(�)�¡(0)ieq, we takeC(0) ! �¡ and therefore need the initial conditions (�(0); �¡(0)). We are consideringthe stationarity state in resonance and at zero temperature, so that

h�ieq = ¡ 1 02+22

0B@ 02 0 02

1CAand therefore in particular

h�zieq = ¡ 02

02+22

h�+ieq = ¡i 0 02+22

:

We have therefore

(�+(0); �¡(0)) = h�+(0)�¡(0)ieq¡h�+ieqh�¡ieq= 1

2h�z+1ieq¡ jh�+ieqj2

= 2

02+22

¡ 022

( 02+22)2

= 24

( 02+22)2;


as well as

(�¡(0); �¡(0)) = h�¡(0)�¡(0)ieq¡h�¡ieqh�¡ieq= ¡h�¡ieq2

= 022

( 02+22)2

and

(�z(0); �¡(0)) = h�z(0)�¡(0)ieq¡h�zieqh�¡ieq= ¡h�¡ieq¡h�zieqh�¡ieq= ¡h�¡ieqh�z+1ieq

= ¡i 2 03

( 02+22)2:

As a last step we use the results for h�z(t)i and h�+(t)i, multiplying them by a factorei!0t, which arises when moving �+ to the Schrödinger picture, while the equations havebeen solved on resonance corresponding to the interaction picture

e¡i!0� h�+(�)�¡(0)ieq = h�+ieqh�¡ieq+(�+(�); �¡(0))= jh�+ieqj2+(�+(�); �¡(0))

= 022

( 02+22)2+ 12e¡

1

2 � [(�+(0); �¡(0))+ (�¡(0); �¡(0))]

+12e¡

3

4 ��cos (��) +

4�sin(��)

�[(�+(0); �¡(0)) ¡ (�¡(0);

�¡(0))]

+i 2�e¡

3

4 �sin(��)(�z(0); �¡(0)):

�After evaluating this quantity one has to take its Fourier transform to come to S(!).

We do not give the general expression, which is quite cumbersome, but consider two lim-iting situations. If the driving is weak, i.e. � 0

4, the power spectrum of the emitted

�eld reads

S(!) '�

�2

�(!¡!0);

as predicted by elastic Rayleigh scattering. In the limit of strong driving � 04we have

instead

S(!) ' 12

02¡ 0

2

�2+(!¡!0)2

+14

3

4 0�

3

4 0

�2+(!¡!0+)2

+ 14

3

4 0�

3

4 0

�2+(!¡!0¡)2

;

so that the spectrum is given by the sum of three Lorentzians, describing three peakswith heights in the ratio 1:3:1 and integrated area below the peaks in the ratio 1:2:1.Besides the central peak at the resonant frequency, one has two side peaks at frequency!0 � . This spectrum is known as Mollow spectrum, a typical quantum feature whichhas been experimentally observed.


DecoherenceWe now provide a very sketchy description of the phenomenon of quantum decoher-

ence for a massive test particle interacting through collisions with a background gas.Now the relevant Hilbert space for the system is L2(R3), while the environment is a gasof identical quantum particles. We have in mind to describe in most simple terms thee�ect of the collision on the dynamics of the otherwise free particle. This situation is rel-evant if one considers an experiment in which interference patterns of massive particlesare observed, letting them propagate through a suitably devised Mach-Zender interfer-ometer. The interaction with the background gas, which can be neglected if the experi-ment is performed at very high vacuum, but is present otherwise, will bring with itself aloss of visibility in the interference fringes. Phenomena of this kind, in which as a conse-quence of interaction with a quantum environment typical quantum coherent e�ects like�nges visibility are suppressed, go under the name of decoherence. We derive in aheuristic phenomenological way the master equation which describes this e�ect. Theresult can be con�rmed on the basis of a microscopic description.

Given that for a dilute gas the collisions can be considered to be independent, a Mar-kovian description naturally applies. The Hamiltonian term can be considered to begiven by the free kinetic term, while we assume that the basic interaction mechanismwith the environment is given by collisions in which the particle exchanges momentumwith the gas. As a result the basic microscopic interaction event leads to a change of thesystem momentum from p to p+ q according to

hpj�jpi ! hp+ q j�jp+ qi= hp��e i~ q�x̂�e¡ i

~ q�x̂��pi;

where we have denoted with x̂ the position operators which are also the generators oftranslations in momentum space. The Lindblad operators describing the microscopicinteraction can therefore be identify with unitary operators describing a momentumtransfer. Since the exchanged momentum in the single collisions is actually a randomvariable, we integrate over all possible momentum transfers, weighted according to a cer-tain probability distribution P(q) to be determined on the basis of some phenomenolog-ical input. Based on these suggestions we therefore introduce the following Lindbladmaster equation

ddt� = ¡ i~

�p̂2

2m; �

�+�

ZdqP(q)

hei

~ q�x̂�e¡i

~ q�x̂¡ �i;

where m is the mass of the test, � is a constant with the dimension of inverse time,quantifying the rate of collisions, that is their number per unit interval of time, and theloss term, generally corresponding to an anticommutator, is simply proportional to theidentity, since the Lindblad operators are unitary. It is worth looking at this masterequation in the position representation. The matrix elements then read

ddthxj�jyi = i

~2m

(�x¡�y)hxj�jyi+�Z

dqP(q)hei

~ q�(x¡y)¡ 1ihxj�jyi:

Neglecting in the �rst instance the contribution due to free motion one notices that inthis basis the action of the dissipative part is simply multiplicative in this basis. Thisallows to simply write down the solution of the master equation in terms of the initialcondition

hxj�(t)jyi = e¡�[1¡�(x¡y)]thxj�(0)jyi;


where �P(x) is the Fourier transform of the probability distribution P(q), also knownas its characteristic function. To see the relevance of this equation for the description ofloss of interference, let us consider the case in which relevant momentum transfers aretypically small, so that that we can approximate

�P(x) ' 1+ i~(x¡ y) � hqiP ¡

12~2(x¡ y)

2hq2iP+ :::;

where we have introduced �rst and second momentum of the probability distribution oftransferred momenta. Assuming as natural isotropy, so that the �rst moment vanishes,we are left with

hxj�(t)jyi ' e¡ �

2~2(x¡y)2hq2iPthxj�(0)jyi;

so that while the diagonal matrix elements of the statistical operator in the position rep-resentation are left untouched, as follows from trace preservation, the o�-diagonal matrixelements in this basis, corresponding to spatial coherence allowing for typical quantumphenomena such as the appearance of interference fringes, are actually suppressed. Thesuppression is more e�ective for far-o� matrix elements, and exponential in time. Thisdynamical behavior, arising just due to interaction with the environment, describes theloss of visibility in a interferometric experiment with massive particles, that is anexample of quantitative description of decoherence. The e�ect of the free evolution onlyleads to a di�erent quantitative description of the same phenomenology.

��


Bibliography

[1] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press,Oxford, 2007)

[2] T. Heinosaari and M. Ziman, The Mathematical Language of Quantum Theory (Cambridge Univer-sity Press, Cambridge, 2011)

[3] A. S. Holevo, Probabilistic and statistical aspects of quantum theory (North-Holland, Amsterdam,1982)

[4] P. Busch, M. Grabowski, and P. Lahti, Operational quantum physics (Springer-Verlag, 1995)

[5] R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications, Vol. 286 of LectureNotes in Physics (Springer, Berlin, 1987)

Acknowledgments.We are grateful to our students Marco Rabbiosi and Carlo Sparaciari for careful

reading of the manuscript.

145

Epilogue

``One of the principal objects of theoretical research in any department ofknowledge is to find the point of view from which the subject appears inthe greatest simplicity.''

(Josiah Willard Gibbs)�

147

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Vorlesungsskript Version - Home INFN Milanovacchini/mqa/lecture_notes.pdf · 2018-09-28 · analysis of course relies on the premise that the statistical experiments are reproducible,