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Information in Individual
Quantum Systems
Dissertation zur Erlangung des akademisches Grades eines
Doktors der technischen Wissenschaften
unter der Leitung vono.Univ.Prof.Dr. Anton Zeilinger
E141
Atominstitut der osterreichischen Universitaten
eingereicht an der Technische Universitat Wien
Naturwissenschaftliche Fakultat
von
Mag. Caslav Brukner9108742
Pulverturmgasse 15/22, 1090 Wien
Wien, am 16. September 1999
Gefordert vom Fonds zur Forderung der wissenschaftlichen Forschung,
Projekt Nr. S6502 und F1506
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Contents
Introduction 5
From quantum theory to an information invariant ... 11
1 Information Acquired in a Quantum Experiment 11
1.1 Unbestimmtheit vs Unbekanntheit in a Quantum Experiment . 12
1.2 Conceptual Inadequacy of the Shannon Information in a Quan-
tum Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.1 An Operational Approach . . . . . . . . . . . . . . . . . . 17
1.2.2 An Axiomatic Approach . . . . . . . . . . . . . . . . . . . 22
1.2.3 A Physical Approach . . . . . . . . . . . . . . . . . . . . . 30
1.3 Measure of Information Acquired in a Quantum Experiment . . . 35
2 Information Content of a Quantum System 43
2.1 A Qubit Carries One Bit . . . . . . . . . . . . . . . . . . . . . . . 44
2.1.1 Complementary Propositions . . . . . . . . . . . . . . . . 44
2.1.2 Invariant Information in a Qubit . . . . . . . . . . . . . . 48
2.2 Two Qubits Carry Two Bits Entanglement . . . . . . . . . . . 53
2.2.1 Pairs of Complementary Propositions . . . . . . . . . . . 53
2.2.2 Invariant Information in Two Qubits . . . . . . . . . . . . 57
2.3 N Qubits Carry N Bits . . . . . . . . . . . . . . . . . . . . . . . . 62
A.1 Information Content of a Classical System . . . . . . . . . . . . . 65
i
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... and back. 71
3 Information and the Structure of Quantum Theory 71
3.1 The Principle of Quantization of Information . . . . . . . . . . . 71
3.2 The Number of Mutually Complementary Propositions . . . . . . 77
3.3 Maluss Law in Quantum Mechanics . . . . . . . . . . . . . . . . 81
3.4 The deBroglie Wavelength . . . . . . . . . . . . . . . . . . . . . . 89
3.5 Dynamics of Information . . . . . . . . . . . . . . . . . . . . . . . 93
3.6 Linearity and Arbitrarily Fast Communication . . . . . . . . . . 99
3.7 Change of Information in Measurement Reduction of the Wave
Packet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
B.1 Continuity of Information Implies Analyticity of Information . . 113
B.2 A General Transformation in the Space of Information . . . . . . 115
Conclusions 117
Preprint from Phys. Rev. Lett. 121
References 127
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Zusammenfassung
In jedem moglichen Quantenexperiment ist eine endliche Anzahl von unter-
schiedlichen Resultaten, z.B. die einzelnen Spinresultate: Spin hinauf und
Spin hinunter, moglich. Bevor das Experiment durchgefuhrt wird, kennt ein
Beobachter nur die spezifischen Wahrscheinlichkeiten aller moglichen einzelnen
Resultate. Wir definieren ein neues Informationma fur eine einzelne Mes-
sung. Dieses basiert auf der Tatsache, da in einer einzelnen Quantenmessung
die einzigen Eigenschaften des Systems, die vor der Durchfuhrung der Messung
definiert sind, die spezifischen Wahrscheinlichkeiten fur alle moglichen einzelnen
Resultate sind.
Nach der Kopenhagener Deutung der Quantenmechanik, die besonders von
Niels Bohr ausgearbeitet wurde, macht es keinen Sinn, von der Eigenschaft
eines Quantensystems unabhangig von dem Versuchsaufbau, in dem sich diese
Eigenschaft manifestiert, zu reden. Dem Beobachter steht es jedenfalls frei, un-
terschiedliche Versuchsanordnungen zu wahlen, die einander sogar vollstandig
ausschliessen konnen, z.B. die Messung der orthogonalen Komponenten des
Spins. Diese Quantenkomplementaritat von Variablen tritt auf, wenn die ent-sprechenden Operatoren nicht kommutieren. Eine Variable, z.B. eine Kompo-
nente des Spins, kann auf Kosten von maximaler Ungewiheit uber die an-
deren orthogonalen Komponenten prazise definiert werden. Wir definieren
den Gesamtinformationsgehalt eines Quantensystems als die Summe der In-
formationmae einzelner Variablen eines vollstandigen Satzes sich gegenseitig
vollstandig ausschlieender (komplementarer) Variablen.
Der Beobachter kann sich entscheiden, einen anderen Satz komplementarer
Variablen zu messen und gewinnt folglich Kenntnis uber eine oder mehrere
Variablen auf Kosten geringerer Kenntnis uber andere. Im Fall der Spinmes-
sungen konnten jene die Projektionen entlang gedrehter Richtungen sein, in de-
nen die Ungewiheit in einer Komponente verringert wird und in einer anderen
Komponente (oder mehreren Komponenten) entsprechend erhoht wird. Intuitiv
erwartet man, da die Gesamtungewiheit, oder gleichwertig die Gesamtinfor-
mation, die in dem System enthalten ist, unter einer solchen Transformation
1
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von einem vollstandigen Satz komplementarer Variablen zu einem anderen un-
verandert bleibt.
Wir zeigen, da die Gesamtinformation eines Systems, die unserem neuen
Ma entsprechend definiert ist, genau diese Invarianzeigenschaft hat. Wir
deuten das Bestehen dieser Eigenschaft der Gesamtinformation als Indiz, da
in der Quantenmechanik die Information der grundlegendste Begriff ist. Im er-sten Teil der vorliegenden Arbeit zeigen wir, gegrundet auf den Ergebnissen der
Quantentheorie, die Gultigkeit der Invarianzeigenschaft der Gesamtinformation
und schlagen Ideen fur das grundlegende Prinzip der Quantenmechanik vor.
Im zweiten Teil argumentieren wir fur ein neues Grundprinzip der Quan-
tenmechanik, das davon ausgeht, da das elementarste System durch ein Bit
an Information gekennzeichnet ist. Ebenso stellt ein zusammengesetztes Sys-
tem, das beispielweise aus zwei Elementarsystemen besteht, zwei Bits dar. Von
diesem Grundprinzip ausgehend, leiten wir dann einige wesentliche Elemente
der logischen Struktur der Quantentheorie ab. Die Gesamtinformation eines
System (bestehend aus einer endlichen Anzahl von Bits) manifestiert sich nurin bestimmten Messungen. Da ein Quantensystem nicht mehr Information tra-
gen kann als in den Bits enhalten ist, ist der Zufallscharakter der einzelnen
Resultate in den anderen (komplementaren) Messungen dann eine notwendige
Konsequenz. Diese Art des Zufallscharakters ist nicht reduzierbar, d.h. er kann
nicht auf verborgene Eigenschaften des Systems zuruckgefuhrt werden. An-
dernfalls wurde das Elementarsystem mehr Information als ein Bit tragen. Die
naturlichste Funktion zwischen der Wahrscheinlichkeit fur das Auftreten eines
spezifischen Resultates und der Laborparameter, die mit dem Grundprinzip,
da ein Elementarsystem nur ein Bit an Information tragt, vereinbar ist, mu
die sinusformige Abhangigkeit sein.
Verschrankung resultiert aus der Tatsache, da Information eines zusam-
mengesetzten Mehrteilchensystems auf gemeinsame Eigenschaften verteilt wer-
den kann. Fur ein Zweiteilchensystem beispielweise erhalten wir maximale Ver-
schrankung dann, wenn die zwei Bits, um gemeinsame Eigenschaften zu spez-
ifizieren, erschopft worden sind, und keine weitere Moglichkeit mehr existiert,
Information in den Einzelteilchen zu verschlusseln.
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Abstract
A new measure of information in quantum mechanics is proposed which takes
into account that for quantum systems, the only feature known before an ex-
periment is performed are the probabilities for various events to occur. The
sum of the individual measures of information for mutually complementary ob-
servations is invariant under the choice of the particular set of complementary
observations and conserved in time if there is no information exchange with an
environment. This operational quantum information invariant results in k bits
of information for a system consisting of k qubits. For a composite system,
maximal entanglement results if the total information carried by the system is
exhausted in specifying joint properties, with no individual qubit carrying any
information on its own.
Our results we interpret as implying that information is the most fundamen-
tal notion in quantum mechanics. Based on this observation we suggest ideas
for a foundational principle for quantum theory. It is proposed here that the
foundational principle for quantum theory may be identified through the as-
sumption that the most elementary system carries one bit of information only.Therefore an elementary system can only give a definite answer in one spe-
cific measurement. The irreducible randomness of individual outcomes in other
measurements and quantum complementarity are then necessary consequences.
The most natural function between probabilities for outcomes to occur and the
experimental parameters, consistent with the foundational principle proposed,
is the well-known sinusoidal dependence.
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Introduction
The ongoing debate about the interpretation of quantum mechanics, including
the meaning of specific phenomena like the measurement problem, indicate that
the foundations of quantum theory are not understood to the same degree as
those of classical mechanics or special relativity. While the basic concepts of
classical mechanics coincide well with our intuition, special relativity is out of
our immediate insight. Yet this theory is based on the principle of relativity,
which asserts that the laws of physics must be the same in all inertial systems
including constancy of the speed of light. However, even as the theory itself
is based on such simple and in part intuitively clear principles it nevertheless
predicts some surprising and even counter-intuitive consequences.
The foundational principles for special relativity imply an invariance of the
specific interval (eigenzeit) between two events with respect to all inertial frames
of reference. Data on pure time intervals obtained with respect to two relatively
moving inertial frames of reference will differ, and so will data on spatial dis-
tances. It is possible however, to form a single expression from time intervals
and space distances that will have the same value with respect to all inertialframes of reference. If the time interval between two distant events is denoted
by t and their space distance from each other by l, an expression involv-
ing a quantity symbolized by s can be derived in which (s)2 equals the
square of the time interval minus the fraction of distance squared over speed
of light squared, (s)2 = (t)2 (l)2/c2. This will have the same value as(t)2 (l)2/c2 with t and l having been obtained in another inertialframe of reference.
Quantum mechanics lacks such invariants and principles to this day. Pos-
sibly the lack of generally accepted invariants and foundational principles for
quantum mechanics is the main reason for the problem in understanding quan-
tum mechanics1 and thus, for the coexistence of philosophically quite different
1In his book [1967] Richard Feynman makes the following statement: There was a time
the newspaper said that only twelve men understood the theory of relativity. I do not believe
there ever was such a time. There might have been a time when only one man did, because
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6
interpretations of the theory. In fact, we have a number of coexisting inter-
pretations utilizing mutually contradictory concepts [Zeilinger, 1996]. A very
incomplete list of the many interpretations of quantum mechanics includes the
original Copenhagen Interpretation [Bohr, 1935], the ManyWorld Interpreta-
tion [Everett, 1957], the Statistical Interpretation [Ballentine, 1970], Bohms
interpretation [Bohm, 1952], the Transactional Interpretation [Cramer, 1986],
Consistent Histories Interpretation [Griffiths, 1984] and Mermins Ithaca inter-pretation [Mermin, 1998(a), 1998(b), 1998(c)].
In any quantum experiment with discrete variables a number of different
outcomes are possible, for example, the individual spin outcomes spin up
and spin down. Before the experiment is performed an experimentalist only
knows the specific probabilities for all possible individual outcomes. In chapter
1 we define a new measure of the experimentalists information for an individual
measurement based on the fact that the only features defined before the mea-
surement is performed are the specific probabilities for all possible individual
outcomes.
The observer is free to choose different experiments which might even com-
pletely exclude each other, for example measurements of orthogonal compo-
nents of spin. This quantum complementarity of variables occurs when the
corresponding operators do not commute. One quantity, for example the z-
component of spin, might be well defined at the expense of maximal uncertainty
about the other orthogonal components. In chapter 2 we define the total infor-
mation content in a quantum system to be the sum over all individual measures
for a complete set of mutually complementary experiments.
The experimentalist may decide to measure a different set of complementary
variables thus gaining certainty about one or more variables at the expense of
losing certainty about other(s). In the case of spin this could be the projections
along rotated directions, for example, where the uncertainty in one compo-
nent is reduced but the one in another component is increased correspondingly.
Intuitively one expects that the total uncertainty or, equivalently, the total in-
formation carried by the system is invariant under such transformation from
one complete set of complementary variables to another one. In chapter 2 we
show that the total information defined according to our new measure has ex-
actly that invariance property. We interpret the existence of that quantum
information invariant as implying that in quantum mechanics information is
the most fundamental notion. In the first part From Quantum Theory to an
he was the only guy who caught on, before he wrote his paper. But after people read the
paper a lot of people understood the theory of relativity in some way or other, certainly more
than twelve. On the other hand, I think I can safely say that nobody understands quantum
mechanics.
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7
Information Invariant ... of the thesis (chapter 1 and 2) we argue, based on the
known features of quantum physics, for the validity of the quantum information
invariant and we suggest ideas for a foundational principle for quantum theory.
In the second part ... and backof the thesis (chapter 3) we will turn the rea-
soning around and, based on the suggested foundational principle for quantum
mechanics, derive some essential features of the logical structure of quantumtheory. In a similar fashion as the foundational principles for special relativity
imply invariance of the specific measure of distance (eigenzeit) in space-time
with respect to all observers in inertial frames of reference, the suggested foun-
dational principle for quantum mechanics will imply invariance of a specific
operational information measure with respect to all possible observers choices
for a complete set of complementary experiments.
By a foundational principle we do not mean an axiomatic formalization of
the mathematical foundation of quantum mechanics, but a foundational concep-
tual principle which answers Wheelers [1983] question Why the Quantum?
This principle is then the reason for some essential features of quantum mechan-ics, like the irreducible randomness of an individual quantum event, quantum
complementarity, sinusoidal relation between probabilities and laboratory pa-
rameters, and entanglement. In this view we will discuss precisely the empirical
significance of the terms involved in formulating quantum theory, particularly
the notion of a quantum state, in a way which leads clearly to an understanding
of the theory. However we are aware of the possibility that this might not carry
the same degree of emotional appeal for everyone. The conceptual groundwork
for the ideas presented here has been prepared most notably by Bohr [1958],
von Weizsacker [1985] and Wheeler [1983].
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From quantum theory to an
information invariant ...
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Chapter 1
Information Acquired in a
Quantum Experiment
In a review article about the role of information in physics W. T. Grandy,
Jr. [1997] writes that ... an unambiguous clear-cut definition of information
remains slippery as that of randomness, say, or complexity. Is it merely a set of
data? Or is it itself physical? If the latter, as Einstein once commented upon
the ether, it has no definite spacetime coordinates. He continuous further in
the text: The difficulty is somewhat similar to that of attempting to explain
the origin and meaning of inertia to beginning students. While the term can
seem a bit obscure in its meaning, there is no ambiguity in defining inertial
mass as its measure, and the concept becomes scientifically useful. Similarly,
the general notion of information becomes a scientific one only if it is made
measurable. The question arises: How to measure information? In particularwe ask: How to measure information acquired in a quantum experiment?
Assume we want to find out the position of the moon in the sky on a full-
moon night. Before we look at the sky we are completely uncertain about the
position of the moon. When we look at the sky, we find out where the moon is
and it is certainly safe to assume that the property of the moon to be there is
independent of whether anyone looks or not. Our ignorance about the position
of the moon given before we look at the sky is the ignorance about a property
already existing in the outside world.
The situation in quantum measurement is drastically different. With the
only exception of a system being in an eigenstate of the measured observable,
an individual quantum event is intrisically random and therefore cannot be
assumed to just reveal a property of the system existing before the experiment
is performed. This we interpret in Sec. 1.1 as implying that the notion of
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12 Chapter 1: Information Acquired in a Quantum Experiment
our ignorance, or information, as to which specific experimental result will be
obtained in an individual run of the experiment plays a more fundamental role
in quantum measurement than in classical measurement.
Based on the fact that in an individual quantum measurement the only fea-
ture defined before the measurement is performed are probabilities for all pos-
sible individual outcomes to occur, we propose a new measure of informationfor an individual quantum measurement in Sec. 1.3. For clarity we emphasize
that our measure of information is not equivalent to Shannons information.
In fact, we show in Sec. 1.2 that because of the completely different root of a
quantum measurement as compared to that of a classical measurement, certain
conceptual difficulties arise when we try to define information gain in a quan-
tum measurement by the notion of Shannons information. While Shannons
information is applicable when a measurement reveals a pre-existing property,
our measure of information takes into account that, in general, a quantum
measurement does not reveal a pre-existing property.
1.1 Unbestimmtheit vs Unbekanntheit in a Quan-
tum Experiment
We begin with a brief survey of the usual textbook examples. Perhaps the
archetypical example is Einsteins recoiling-slit experiment [Bohr, 1949]. By
this example Einstein hoped to give a gedanken double-slit experiment which
would yield both which-path information and also show the wave-like interfer-
ence phenomenon. In a famous paper [1949], Bohr analyzed two arrangements
related to the recoiling-slit experiment. In the first arrangement, the diaphragm
placed in front of the diaphragm pierced with two slits can recoil (Fig. 1.1a)
and reveal through which slit of the second diaphragm the photon reached the
screen, in as much as only one of the momenta of a photon passing through one
or the other slit is consistent with a known amount of recoil momentum. In
the second arrangement in Fig. 1.1b, the diaphragm is fixed so that the path
can not be determinated. One finds that only in the latter arrangement an
interference pattern is exhibited. Bohr concluded ... we are presented with a
choice of either tracing the path of a particle or observing interference effects.
Another example along these lines is Feynmans [Feynman et al., 1965] ver-sion of Einsteins gedanken experiment. In this scheme the interfering electron
is observed by light-scattering. The scattering of a photon is used to detect
the electron position just behind the slits, revealing through which slit the elec-
tron reached the screen. Feynman explained that this observation procedure
destroys the interference pattern. He concluded his analysis with the following
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1.1 Unbestimmtheit vs Unbekanntheit in a Quantum Experiment 13
a) b)
Figure 1.1: Two mutually exclusive experimental arrangements to observe the in-terference pattern (Fig. a) and the path of the particle (Fig. b) in the double slit
experiment. The figures are taken from [Bohr, 1949]. If the diaphragm with two slits is
fixed an interference pattern is exhibited as given in Fig. a). In the experimental situa-
tion in Fig. b) when the diaphragm can recoil no interference pattern is observed. Bohr
[1949] writes: Since, however, any reading of the scale, in whatever way performed,will involve an uncontrollable change in the momentum of the diaphragm, there will
always be, in conformity with the indeterminacy principle, a reciprocal relationship
between our knowledge of the position of the slit and the accuracy of the momentum
control. The lack of our knowledge of the position of the slit excludes then the
appearance of the interference phenomena.
statement: If an apparatus is capable of determining which hole the electron
goes through, it cannot be so delicate that it does not disturb the pattern in
an essential way. No one has ever found (or even thought of) a way around the
uncertainty principle.
In the experimental situations discussed so far, as in most other usual text-
book examples, the which-path information is obtained, exposing the interfering
particle to uncontrollable scattering effects. This initiated a number of miscon-
ceptions being put forward in the literature. According to the most significant
misconception, loss of interference is due to an uncontrollable transfer of en-
ergy and/or momentum to the particle associated with any attempt to observe
the particles path. Unavoidable disturbances might again be because of the
intrinsic clumsiness of any macroscopic measuring apparatus. Over the last
few years experiments were considered and some already performed, where thereason why no interference pattern arises is not due to any uncontrollable dis-
turbance of the quantum system or the clumsiness of the apparatus. Rather the
lack of interference is due to the fact that the quantum state is prepared in such
a way as to permit path information to be obtained, in principle, independent
of whether the experimenter cares to read it out or not.
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14 Chapter 1: Information Acquired in a Quantum Experiment
Figure 1.2: An arrangement for two-particle interferometry. The source emits two
particles in the entangled state (1.1). Particle 1 traverses the Mach-Zehnder interfer-ometer starting with the beams A and B while particle 2 traverses the Mach-Zehnder
interferometer starting with the beams C and D. Phase shifters in both interferometers
permit continuous variations of the phases 1 and 2.
One line of such research considers the use of pairs of particles which are
strongly entangled. Consider a setup where a source emits two particles with
antiparallel momenta which then feed two Mach-Zehnder interferometers [Horne
et al., 1989], [Rarity and Tapster, 1990], [Herzog et al., 1995] as shown in Fig.
1.2. Then whenever particle 1 is found in beam A, particle 2 is found in beam
C and whenever particle 1 is found in beam B particle 2 is found in beam D.The quantum state is
| = 12
(|A1|C2 + |B1|D2). (1.1)
Will we now observe an interference pattern for particle 2, i.e. the well-known
sinusoidal variation of the intensities registered in the detectors U2 and L2upon variation of the phase 2? The answer has to be negative because by
simply placing detectors in the beams A and B of particle 1 we can determine
which path particle 1 took. The lack of interference can easily be calculated
starting from the state (1.1). Yet, if we recombine the two paths of particle 1
as indicated in Fig. 1.2, and if we register both particle 1 in either detector U1or L1 and particle 2 in either detector U2 or L2, we have forgone any possibility
of obtaining path information. Therefore we conclude an interference pattern
should arise in coincidence counts between the detectors for particle 1 and for
particle 2 shown in Fig. 1.2. This indeed follows from quantum mechanical
calculations [Horne et al., 1989].
Another independent approach to complementarity in an interference ex-
periment considers the use of micromasers in atomic beam experiments [Scully
et al., 1991]. Typically in such an experiment, an atom passes through a cavitysuch that it exchanges exactly one photon with the cavity without changing
momentum. Thus by investigating the cavity, one has information on whether
or not an atom passed through it without influencing the momentum of the
atom. Now, if we place one cavity into the each of two paths of the interference
experiment, we may obtain information on which path the atom took. The
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1.1 Unbestimmtheit vs Unbekanntheit in a Quantum Experiment 15
interference pattern does not arise. It is the mere possibility of obtaining path
information which guarantees that no interference occurs1. On the other hand,
we can read the information in the micromasers in such a way as to erase all
information on which micromaser the photon has been stored in. Then we have
just the information that the atom passed through the apparatus, but not along
which path. In this case the atoms counted in coincidence with the photons are
members of an ensemble defining an interference pattern.
These two experiments underline clearly that complementary does not origi-
nate in some uncontrollable disturbance of pre-assigned properties of a quantum
system in a measurement process. In fact, as theorems like those of Bell [Bell,
1964] and Greenberger, Horne and Zeilinger [Greenberger et al. 1989, 1990]
show, it is in principle not possible to assign to a quantum system simultane-
ously properties that both correspond to complementary measurements, and
which in order to be in agreement with special relativity, have to be local. The
principle impossibility of local realism will now be briefly demonstrated for our
example of the two-particle interference experiment given in Fig. 1.2.
As the two particles in our example might be widely separated, it is nat-
ural to assume validity of the locality condition suggested by EPR [Einstein,
Podolsky and Rosen, 1935]: Since at the time of measurement the two sys-
tems no longer interact, no real change can take place in the second system in
consequence of anything that may be done to the first system. Then, whether
detector U2 or L2 for a specific phase 2 is triggered must be independent of
which measurement we actually perform on the other particle (e.g, indepen-
dent of the phase 1) and even independent of whether we care to perform
any measurement at all on that particle. This assumption implies that certain
combinations of expectation values have definite bounds. The mathematicalexpression of that bound is called Bells inequality, of which many variants ex-
ists. For example, a version given by Clauser, Horne, Shimony and Holt [1969]
is
|E(1, 2) E(1, 2)| + |E(1, 2) + E(1, 2)| 2 (1.2)
where
E(1, 2) = (1.3)P++(1, 2) + P(1, 2) P+(1, 2) P+(1, 2).
1Scully et al. [1991] wrote: ... it is simply the information contained in a functioning
measuring apparatus that changes the outcome of the experiment, and not uncontrollable
alternations of the spatial wave function, resulting from the action of the measuring apparatus
on the system under observation.
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16 Chapter 1: Information Acquired in a Quantum Experiment
For the quantum state (1.1) this becomes
EQM(1, 2) = cos(2 1),
where we suppose a phase shift of i for reflection and 1 for transmission at
the beam splitter. Here we assume that particle 1 gives result + (
) when
it triggers detector U1 (L1) and particle 2 gives result + () when it triggersdetector U2 (L2). Then, e.g. P++(1, 2) is the joint probability that particle 1
gives + and particle 2 gives +. Maximal violation occurs for 1 = 45, 2 = 0,
1 = 135, 2 = 90, where the left-hand side of Eq. (1.2) will be 2
2 in clear
violation of the inequality. Thus, the assumption of local realism is in conflict
with quantum physics itself.
From this we learn that we cannot speak of complementarity as a conse-
quence of some disturbance of a system in the measurement if there are no
objective properties to disturb2. An important feature of the analysis so far is
that we have to base our concept of complementarity on the much more funda-mental concept of information. Any firm foundation of complementarity has to
make recourse to the property of mutual exclusiveness of different classes of in-
formation of a quantum system. As stated by Pauli [1958] in the analysis of the
uncertainty relations3: ... diese Relationen enthalten die Aussage, da jede
genaue Kenntnis des Teilchenortes zugleich eine prinzipielle Unbestimmtheit,
nicht nur Unbekanntheit des Impulses zur Folge hat und umgekehrt. Die Un-
terscheidung zwischen (prinzipieller) Unbestimmtheit und Unbekanntheit und
der Zusammenhang beider Begriffe sind fur die ganze Quantentheorie entschei-
dend.
We note that a view of information as the most fundamental concept in
quantum mechanics also leads to the most natural understanding of new phe-
nomena in quantum computation [Barenco et al., 1995(a)], entanglement swap-
ping [Zukowski et al., 1993], [Pan et al., 1998], quantum cloning [Wootters and
Zurek, 1982], [Buzek and Hillery, 1996] and quantum communication such as
quantum dense coding [Mattle et al., 1996], quantum cryptography [Bennett et
al., 1992] and quantum teleportation [Bennett et al., 1993], [Bouwmeester et.
al, 1997].
2Bohr dislikes phrases like disturbing phenomena by observations exactly because of their
potential for confusion. He stresses [Bohr, 1958] the use of the word phenomenon exclusively
to refer to observations made under specific circumstances, including an account of the wholeexperimental arrangement.
3Translated:... this relations contain the statement that any precise knowledge of the
position of a particle implies a fundamental indefiniteness, not just an unknownness, of the
momentum for a consequence and vice versa. The distinction between (fundamental) indef-
initeness and unknownness, and a connection of these two notions is decisive for the whole
quantum theory.
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1.2 Conceptual Inadequacy of the Shannon Information ... 17
1.2 Conceptual Inadequacy of the Shannon Informa-
tion in a Quantum Measurement
Shannons measure of information is generally considered to be very useful to
describe information in a physical observation. Here we will see that, while this
is rather natural in classical physics, it becomes problematic and even untenablein quantum physics.
There are various ways to motivate the Shannon measure of information. In
an operational approach Shannons information is introduced as the expected
minimal number of binary questions, i.e. questions with yes or no an-
swers only, required to discern the outcome of an experiment. In an axiomatic
approach the Shannon measure is uniquely specified by Shannons postulates
which establish some intuitively clear relations between individual amounts of
information gained in different individual observations. And in a physical ap-
proach Shannons information is characterized in terms of some natural prop-
erties which are essential from the point of view of the physics considered.
When investigating these three approaches in the next sections we will no-
tice that each approach contains an element that escapes complete and full
description in quantum mechanics. This element is always associated with the
objective randomness of individual quantum events and with quantum comple-
mentarity.
1.2.1 An Operational Approach
For classical observations Shannons information can be strengthened through
an operational approach to the question. To carry this out, consider the fol-
lowing example. An urn is filled with colored balls. The proportions in which
the different colors are present is known. Now the urn is shaken, and we draw
a single ball. To what extent can we predict the color of the drawn ball? If
all the balls in the urn are of the same color, we can completely predict the
outcome of the draw. On the other hand, if the various colors are present in
equal proportions, we are completely uncertain about the outcome. One can
think of these situations as extreme cases on a varying scale of predictability.
As a specific example consider an urn containing balls of four colors: black,
white, red, and green, with the proportions p1 =12 , p2 =
14 , p3 =
18 and p4 =
18 ,
respectively. Suppose now that one wishes to learn the color of the drawn ball
by asking questions to which only yes or no can be given as an answer. Of
course, the number of questions needed will depend on the questioning strategy
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18 Chapter 1: Information Acquired in a Quantum Experiment
Figure 1.3: Binary question tree to determine the color of a drawn ball. The pro-portions in which black, white, red and green colors are present are p1 =
1
2, p2 =
1
4,
p3 =1
8and p4 =
1
8, respectively.
adopted. In order to make this strategy the most optimal, that is, in order
that we can expect to gain from each yes-or-no question maximal information,
we evidently have to ask questions whose answers will strike out half of the
possibilities.
Indeed, a good question to start with is to ask Is the color of the drawnball black? (Fig. 1.3), the virtue being that, regardless of the answer yes
or no, we will be able to strike out a weighted half of the possibilities. If
the answer is yes, then we are done. If the answer is no, one may divide
the set that remains after this first round into two parts of equal probability
{white} and {red, green} and proceed by posing the question Is the color ofthe drawn ball white?. Again, if the answer is yes, we are done, and if the
answer is no we proceed in a similar fashion until the identity of the outcome
is at hand. A particular outcome is specified by writing down, in order, the
yess and nos encountered in travelling from the root to the specific leaf of the
tree schematically depicted in Fig. 1.3. It is easy to see that following theabove optimal strategy the mean minimal number of binary questions needed
to determine the color of the drawn ball is
p1 1 +p2 2 + (p3 +p4) 3 = 12
1 + 14
2 +
1
8+
1
8
3 = 7
4.
Notice that this may be written as
1
2
log1
2 1
4
log1
4 1
8
log1
8 1
8
log1
8
=4
i=1pi logpi.
where the logarithm is taken to base 2.
Now of course for an arbitrary probability distribution p1, p2, p3 and p4over a set of colors, a division into two sets of equal probability is not always
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1.2 Conceptual Inadequacy of the Shannon Information ... 19
possible. One may then consider a generalized situation where we draw a ball
N times without replacing the drawn ball. We assume again that we wish to
learn the colors of N drawn balls by asking questions to which only yes or
no can be given as an answer. Now, however, questions of a mixed type may
be asked, like Is the color of the first drawn ball black or white, of the second
drawn ball red, ..., and of the Nth black or white or green?. In this manner
it becomes easier to find questions for which the probability of yes and noare approximately equal, and thus the total number of questions needed can be
reduced.
Suppose p1N, p2N, p3N and p4N are all integers, then the probability of
obtaining the sequence containing p1N black balls, p2N white balls, p3N red
balls and p4N green balls is [Shannon, 1949]
p(sequence) = pNp11 pNp22 p
Np33 p
Np44 =
1
2NH
where
H = 4
i=1
pi logpi (1.4)
is the Shannon information expressed in bits when the logarithm is taken to
base 2. Such a sequence is called typical sequence4. Notice that a particular
typical sequence is specified by the particular order of the balls distinguishable
by the particular color sequence. The total number of typical sequences can
be obtained as the number of distinguishable permutations of N balls made up
of 4 groups of black, white, red and green balls indistinguishable within each
group. If N is sufficiently large then
N!
(p1N)!(p2N)!(p3N)!(p4N)! 2NH, (1.5)
where we use the Stirling approximation N! 2N NNeN. Hence, thetypical sequences all have equal probability, and there are 2NH of them.
Let us now turn back to our problem. We wish to learn colors of N drawn
balls by asking questions to which only yes or no can be given as the answer.4To be specific, we define the set of typical sequences to be all sequences such that
2N(H+) p(sequence) 2N(H) > 0.
Now, it can be shown that the probability that N outcomes actually form a typical sequence
is greater than 1 , for sufficiently large N, no matter how small might be.
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20 Chapter 1: Information Acquired in a Quantum Experiment
Figure 1.4: Binary question tree to determine the specific sequence of outcomes (colorof the drawn balls) in a sufficiently large number N of experimental trials (numberof drawings). An urn is filled with black and white balls with proportions p1 and
p2, respectively. The expected number of questions needed to determine the actual
sequence of outcomes is N H, where H = p1 logp1 p2 logp2.
If we address this problem in a piece-wise manner, determining the colors of
the drawn balls one after another, the number of questions needed will just be
N times that needed for a single ball.
However we may use another strategy. Suppose N is sufficiently large that
the sequence of N drawn balls contains close to p1N black balls, p2N whiteballs, p3N red balls and p4N green balls. In other words, suppose N drawn
balls form a typical sequence. Now, in order to learn the colors of the drawn
balls we need only to identify which particular typical sequence is actually
drawn. Since there are 2NH possible typical sequences and all of them have
equal probability to be drawn, the minimal number of yes-no questions needed
is just NH. Or equivalently, the Shannon information5 expressed in bits is the
minimal number of yes-no questions necessary to determine which particular
sequence of outcomes occurs, divided by N [Feinstein, 1958], [Uffink, 1990].
This is known as the noiseless coding theorem. An explicit example with an
urn containing balls of two different colors is given in Fig. 1.4. A generalization
5The Shannon information therefore refers to the information about an individual outcome
of an experiment. This should be contrasted to the cases where the notion of information refers
to knowledge about an unknown parameter in a probability distribution [Fisher, 1925], or the
information for discriminating between two probability distributions [Kullback, 1959], or the
information that one event provides about another event [Gelfand and Yaglom, 1957].
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1.2 Conceptual Inadequacy of the Shannon Information ... 21
for the probability distribution p1, p2, ..., pn over a finite set of n colors may
easily be obtained.
We now analyze Shannons notion of information in a quantum measure-
ment. In particular we consider a beam of photons prepared with vertical
polarization and analyzed by a filter polarized at an angle of 45 from the verti-
cal position. Each individual photon, when it encounters the polarization filter,has exactly two equally probable options: to pass straight through the filter
(we call this the outcome 1) or to be absorbed by the filter (the outcome
0). Now suppose we perform the polarization experiment a sufficiently large
number N of times so that the sequence of actual outcomes forms a typical
sequence.
We observe a particular sequence of 1s and 0s. An individual outcome
observed in a single experimental trial is fundamentally random and cannot be
assumed to reveal the property of an individual photon, assigned before the
measurement is performed, to pass through the filter or to be absorbed by the
filter. The principal indefiniteness, in the sense of fundamental nonexistenceof a detailed description of and prediction for the individual quantum event
resulting in the particular measurement result, implies that the particular out-
come sequence of 1s and 0s specified by writing down, in order, the yess and
nos encountered in a row of yes/no questions asked is not defined before the
measurement is performed. This implies that Shannons information defined
as the number of yes/no questions needed to determine the particular order of
1 and 0s in the actual sequence of outcomes cannot be assumed to describe
our ignorance about the future measurement results that is given before the
measurements are performed and that is then removed after the measurements
are performed, because no individual outcome and consequently no particularorder of 1s and 0s we observe in the sequence of measurements is defined before
the measurements are performed.
Of course, after the measurement is performed and its actual result becomes
known the information necessary to specify the measurement result is quantified
by the Shannon measure of information. Yet, this information has no reference
to the particular experimental situation given before the experiment is per-
formed and therefore it is not appropriate to define the information about the
system that is gained by the performance of the experiment. In the sense that
an individual quantum event manifests itself only in the measurement process
and is not precisely defined before measurement is performed, we may speak
of a creation of Shannons information in the measurement. In our explicit
example, the amount of created information is maximal because vertical po-
larization and polarization at 45 are maximally complementary attributes. Itis interesting to contrast this with Shannons [1949] writing of information as
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22 Chapter 1: Information Acquired in a Quantum Experiment
being produced by a source.
The Shannon information is surely adequate for the situation in classical
physics where we can always mentally split the ensemble into its constituents
and where the stochastic behavior of the whole ensemble follows from the be-
havior of its intrinsic different individual constituents which can be thought of
as being defined to any precision. In classical physics, this can be done evenin situations where we have no way to distinguish the individual constituents
and their behavior experimentally. If we perform a sequence of measurements
on the ensemble, a particular order of individual events that is recorded is
predetermined and originates in the intrinsic properties individual constituents
possess before measurements. The Shannon information may then be assumed
to measure the information necessary to reveal the property of an individual
system of the ensemble given before measurements are performed. Again this
cannot be assumed in a quantum measurement, because a quantum measure-
ment, with the only exception being that of the system in an eigenstate of the
measured observable, changes the state of the system into a new state in a fun-
damentally unpredictable way, and thus cannot be claimed to reveal a property
existing before the measurement is performed. In fact, as theorems like those
of Kochen-Specker [Kochen and Specker 1967] show, in quantum mechanics it
is not possible, not even in principle, to assign to a quantum system properties
corresponding to all possible measurements.
1.2.2 An Axiomatic Approach
An important reason for preferring the Shannon measure of information inthe literature lies in the fact that it is uniquely characterized by Shannons
intuitively reasonable postulates, and that alternative expressions should be
rejected for that reason. This has been expressed strongly by Jaynes [1957]
in words: One ... important reason for preferring the Shannon measure is
that it is the only one that satisfies ... [Shannons postulates]. Therefore one
expects that any deduction made from other information measures, if carried
far enough, will eventually lead to contradiction. A good way to continue our
discussion is by reviewing how Shannon, using his postulates, arrived at his
famous expression. He writes [1949]:
Suppose we have a set of possible events whose probabilities of occurrenceare p1, p2,...,pn. These probabilities are known but that is all we know con-
cerning which event will occur. Can we find a measure of how much choice is
involved in the selection of the event or how uncertain we are of the outcome?
If there is such a measure, say H(p1, p2,...,pn), it is reasonable to require of
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1.2 Conceptual Inadequacy of the Shannon Information ... 23
12
13
13
23
16
13
16
121
2
12
Figure 1.5: Decomposition of a choice from three possibilities. Figure taken from[Shannon, 1949].
it the following properties:
1. H should be continuous in the pi.
2. If all the pi are equal, pi =1n
, then H should be a monotonic increas-
ing function of n. With equally likely events there is more choice, or
uncertainty, when there are more possible events.
3. If a choice be broken down into two successive choices, the original H
should be the weighted sum of the individual values of H. The meaning of
this is illustrated in Fig. 1.5. At the left we have three possibilities p1 =12 ,
p2 =13 , p3 =
16 . On the right we first choose between two possibilities
each with probability 12 , and if the second occurs make another choice
with probabilities 23 ,13 . The final results have the same probabilities as
before. We require, in this special case, that
H
1
2,
1
3,
1
6
= H
1
2,
1
2
+
1
2H
2
3,
1
3
.
The coefficient 12 is the weighing factor introduced because this second
choice occurs half the time.
Shannon then shows that only the function (1.4) satisfies all three postulates.
It is clear from the way Shannon formulated the problem, that H is in-
troduced as an uncertainty about the outcome of an experiment based on a
given probability distribution. The uncertainty arises, of course, because the
probability distribution does not enable us to predict exactly what the actual
outcome will be. This uncertainty is, of course, removed when the experiment is
performed and its actual outcome becomes known. Thus, we may think of H asthe amount of information that is gained by the performance of the experiment.
We now turn to the discussion of Shannons postulates. While the first two
postulates are purely qualitative and natural for every meaningful measure of
information, the last postulate might appear to have no immediate intuitive
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24 Chapter 1: Information Acquired in a Quantum Experiment
appeal. The third Shannon postulate originally formulated as an example was
reformulated as an exact rule by Faddeev [1957]: For every n 2
H(p1,..,pn1, q1, q2) =H(p1,..,pn1, pn)+pnH
q1pn
,q2
pn
, (1.6)
where pn = q1 + q2.
Without physical interpretation the recursion postulate (1.6) is merely a
mathematical expression which is certainly necessary for the uniqueness of the
function (1.4) but has no further physical significance. We adopt the following
well-known interpretation [Uffink, 1990], [Jaynes, 1996]. Assume the possible
outcomes of the experiment to be a1,...,an and H(p1,...,pn) to represent the
amount of information that is gained by the performance of the experiment.
Now, decompose event an into two distinct events an b1 and an b2 (denotes and, thus a b denotes a joint event). Denote the probabilities ofoutcomes an b1 and an b2 by q1 and q2, respectively. Then the left-hand sideH(p1,...,pn1, q1, q2) of Eq. (1.6) represents the amount of information that isgained by the performance of the experiment with outcomes a1,...,an1, an b1, an b2. When the outcome an occurs, the conditional probabilities forb1 and b2 are
q1pn
and q2pn
respectively and the amount of information gained
by the performance of the conditional experiment is Hq1pn
, q2pn
. Hence the
recursion requirement states that the information gained in the experiment
with outcomes a1,...,an1, an b1, an b2 equals the sum of the informationgained in the experiment with outcomes a1,...,an and the information gained
in the conditional experiment with outcomes b1 or b2, given that the outcome
an occurred with probability pn. This interpretation implies that the third
postulate can be rewritten as
H(p(a1),...,p(an1), p(an b1), p(an b2)) (1.7)= H(p(a1),...,p(an1), p(an))+p(an)H(p(b1|an), p(b2|an)),
where
p(an) = p(an b1) +p(an b2),p(an b1) =p(an)p(b1|an) and p(an b2) =p(an)p(b2|an).
Here p(bi|an) i = 1, 2 denotes the conditional probability for outcome an giventhe outcome bi occurred andp(anbi) denotes the joint probability that outcomean bi occurs.
If we analyze the generalized situation with n outcomes ai of the first ex-
periment A, m outcomes bj of the conditional experiment B and mn outcomes
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1.2 Conceptual Inadequacy of the Shannon Information ... 25
aibj of the joint experiment AB, we may then rewrite the recursion postulatein a short form as
H(A B) = H(A) + H(B|A) (1.8)
where H(B|A) = nj p(aj)H(b1|aj ,...,bm|aj) is the average of information gainedby observation B given that the conditional outcome aj occurred weighted by
probability p(aj) for aj to occur.
It is essential to note that the recursion postulate is inevitably related to
the manner in which we gain information in a classical measurement. In fact, in
classical measurements it is always possible to assign to a system simultaneously
attributes corresponding to all possible measurements, here ai, bj and ai bj.Also, the interaction between measuring apparatus and classical system can be
thought to be made arbitrarily small so that the experimental determination
of A has no influence on our possibility to predict the outcomes of the possible
future experiment B. In conclusion, the information expected from the jointexperiment A B is simply the sum of the information expected from the firstexperiment A and the conditional information of the second experiment B with
respect to the first, as predicted by Eq. (1.8).
In contrast we know that in a quantum measurement it is not possible to
assign to a system simultaneously complementary attributes, like position and
momentum, or the path of the system and the position of appearance in the
interference pattern in the double-slit experiment, or the spin values along or-
thogonal directions. Therefore Shannons crucial third postulate (1.8) necessary
for uniqueness of Shannons measure of information is not well-defined in quan-
tum mechanics when A and B are measurements of mutually complementary
attributes. Consequently, the Shannon measure loses its preferential status with
respect to alternative expressions when applied to define information gain in
quantum measurements.
Here a certain misconception might be put forward that arises from a certain
operational point of view. According to that view, for example, complementar-
ity between interference pattern and information about the path of the system
in the double-slit experiment arises from the fact that any attempt to observe
the particle path would be associated with an uncontrollable disturbance of the
particle. Such a disturbance in itself would then be the reason for the loss of theinterference pattern. In such of view it would be possible to define Shannons
information for all attributes of the system simultaneously, and the third Shan-
non postulate would be violated because of the unavoidable disturbance of the
system occurring whenever the subsequently measured property B is incompat-
ible with the previous one A. Yet, this is a misconception not only because it
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26 Chapter 1: Information Acquired in a Quantum Experiment
was shown [Bell, 1964], [Greenberger et al. 1989, 1990] that it is in principle im-
possible to assign to a quantum system simultaneously observation-independent
properties (which in order to be in agreement with special relatively have to be
local) but also because some experiments have already been performed [Herzog
et al., 1995] where the reason why no interference pattern arises is not due to
an uncontrollable disturbance of the quantum system (see also Sec. 1.1).
We next introduce two requirements that are immediate consequences of
Shannons postulate and in which all the probabilities that appear are well-
defined in quantum mechanics. We will show that the two requirements are
violated by the information gained in quantum measurements.
1. Every new observation reduces our ignorance and increases our knowledge.
In his work Shannon [1949] offers a list of properties to substantiate that
H is a reasonable measure of information. He writes: It is easily shown
that
H(A B) H(A) + H(B)
with equality only if the events are independent (i.e., p(aibj) = p(ai)p(bj)).The uncertainty of a joint event is less than or equal to the sum of the
individual uncertainties. He continues further in the text: ... we have
H(A) + H(B) H(A B) = H(A) + H(B|A).
Hence,
H(B)
H(B
|A). (1.9)
The uncertainty of B is never increased by knowledge of A. It will be
decreased unless A and B are independent events, in which case it is not
changed (we have changed Shannons notation to coincide with that of
our work).
2. Information is indifferent on the order of acquisition. The total amount
of information gained in successive measurements is independent of the
order in which it is acquired, so that the amount of information gained
by the observation of A followed by the observation of B is equivalent to
the amount of information gained from the observation of B followed by
the observation of A
H(A) + H(B|A) = H(B) + H(A|B). (1.10)
This is an immediate consequence of the recursive postulate which can
be obtained when we write the recursion postulate in two different ways
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1.2 Conceptual Inadequacy of the Shannon Information ... 2700110 00 01 11 1 white plastic ballblack plastic ballwhite wooden ball 0 00 01 11 1white
color
1/3
black2/30 01 1 01 01 0011whiteblackcolor 1/21/2 plasticwoodcomposition
plastic1/2
wood
composition
1/2
1
0
a)
plastic
wood
composition
3/4
1/4
white
black
color
1
0
b)
Figure 1.6: Indifference of information to the order of its acquisition in classicalmeasurements. A box is filled with balls of different compositions (plastic and wooden
balls) and different colors (black and white balls). Now, the box is shaken. In Fig a)
we first draw a ball asking about the color of the drawn ball and gain H(color) = 1 bit
of information. Subsequently, we put the black and white balls in separate boxes, draw
a ball from each box separately and ask about the composition of the drawn ball. We
gain Hbl(comp.) = 0 bits for the black balls and Hwh(comp.) = 1 bit for the white balls.
In Fig. b) we pose the two questions in the opposite order. We firstly ask about the
composition of the drawn ball and gain H(comp.) = 0.81 bit. In a conditional drawing
we ask about the color of the drawn ball and gain Hwo(color) = 0 bits for wooden
balls and Hpl(color) = 0.92 bits for the plastic balls. The total information gained is
independent on the order of the two questions asked, i.e. H(color)+1/2Hbl(comp.)+
1/2Hwh(comp.)=H(comp.)+1/4Hwo(color)+3/4Hpl(color)= 1.5.
depending on whether the observation of A is followed by the observa-
tion of B or vice versa. An explicit example for a sequence of classical
measurements is given in Fig. 1.6.
Are these two requirements satisfied by information gained in quantum mea-
surements? Consider a beam of randomly polarized photons. Filters F, Fand F are oriented vertically, at +45, and horizontally respectively, and canbe placed so as to intersect the beam of photons (Fig. 1.7). If we insert filter Fthe intensity at the detection plate will be half of the intensity of the incoming
beam. The outgoing photons are now all with vertical polarization. Notice
that the function of filter F cannot be explained as a sieve that only letsthose photons pass that are already with horizontal polarization in the incoming
beam. If that were the case, only a certain small number of randomly polarized
incoming photons would be with horizontal polarization, so we would expect a
much larger attenuation of the intensity of the beam as it passes the filter.
Insertion of filters F and F correspond to the measurements of A polar-ization at +45 and B horizontal polarization, respectively. Now, when filterF is inserted behind the filter F, the intensity of the outgoing beam dropsto zero. None of the photons with vertical polarization can pass through the
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28 Chapter 1: Information Acquired in a Quantum Experiment
Figure 1.7: New observation (of polarization at 45) reduces our knowledge (of thevertical/horizontal) polarization) at hand from a previous observation. Filters F, Fand F are oriented vertically, at +45
and horizontally, respectively. If filter F is
inserted behind the filter F (Fig. a), no photons are observed at the detector plate.
In that case we have complete knowledge of the vertical/horizontal polarization of the
photon. After filter F is inserted between F and F (Fig. b), a certain number of
photons will be observed at the detection plate. Here acquisition of information about
the polarization of the photon at 45 leads to a decrease of our knowledge aboutvertical/horizontal polarization of the photon.
horizontal filter as shown in Fig. 1.7a. In this case we have complete knowledge
of the property B, i.e. H(B) = 0. Notice that a sieve model where F (F)only lets those photons pass that have already horizontal (vertical) polarization
in the incoming beam could explain this behaviour. Now, after filter F isinserted between F and F, a certain intensity will be visible at the detec-tion plate, exactly 14 of the intensity of the beam passed through F as shownin Fig. 1.7b. In that case, a certain number of photons that passed through
F will also pass through F. Therefore, acquisition of information about thepolarization of the photon at 45 leads to a decrease of our knowledge abouthorizontal polarization of the photon implying H(B|A) > 0. Consequently,0 = H(B) H(B|A) > 0 which clearly violates requirement (1.9). Now,imagine after F we insert the filter F in Fig. 1.7a (this, of course, doesnot make any essential change compared with the situation without the addi-
tional filter). We may consider this new experimental situation as a sequence
of measurements BA. Now, information gained in the sequence BA in Fig.
1.7a differs from the information gained in the sequence AB in Fig. 1.7b, i.e.
0 = H(B) + H(A|B) = H(A) + H(B|A) > 0, thus violating the requirement(1.10). Another independent example where requirement (1.10) is violated is
given in Fig. 1.8.
Here we have an effect which cannot be explained by a sieve model. Classical
experience suggests that the addition of a filter should only be able to decrease
the intensity of the beam getting through. In a sieve model where the filter
does not change the object, adding a new filter will always reduce the intensity
of the beam. For completeness we note that a classical wave model can explain
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1.2 Conceptual Inadequacy of the Shannon Information ... 29
Figure 1.8: Dependence of information on the order of its acquisition in successivequantum measurements. A spin-1/2 particle is in the state |z+ spin-up along the z-axis. Spin along the x-axis and spin along the direction in the x-z plane tilted at an angle
from the z-axes are successively measured, in the order in Fig. a) and opposite to that
in Fig. b). Whereas we obtain an equal portion H(cos2(/4/2), sin2(/4/2)) ofinformation in the conditional (subsequent) measurement both in Fig. a) and in Fig.
b), the amounts of information H(cos2 /2, sin2 /2) and H(12
, 12
) = 1 we gain in the
first measurement in Fig. a) and in the first measurement in Fig. b) respectively, can
be significantly different. Specifically for 0 we have complete knowledge aboutspin along the direction at the angle in Fig. a) but absolutely no knowledge about
the spin along the x-axis in Fig. b). We emphasize that we do not assume any specific
functional dependence for the measure of information H.
the increase of the intensity of the wave transmitted through the filters.
In contrast to the sieve model where adding a new filter just add some new
knowledge of the object and never decrease our knowledge at hand from the
previous measurements, a quantum measurement can decrease our knowledge
collected from previous measurements. This originates from the distinction
between maximal and complete information in quantum physics. In clas-
sical physics the maximal information about a system is complete. In quantum
physics the maximal information, represented by the state vector, is never com-
plete in the sense that all possible future measurement results are precisely
defined. Yet, we do not hesitate to emphasize that it certainly is complete in
the sense that it is not possible to have more information about a system than
what can be specified in its quantum state. In fact, the state vector represents
that information which is necessary to arrive at the maximum possible set of
probabilistic prediction for all possible future observations of the system.
In our explicit example the state vector of the polarization of a photon can
be expressed as | = a|+b| (a and b are complex numbers) in the basis ofvertical | and horizontal | polarization. The probability to observe verticalpolarization is |a|2 and the probability to observe horizontal polarization is |b|2.Measurement of vertical/horizontal polarization will change the state to an
eigenstate associated with the result of the measurement. In our example if
measurement by filter F results in vertical polarization, then the state changes
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to | and when the polarization is measured again with respect to the samebasis by F, it will return vertical polarization with probability one. Thus,no photon will have the property of horizontal polarization as indicated in Fig.
1.7a implying H(B) = 0. In Fig. 1.7b, a photon passing through F withthe state | will pass filter F with a probability of 1/2, and so 50% of thephotons will pass through F. A photon passing through F changes the state
from | to | =12(| + | ), indicating gain of the new knowledge
(about polarization at 45) at the expense of unavoidable and irrecoverableloss of the prior knowledge (about vertical/horizontal polarization). As before,
this photon will pass F with a probability of 1/2. Thus, the probability for aphoton to pass the sequence of filters FF is 1/4 implying H(B|A) = 0.56.
Being a summary representation of the observers in general probabilistic
predictions for future observations, the quantum state normally changes in a
measurement process into one of the new states defined by the measurement
apparatus. After the measurement the old summary of the observers informa-
tion is at least partially lost and a new one, established to be in accord with the
change of the state, is indifferent to the knowledge collected from the previous
measurements in the whole history of the system. Such a view was assumed by
Pauli [1958] who writes6: Bei Unbestimmtheit einer Eigenschaft eines Systems
bei einer bestimmten Anordnung (bei einem bestimmten Zustand des Systems)
vernichtet jeder Versuch, die betreffende Eigenschaft zu messen, (mindestend
teilweise) den Einflu der fruheren Kenntnisse vom System auf die (eventuell
statistischen) Aussagen uber spatere mogliche Messungsergebnisse.
1.2.3 A Physical Approach
A specific measure of information becomes a meaningful concept in physics only
when it can be characterized by the properties which naturally follow from the
physics considered. Such a property can be, for example, invariance of the
total information content of the system under variation of modes of observa-
tion or conservation of the total information in time if there is no information
exchange with an environment. We will show that for a quantum system the
total information defined according to Shannons measure does not have these
properties.
The classical world appears to be composed of particles and fields, and thenature of each one of these constituents could be specified quite independently
6Translated: In the case of indefiniteness of a property of a system for a certain experi-
mental arrangement (for a certain state of the system) any attempt to measure that property
destroys (at least partially) the influence of earlier knowledge of the system on (possibly
statistical) statements about later possible measurement results.
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1.2 Conceptual Inadequacy of the Shannon Information ... 31
of the particular phenomenon discussed or of the experimental procedure a
physicist chooses. In other words, any concept introduced in classical physics is
totally noncontextual. In particular, the total information content of a classical
pointlike system (with no rotation and inertial degrees of freedom) defined as
Shannons information associated with the probability distribution over the
phase space is independent of the specific set of variables (such as position and
momentum, or angle and angular momentum, etc.) considered and conserved intime if there is no information exchange with an environment7. Operationally
the total information content of a classical system can be obtained in the joint
measurement of position and momentum, or in successive measurements in
which the observation of position is followed by the observation of momentum
or vice versa8.
In quantum physics any concept is limited to the description of phenom-
ena taking place within some well-defined experimental context, that is, always
restricted to a specific experimental procedure the physicist chooses. This im-
plies the question: How to define the total information content of a quantum
system if in order to be in reasonable agreement with common sense it has to
be invariant under variation of modes of observation and conserved in time if
there is no information exchange with an environment?
For a given density matrix the von Neumann entropy
S() = T r( log ) (1.12)
is widely accepted as a suitable definition for an information content of a quan-
tum system. For a system described in N-dimensional Hilbert space this ranges
from log N for a completely mixed state up to 0 for a pure state. Also, the von
Neumann entropy is invariant under unitary transformations UU+. Thatis, it is invariant under the change of the representation (basis) of and also
conserved in time if there is no information exchange with an environment.
However, we observe that any function9 of the form T r(f()) can possess these
7We discuss this in detail in Appendix A.1. Here, we note that given the probability
distribution (r, p, t) over the phase space the total lack of information of a classical system
is defined by [Jaynes, 1962]
Htotal(t) =
d3rd3p(r, p, t)log
(r, p, t)
(r, p), (1.11)
where a background measure (r, p) is an additional ingredient that has to be added to theformalism to ensure invariance under variable change when we consider continuous probability
distributions. The conservation of Htotal in time for a system with no information exchange
with an environment is implied by the Hamiltonian evolution of a point in the phase space.8In full analogy with (1.10) we may write Htotal(r, p) = H(r) + H(p|r) = H(p) + H(r|p).9The operator f() is identified by having the same eigenstates as and the eigenvalues
f(wj), equal to the function values taken at the eigenvalues wj of .
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32 Chapter 1: Information Acquired in a Quantum Experiment
properties for a suitably defined function f and can, therefore, serve as indices
of the measure of the information content of a system. We also observe that the
von Neumann entropy is a property of the quantum state as a whole without
explicit reference to information contained in individual measurements. The
question arises: How to define and how to obtain information content of a
quantum system operationally? Here we ask precisely: What set of individual
measurements should we perform and how to combine individual measures ofinformation gained in different individual measurements to arrive at the total
information content of a quantum system?
We observe that, unlike the classical case, information carried by a quantum
system cannot be obtained through a set of successive measurements in a con-
sistent way, because information gained in successive quantum measurements
depends on the order of its acquisition (see Fig. 1.8 and discussion above). This
suggests that any attempt to obtain the total information content of a quan-
tum system has to be related to the specific set of different possible experiments
performed on an ensemble of equally prepared systems.
For a quantum system in the state different experiments correspond to
different probabilities for possible outcomes and therefore to different Shannon
information. How are individual measures of information obtained in different
individual experiments related to the total information carried by a quantum
system? It can be shown that the optimal experiment, which minimizes Shan-
nons information, is the one which corresponds to the orthonormal basis |iformed by the eigenvectors of the density matrix : |i = wi|i. The cor-responding Shannon information is then equal to the von Neumann entropy,
i.e.
H = i
wi log wi = T r( log ). (1.13)
Clearly this is invariant under unitary transformations. Again this implies
invariance of H under the change of the representation basis of and also its
conservation in time if there is no information exchange with an environment.
That is, if we perform the optimal experiments both at time t0 and at some
future time t, the Shannons information measures associated to the optimal
experiments at the two times
H(t) = i
wi(t)log wi(t) = i
wi log wi = H(t0) (1.14)
will be the same. Here, the eigenvalues of the density matrix at time t are wi(t).
However, without the additional knowledge of the eigenbasis of the density
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1.2 Conceptual Inadequacy of the Shannon Information ... 33
matrix we cannot find the optimal experiment and determine experimentally
the Shannon information associated. Also, all the statistical predictions that
can be made for the optimal measurement are the same as if we had an ordi-
nary (classical) mixture, with fractions wi of the systems giving with certainty
results that are associated to the eigenvectors |i. In this sense the optimalmeasurement is a classical type measurement and therefore in this particular
case, and only then, Shannons measure defines the information gain in a mea-surement appropriately. It is thus not surprising that Shannons measure is
useful only when applied to measurements which can be understood as classical
measurements. Again the question arises: How to combine individual measures
of information obtained in different individual measurements in order to arrive
at the information content of a quantum system if the individual measurements
are incompatible with the density operator (non-optimal measurements)?
One may be tempted to define the total information content of a quantum
system in a constructive fashion, namely as a sum of individual measures of
information over a complete set of mutually complementary experiments. These
are experiments with the property that complete knowledge of the outcome
in one of the experiments excludes any knowledge of the outcomes in others.
For example, a set of measurements of (1) vertical/horizontal polarization, (2)
polarization at +45/45, and (3) left/right circular polarization is a completeset of mutually complementary measurements for photons polarization.
Consider a photons polarization state | = cos | + sin | . We sum-marize individual measures of Shannons information for the mutually comple-
mentary observations (1), (2) and (3) and obtain
Htotal = H1 + H2 + H3
= cos2
2log cos2
2+ sin2
2log sin2
2(1.15)
+1 sin
2log
1 sin 2
+1 + sin
2log
1 + sin
2
for the total Shannon information carried by the photons polarization. Our
result clearly depends on the parameter and thus is not invariant under uni-
tary transformations. This further associates certain features with our candi-
date Htotal for the total information carried by the photons polarization that
strongly disagrees with our intuitive appeal. Firstly, Htotal is not equal for eachpolarization state of the same purity. Secondly, Htotal is not specified by the
polarization state alone but depends on the particular set of mutually com-
plementary observations. If we choose another set of mutually complementary
observations, e.g., (1) polarization along the direction at an angle with re-spect to the vertical direction, (2) polarization along the direction at an angle
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34 Chapter 1: Information Acquired in a Quantum Experiment
( + 45) with respect to the vertical direction, and (3) left/right circular po-larization, the total information carried by photons polarization might not be
the same (it depends on the particular value of the angle ). And thirdly, Htotalis not conserved in time for a system isolated from its environment completely.
In this section we have stressed some conceptual difficulties arising when
we apply Shannons notion of information to define information gain in a quan-tum measurement. Investigating three different approaches to the concept of
Shannons information we argued that these difficulties arise whenever it is not
possible, not even in principle, to assume that attributes observed are assigned
to the system before the observation is performed. The question arises: Are
there other physical situations where the use of Shannons measure of informa-
tion might be justified in quantum mechanics? Obviously, there are.
Suppose that there is a set of different possible preparations of the initial
state and that the a prioriprobabilities for the different preparations are known
to the observer. The observer is not told which one of the states is actually
implemented. Suppose now that the observer wants to determine the actualstate. Here the observers ignorance about the possible prepared states can be
quantified by Shannons measure of information because the possible states, in
principle, can be thought of as being objectively present before the measurement
is performed.
We briefly review an explicit example analyzed by Peres [1995]. Let n1, n2and n3 denote three unit vectors defined in a plane separated by angles of 120
.Consider a spin-1/2 particle and define normalized states |i by ni|i = |i(i=1,2,3). The spin-1/2 particle can be prepared in one of three states |idefined above, and these three preparations have equal a priori probability, i.e
H = log 3. Which one of these states is actually prepared? Since the states
are not orthogonal the answer cannot be unambiguous. The procedure giving
the maximal possible information (that is reducing H as much as possible)
is obtained in a POVM (positive-operator-valued measurement) by ruling out
one of the three allowed states, and leaving equal a posteriori probabilities for
the two others. The value of H is reduced to log 2 = 1 , so that the actual
information gain is log(3/2).
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1.3 Measure of Information Acquired in a Quantum Experiment 35
1.3 Measure of Information Acquired in a Quantum
Experiment
Quantum mechanics is an intrinsically probabilistic description of Nature. All
an experimentalist can know before a quantum experiment is performed are
the probabilities for all possible outcomes to occur. In general, which specificoutcome occurs is objectively random. We define a new measure of information
for an individual measurement which is based on the fact that the probabilistic
predictions an experimentalist can make have no empirical significance for any
individual experiment but only as predictions about the number of occurrences
of a specific outcome in future repetitions of the experiment.
Consider a stationary experimental arrangement with two detectors, where
only one detector fires at a time, i.e. in each experimental trial. Detector 1,
say, fires (we call this the yes outcome) with probability p. If it does not
fire (the no outcome) the other detector will fire with probability q = 1
p.
When exactly one detector has fired, the experiment is over. Examples wouldbe the Stern-Gerlach experiment with a spin-1/2 particle or an interference
experiment with an interferometer of the Mach-Zehnder type.
Knowing the probabilities for the two outcomes to occur all an experimenter
can predict is how many times a specific detector fires. In making her prediction
she has only a limited number of systems to work with. Then, because of the
statistical fluctuations associated with any finite number of experimental trials,
the number of occurrences of a specific outcome in N future repetitions of the
experiment is not precisely predictable. In N independent experimental trials,
the particular ordered sequence of results yes,no,no ... yes containingyes exactly n times and no exactly N n times occurs with probability
p (1 p) (1 p) p = pn(1 p)Nn. (1.16)
The various different permutations of the sequence are independent events, and
so we can add their probabilities to obtain10
PN(n) =
N
n
pn(1 p)Nn, (1.17)
10We are ignorant about different possible orders of individual outcomes because, in quan-
tum measurement the particular order of individual outcomes is not defined before the ex-
periment is performed. In contrast, classical measurements reveals pre-existing properties of
individual systems and therefore the particular sequence of individual outcomes is of impor-
tance. Information that is gained about a particular sequence observed is adequately defined
by Shannons measure of information (see Sec. 1.2).
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36 Chapter 1: Information Acquired in a Quantum Experiment
the probability that from N independent experimental trials we observe n times
yes and N n times no. This is known as the binomial distribution [Gne-denko, 1976]. Note that if one bets on a specific result, e.g. that the number of
yes outcomes will be the one with highest probability, which is nmax pN,the probability of success still depends on p. With an inner probability of
p = 0.5, the probability of 5 yes outcomes in 10 trials is only 0.25, but with
one where p = 0.9 the probability of 9 yes outcomes in 10 trials is 0.39. Itis a peculiar feature of the binomial distribution, that the future number of
occurrences is less specified when p is around 0.5.
An experimenters uncertainty11, or lack of information, in the value n is
given by the mean-square-deviation defined as the expectation of the square of
the deviation of n from the mean value pN [Gnedenko, 1976]
2 :=Nn=1
PN(n)(n pN)2 = p(1 p)N. (1.19)
In fact, if is small, then each term in the sum in Eq. (1.19) is small. A
value n for which |n pN| is large must therefore have a small probabilityPN(n). In other words, in the case of small , large deviations of the number
of occurrences of the yes outcome from the mean pN are improbable. In
this case an experimenter knows the future number of occurrences with a high
certainty. Conversely, a large variance indicates that not all highly probable
values of n lie near the mean pN. In that case experimenter knows much less
about the future number of occurrences.
For a sufficiently large number N of experimental trials, the confidence
interval within which the number of occurrences of the yes outcome can be
found in 68% of cases is given as [Gnedenko, 1976]
(pN ,pN+ ). (1.20)
Therefore, if an observer just plans to perform the experiment N times, he
knows in advance, before the experiments are performed and their outcomes
11Since the binomial distribution has a finite deviation, it fulfills Chebyshevs inequality
[Gnedenko, 1976]:
Prob{|n pN| > k} 1k2
. (1.18)
This inequality means tha