Eur. Phys. J. D 41, 589–598 (2007)DOI: 10.1140/epjd/e2006-00265-1 THE EUROPEAN

PHYSICAL JOURNAL D

Maximum confidence measurements and their opticalimplementation

S. Croke1,2,a, P.J. Mosley3, S.M. Barnett1, and I.A. Walmsley3

1 Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK2 Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK3 Clarendon Laboratory, Oxford University, Parks Road, Oxford, OX1 3PU, UK

Received 19 June 2006 / Received in final form 5 October 2006Published online 8 December 2006 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2006

Abstract. Perfect discrimination between non-orthogonal states is forbidden by the laws of quantum me-chanics. Several strategies to discriminate optimally between states of an arbitrary set exist, and many ofthese have been implemented in experiments using optical polarisation. In this paper we discuss maximumconfidence measurements and their recent optical implementation.

PACS. 42.50.Xa Optical tests of quantum theory – 03.67.Hk Quantum information

1 Introduction

In classical mechanics, a system is described by variablessuch as position and momentum, which can in principle bemeasured arbitrarily accurately to obtain complete infor-mation about the state of the system. Thus, when classi-cal systems are used to carry information, it is possible, atleast in theory, to distinguish perfectly between any givenset of signal states, although in reality a noisy signal maymake this impracticable. In quantum mechanics however,the state of a system is not itself an observable. Further-more, the act of measuring a quantum system changesthe state, destroying the information previously containedtherein. As a consequence, the accuracy with which it ispossible to discriminate between states of an arbitrary setis limited by fundamental laws of quantum mechanics.

Perfect discrimination is possible only when the sig-nal states form a mutually orthogonal set. However, thereare certain advantages to be gained in exploiting quan-tum effects arising from the use of non-orthogonal signalstates. For example, cryptographic protocols using non-orthogonal states are provably secure against attack byeavesdroppers [1,2]. Also, the information carrying capa-bility for certain noisy channels is maximised by usingnon-orthogonal states [3]. It is important therefore, to un-derstand how to discriminate optimally between states ofan arbitrary set [4]. Optimality, of course, lies in the eye ofthe beholder. It requires a criterion against which the sys-tem performance can be judged. For example, is it worseto get a wrong answer or miss a right one? Possibly thesimplest definition of optimality is a measurement whichminimises the probability of incorrectly identifying the

a e-mail: sarah@phys.strath.ac.uk

state [5–7]. However, error-free or unambiguous discrimi-nation is possible between two non-orthogonal states, if weare prepared to accept the possibility of an inconclusiveresult [8]. This strategy can be extended to larger numbersof states [9], but is only applicable to linearly independentsets [10]. It has recently been shown however, that an anal-ogous strategy — one which achieves maximum confidencethat when a state is identified it was indeed present — ispossible for linearly dependent states [11]. Other defini-tions of optimal outcomes include maximising the mutualinformation shared by the receiving and transmitting par-ties [12,13], and maximising the fidelity between the statereceived and one transmitted on the basis of the measure-ment result [14,15] as well as minimising the a posterioriprobability of error [16]. Variants on the quantum statediscrimination problem have also been considered, includ-ing assigning the state of a system to one of two comple-mentary subsets of a set of non-orthogonal states [17], andthe special case where one of the subsets contains a singlestate, known as quantum state filtration [17–19].

Experimental demonstrations of optimal strategies todate have predominantly used the polarisation state oflight as a two-level system or qubit [20]. This is largelydue to the relative ease with which polarisation statescan be created and manipulated in the laboratory usingreadily available linear optical components. One alterna-tive is the construction of a multi-level optical system bymaking use of different ports of a multi-rail interferom-eter [18]. This approach, which also uses linear opticalelements to manipulate input states, was used to demon-strate unambiguous discrimination between three non-orthogonal states, and between a pure and a mixed statein three dimensions [21]. Examples of optimal minimum

590 The European Physical Journal D

error, mutual information, and unambiguous discrimina-tion measurement strategies have been demonstrated inexperiments on optical polarisation [22–26]. We have re-cently implemented a maximum confidence measurementbetween three equiprobable symmetric qubit states in thisway [27]. In this paper we will discuss maximum con-fidence measurements and their optical implementation.The remainder of the paper is organised as follows. InSection 2 we discuss how the maximum confidence mea-surement is constructed, and compare it to other strate-gies. In Section 3 we present our experimental design andits implementation. We discuss the experimental resultsand sources of error in Section 4, before concluding inSection 5.

2 Maximum confidence measurements

In quantum state discrimination, it is usually assumedthat both the set of possible states {ρi}, and the a prioriprobabilities {pi} that each state is prepared are known,although this is not always the case [28]. The prob-lem then is to choose the measurement to determinebest which state was actually prepared. A conventional,von Neumann measurement gives outcomes correspondingto one of a set of mutually orthogonal states. The numberof outcomes is therefore restricted by the dimensions of thestate space of the system. More general measurements arepossible however, and any measurement in quantum me-chanics can be described mathematically by a probabilityoperator measure (POM) [5], also known as a positive op-erator valued measure [29]. Thus the optimisation can becast as a purely mathematical problem whose convex na-ture allows globally optimal solutions to be found. In thePOM formalism, measurement results {ωi} are associatedwith operators {Πi}, which completely specify the mea-surement. The operator Πj is defined by the probabilityof occurence of the associated measurement result

P (ωj |ρ) = Tr(ρΠj) (1)

for a measurement on a system in state ρ. In order to forma physically realisable measurement, these operators mustbe non-negative, and satisfy a completeness condition:

Πi ≥ 0,∑

i

Πi = I. (2)

Although in general the result of a measurement will notreveal with certainty which state was prepared, it doesprovide information which allows us to modify our de-scription of the state via Bayes’ rule. Thus if outcomeωj is obtained, the probability distribution for the inputstates {ρi} becomes:

P (ρi|ωj) =P (ρi)P (ωj |ρi)

P (ωj)=

piTr(ρiΠj)Tr(ρΠj)

(3)

where ρ =∑

i piρi. Thus this represents the informationwe now have about the state given knowledge of the mea-surement result ωj. It is usual to associate each input state

with a particular measurement result, so that when resultωj is obtained, we take this to imply that the state of thesystem was ρj . The probability that we are correct to doso is the quantity

P (ρj |ωj) =pjTr(ρjΠj)Tr(ρΠj)

, (4)

and is maximised by the maximum confidence measure-ment. This conditional probability is therefore interpretedas our confidence in identifying state ρj as a result of ob-taining outcome ωj . Before discussing how the maximumconfidence measurement is constructed, it is worth com-menting that, having identified this conditional probabil-ity as a quantity of interest, there are still several opti-mality conditions which may be applied to form distinctmeasurement strategies. Note that the average of thisquantity, over all measurement outcomes, is given by:

Pcorr =∑

j

P (ωj)P (ρj |ωj) =∑

j

P (ρj)P (ωj |ρj), (5)

and represents the overall probability of correctly iden-tifying the state, the figure of merit maximised by theminimum error measurement. Another possibility is tomaximise the smallest value of P (ρj |ωj) for a given setof states, i.e. apply a worst-case optimality condition [16].The maximum confidence measurement however, opti-mises this probability for each state in a set independently.Thus, for each input state ρj , we look for the positive op-erator Πj which maximises the conditional probability inequation (4). The operators {Πj} are treated as indepen-dent, essentially relaxing the completeness condition inequation (2). The result of this is that ultimately an incon-clusive outcome may be needed in order to form a physi-cally realisable measurement, but the remaining outcomeseach achieve the maximum possible value of P (ρj |ωj) forthe corresponding state ρj . Thus whenever outcome ωj isobtained, we can be as confident as possible that ρj wasindeed prepared. Note that as the operator Πj appears inthe denominator and the numerator, it can only be de-termined up to an arbitrary multiplicative factor. Thesefactors can always be chosen such that Π? ≥ 0, where

Π? = I −∑

j

Πj (6)

is the POM element corresponding to the inconclusive re-sult. Thus the problem is reduced to finding the positiveoperator Πj which maximises the right hand side of equa-tion (4).

As pointed out in [11], a closed-form solution for anarbitrary set of states is made possible by the ansatz:

Πj = cj ρ−1/2Qj ρ

−1/2, (7)

where Qj is a positive operator with unit trace, and hencecj represents the probability of occurrence of outcome ωj ,cj = P (ωj). With this definition equation (4) becomes

P (ρj |ωj) = pjTr(ρ−1/2ρj ρ−1/2Qj)

= pjTr(ρj ρ−1)Tr(ρ′jQj) (8)

S. Croke et al.: Maximum confidence measurements and their optical implementation 591

where ρ′j = ρ−1/2ρj ρ−1/2/Tr(ρj ρ

−1). ρ′j and Qj are bothpositive, trace-1 operators, and thus can be thought of asdensity operators. P (ρj |ωj) is therefore maximised if Qj

corresponds to the state with which ρ′j has largest overlap,i.e. is a projector onto the pure state

Qj = |λ′maxj 〉〈λ′max

j |, (9)

where |λ′maxj 〉 is the eigenket of ρ′j corresponding to its

largest eigenvalue λ′maxj . The limit is then given by

[P (ρj |ωj)]max = pjTr(ρj ρ−1)λ′max

j . (10)

We can also easily find the corresponding POM elementby applying the transformation in equation (7):

Πj = cj ρ−1/2|λ′max

j 〉〈λ′maxj |ρ−1/2. (11)

If the state ρj is pure then this simplifies to

Πj ∝ ρ−1ρj ρ−1. (12)

Although the optimal measurement is not uniquely de-fined — some arbitrariness remains in the choice of con-stants of proportionality — the maximum confidencestrategy is completely specified by the above. The strat-egy is defined by the criterion that whenever we iden-tify state ρj , we do so as confidently as is possible. Thefact that it is always possible to construct a measurementwhich achieves this maximum confidence for each statein a given set is precisely because the optimality of thePOM elements is independent of their respective weights.Furthermore, as the elements are treated completely inde-pendently in the derivation it is clear that the optimalityof a given element Πj is independent of how we chooseto construct the other elements in the POM to satisfythe completeness relation. Consider therefore the quan-tum state filtration problem — the question of whetherthe system is in state ρj or simply any one of the otherpossible states {ρi}, i �= j. This may seem less demandingthan the problem of discriminating between all the statesequally. Indeed in the minimum error approach, the prob-ability of making an error in the filtration problem canbe smaller than that in the corresponding discriminationproblem [17]. It is clear from the argument above however,that in the maximum confidence approach, the confidencein identifying state ρj cannot be increased by consideringthis more restrictive problem. The limit in equation (10)is dependent only on the geometry of the set. In this sensethe limit is a measure of how distinguishable ρj is in thegiven set.

If the states {ρi} are linearly independent, the limitis unity and the state can be distinguished without er-ror. Thus unambiguous discrimination is an example of amaximum confidence measurement. In unambiguous dis-crimination the constants of proportionality cj are oftenchosen to minimise the probability of occurrence of theinconclusive result, although this may not always be themost suitable criterion. For example, in the case of two

non-orthogonal states, if the a priori probability of occur-rence for one of the states is sufficiently small, the mea-surement which minimises the probability of obtaining theinconclusive result never identifies this state [30]. Thus ifwe wish to have at least some probability of identifyingeach state, we may be prepared to accept an inconclusiveoutcome which occurs more often than is strictly neces-sary. For a maximum confidence measurement to discrimi-nate between states of an arbitrary set, we also need someother criteria to choose the constants of proportionality. Insome cases, as with unambiguous discrimination, we maychoose to minimise the probability of occurrence of the in-conclusive outcome [11]. Indeed, for a linearly dependentset the number of operators Πj will be greater than thedimensionality of the space spanned by the states, andit may be possible to choose the constants such that aninconclusive outcome is not necessary. However, it is in-teresting to note that, unlike unambiguous discrimination,the quantity [P (ρj |ωj)]max will in general be different fordifferent states ρj . It may sometimes be useful to choosea measurement which only ever identifies the states forwhich this limit is greater than a certain threshold value.For example, it was pointed out by Sun et al. [18], that un-ambiguous discrimination is possible between a pure and amixed state of a two-level system (we shall refer to theseas ρpure and ρmixed respectively). This is achieved by avon Neumann measurement with outcomes correspondingto states along and orthogonal to ρpure. The result corre-sponding to a measurement along ρpure is interpreted asinconclusive, while the result corresponding to the stateorthogonal to ρpure tells us with certainty that the statewas ρmixed. Within the framework of maximum confidencemeasurements it is possible to construct a measurementwhich sometimes identifies ρmixed with certainty, some-times identifies ρpure as confidently as possible, and some-times gives an inconclusive result. However, although theprobability of occurrence of the inconclusive result maybe smaller in this case, we may wish to only allow unam-biguous results, and thus choose the former. Therefore thequestion of how to construct a maximum confidence strat-egy which is optimal for a given situation is dependent onthe situation and the information desired.

Finally in this section, we compare maximum confi-dence measurements and minimum error measurements.As already noted, the two strategies can be thought ofas applying a different optimality condition to the samequantity — the conditional probability P (ρj |ωj). Theminimum error measurement, however, optimises an av-erage over all measurement outcomes, and the POM el-ements must always form a complete set (clearly therewill never be an inconclusive outcome, as if this were thecase the expression in equation (5) could be increasedby the addition of a term of the form P (ω?)P (ρj |ω?)).As a result, finding the optimal minimum error measure-ment is in general a difficult problem. In fact, althoughthe necessary and sufficient conditions which the optimalmeasurement must satisfy are known [6,7], the measure-ment itself is known only in a limited number of spe-cial cases [5,7,31–34]. As we have seen, the maximum

592 The European Physical Journal D

confidence measurement will in general have an inconclu-sive outcome. In the special case where the maximum con-fidence is the same for all states in a set, and the POMelements in equation (11) form a complete set, it may beseen that the two strategies coincide. More generally, itis clear from examination of equation (5) that a limit tothis quantity is given by the largest value of the maximumconfidence for a given set.

In certain symmetric cases the square root measure-ment [31,35], given by

Πj = pj ρ−1/2ρj ρ

−1/2, (13)

is the optimal minimum error measurement. Interestingly,the maximum confidence POM elements have a similarform to the above, but the square root measurement hasthe advantage that it is always possible to form a POMfrom these elements as

∑

j

Πj = ρ−1/2ρρ−1/2 = I. (14)

It has proved useful, in the absence of a general formulafor the minimum error POM to investigate what informa-tion such a measurement can provide about a system inone of the states {ρj}. Indeed this measurement is consid-ered a ‘pretty good’ measurement even in cases where itis not optimal [34,35]. As the maximum confidence POMelements will not always form a complete measurement,elements of this form cannot in general be applied to theminimum error problem. The exception is when ρ ∝ I,and the states are all pure, in which case the square rootmeasurement and the maximum confidence measurementcoincide.

3 Experimental design

3.1 Outline of experiment

Our experiment was designed to discriminate with max-imum confidence between three equiprobable (pi =1/3, i = 0, 1, 2) symmetric polarisation states given byρi = |Ψi〉〈Ψi| where

|Ψ0〉 = cos θ|R〉 + sin θ|L〉,|Ψ1〉 = cos θ|R〉 + e2πi/3 sin θ|L〉,|Ψ2〉 = cos θ|R〉 + e−2πi/3 sin θ|L〉, (15)

and |R〉, |L〉 denote right and left circular polarisationstates respectively. This example was considered in [11]for a general two-level system with orthonormal base kets|0〉, |1〉. For our choice of |R〉, |L〉 as basis states, theinput states correspond to elliptical polarisations lyingon the same latitude of the Poincare sphere [36] (seeFig. 1). In terms of the horizontal and vertical polar-isations (|H〉, |V 〉), we define |R〉 = (i|H〉 + |V 〉)/√2,|L〉 = (|H〉 + i|V 〉)/√2.

We will first outline the optimal measurement, andthen discuss how this measurement can be implemented

R

H

V

LFig. 1. (Colour online) Poincare sphere representation ofstates. The north and south poles correspond to right (R) andleft (L) circular polarisation respectively, while linear polarisa-tions lie in the equatorial plane. The input states (red - lyingin the northern hemisphere) and the states associated withthe outcomes of the optimal maximum confidence (green - inthe southern hemisphere) and minimum error (blue - on theequatorial plane) POMs are shown.

optically. For this set the a priori density operator, ρ =cos2 θ|R〉〈R|+sin2 θ|L〉〈L|, is diagonal in the |R〉, |L〉 basis,and it is straightforward to calculate the optimal POMelements using equation (12). Thus the POM elementsachieving maximum confidence are given by Πi ∝ |φi〉〈φi|,where

|φ0〉 = sin θ|R〉 + cos θ|L〉,|φ1〉 = sin θ|R〉 + e2πi/3 cos θ|L〉,|φ2〉 = sin θ|R〉 + e−2πi/3 cos θ|L〉. (16)

The confidence that ρj was indeed present when outcomeωj is obtained may then be written:

P (ρj |ωj) =|〈φj |Ψj〉|2∑i |〈φj |Ψi〉|2 , (17)

and has the value 2/3 for all j = 0, 1, 2. The states |φi〉 arethe reflection of the input states in the equatorial planeof the Poincare sphere, and are shown in Figure 1. Thiscorresponds to a measurement with outcomes associatedwith elliptical polarisations with opposite handedness tothe input states.

Following [11], we choose the constants of proportion-ality such that the probability of occurence of the incon-clusive result is minimised. Thus our optical apparatuswas designed to implement the POM given by:

Πi = (3 cos2 θ)−1|φi〉〈φi|, i = 0, 1, 2,

Π? = (1 − tan2 θ)|R〉〈R|, (18)

S. Croke et al.: Maximum confidence measurements and their optical implementation 593

where Π? denotes the POM element associated with theinconclusive result. The probability of obtaining this resultis P (?) = cos 2θ. It can be seen that the inconclusive resultis equally likely to occur for any of the input states, andthus gives no information about the state.

For comparison purposes, we note that the minimumerror measurement for this set is given by the square rootmeasurement discussed above, and may be written Πi =(2/3)|φME

i 〉〈φMEi | where

|φME0 〉 =

1√2(|R〉 + |L〉)

|φME1 〉 =

1√2(|R〉 + e2πi/3|L〉)

|φME2 〉 =

1√2(|R〉 + e−2πi/3|L〉). (19)

These are the projections of the input states onto the equa-torial plane of the Poincare sphere, and are also shown inFigure 1. Thus the outcomes of this measurement are eachassociated with the linear polarisation specified by the ori-entation of the major axis of the polarisation ellipse of thecorresponding input state. For this measurement the con-fidence figure of merit may be written in a similar form tothat in equation (17) and is given by (1 + sin 2θ)/3 for allinput states.

The optical network used to realise this measurementis shown in Figure 2. Quarter waveplates (QWP) and halfwaveplates (HWP) are used to perform unitary transfor-mations, while polarising beamplitters (PBS) are used toseparate orthogonal modes. POMs require an expansion ofthe Hilbert space beyond that of the states of interest [29].The use of an interferometer in our set-up provides fourorthogonal modes which can then be separated at PBS3-5 to realise a four outcome measurement. Note that theinput states are states of a two-level system, and that theextra two orthogonal modes are provided by the vacuuminput to PBS2, and it is this that provides the Naimarkextension of the Hilbert space. The appropriate mixingof modes to perform the particular measurement we areinterested in is achieved using HWP4-7 and QWP2-3.

Our design groups together the four outputs in pairs,so that two orthogonal modes in output arm A corre-spond to results ω?, ω0, while two in arm B correspondto results ω1, ω2. Thus the interferometer in our set-upperforms the measurement described by the 2-elementPOM {Π? + Π0, Π1 + Π2}. The remaining unitary trans-formations and separation of modes necessary to per-form the full four outcome measurement are performed byHWP7, QWP3 and PBS4-5. The photodetectors PD0-2,?measure the number of photons in the two orthogonalmodes {H , V } incident on each detector. As one of thesemodes is always empty, the detectors realise the projec-tors πi = |Hi〉〈Hi| (i = 1, ?) and πi = |Vi〉〈Vi| (i = 0, 2).The mode transformation from the interferometer inputto these detector modes then means that |Hi〉 ∝ |φi〉.

PD0

QWP2HWP4

HWP5

HWP6

PBS2

PBS3

HWP7 QWP3

PBS4PBS5

PD? PD1

PD2

M2

M1

Arm A Arm B

Upperarm

Lowerarm

Fig. 2. Optical network to realise the maximum confidencemeasurement. PBS2–5 = polarising beamsplitters, HWP4–7 =zero-order half waveplates, QWP2–3 = zero-order quarterwaveplates, PD0–2, PD? = amplified photodiodes. Dotted linesrepresent the vacuum inputs to PBS2,4-5.

3.2 Performing a general two outcome measurement

Before discussing the details of our measurement, it isuseful to first consider how an arbitrary two outcomemeasurement on an input polarisation state may be imple-mented using linear optical elements. The simplest mea-surement of this kind is a von Neumann measurementwhich assigns the input state to one of two orthogonalmodes, and may be implemented using a polarising beam-splitter [22]. More general two outcome measurements arepossible however, and may be expressed in the POM for-malism by means of a two-element POM {ΠA, ΠB}. Anysuch POM may be written

ΠA = λ0|λ0〉〈λ0| + λ1|λ1〉〈λ1|,ΠB = (1 − λ0)|λ0〉〈λ0| + (1 − λ1)|λ1〉〈λ1| (20)

where 0 ≤ λ0, λ1 ≤ 1, and |λ0〉, |λ1〉 represent two or-thonormal polarisation states. λ0, λ1 (|λ0〉, |λ1〉) are theeigenvalues (eigenkets) of ΠA. Any such measurement maybe implemented using an interferometer, as shown in ourset-up. The measurement is completely defined by theprobabilities of obtaining outcomes ωA, ωB for any giveninput state, and therefore any optical network realisingthese probabilities is a valid implementation of the POM.

It is assumed that the transmitted and reflected beamsat each beamsplitter transit identical optical path lengths.

594 The European Physical Journal D

We also follow the convention that the orientations of thewaveplates are specified by the angle the fast axis makeswith the horizontal, measured anti-clockwise when viewedin the direction of propagation. The action of a half wave-plate at an angle A/2 may then be expressed using Jonesmatrix notation [37] as

Λ1/2(A/2)( |H〉|V 〉

)=

(cosA sin AsinA − cosA

) ( |H〉|V 〉

). (21)

Similarly, the action of the quarter waveplates in our ap-paratus, which are all oriented at either ±45◦, may beexpressed

Λ1/4(±45◦)( |H〉|V 〉

)=

1√2

(1 ±i±i 1

) ( |H〉|V 〉

). (22)

For simplicity, in this discussion we will consider pure in-put states, and assume for the moment that PBS2-3 trans-mit |λ0〉 and reflect |λ1〉. Thus for any given input state|Ψ〉, the components along |λ0〉 and |λ1〉 are separated atPBS2. HWP5-6 are used to rotate the polarisations in eacharm, so that the magnitude of |λ0〉 in the lower arm, and|λ1〉 in the upper arm are reduced by factors λ

1/20 , λ

1/21

respectively. Thus HWP5 should be oriented at γ0/2 =arccos(λ1/2

0 )/2 and HWP6 at γ1/2 = arccos(λ1/21 )/2. The

polarisations are then recombined at PBS3. Thus, the in-put state

|Ψ〉 = a0|λ0〉 + a1|λ1〉 (23)

where |a0|2 + |a1|2 = 1, evolves to

|Ψ〉 → a0|λ0L〉 + a1|λ1U 〉→ a0(λ

1/20 |λ0L〉 + (1 − λ0)1/2|λ1L〉)

+a1((1 − λ1)1/2|λ0U 〉 − λ1/21 |λ1U 〉)

→ a1eiφ(1 − λ1)1/2|λ0B〉 + a0(1 − λ0)1/2|λ1B〉

+a0λ1/20 |λ0A〉 − a1e

iφλ1/21 |λ1A〉 (24)

at PBS2, HWP5-6, PBS3 respectively, where we have al-lowed for a phase φ to be introduced between the arms ofthe interferometer. The subscripts U , L refer to the upperand lower arms of the interferometer, while the subscriptsA, B refer to the output arms A and B, as shown in Fig-ure 2. The probability that the photon is found in outputarm A or B is given by the modulus squared of the am-plitude of the state in each arm, and may be written:

P (A) = |a0|2λ0 + |a1|2λ1

P (B) = |a0|2(1 − λ0) + |a1|2(1 − λ1). (25)

Thus the measurement {ΠA, ΠB} is realised by this ap-paratus. Note that, perhaps surprisingly, the probabili-ties are independent of the phase of the interferometer,which only affects the polarisation of the output states,and not their magnitude. Any more complicated measure-ment may be realised as a series of measurements suchas this one. This design is similar to that of Ahnert andPayne [38], but has the advantage of requiring fewer opti-cal elements, and thus is easier to implement.

3.3 Experimental design

In our set-up, ΠA = Π?+Π0, ΠB = Π1+Π2. The operatorΠA is:

ΠA =13(I+2(1− tan2 θ)|R〉〈R|+tan θ(|R〉〈L|+ |L〉〈R|)).

(26)so that

ΠA

( |R〉|L〉

)=

(1 − 2

3 tan2 θ 13 tan θ

13 tan θ 1

3

) ( |R〉|L〉

). (27)

Diagonalising this gives

λ0,1 = (6 cos2 θ)−1(1 + 3 cos 2θ ±√

1 + 3 cos2 2θ)|λ0〉 = cosα|R〉 + sinα|L〉|λ1〉 = − sinα|R〉 + cosα|L〉, (28)

where cos 2α = 2 cos 2θ/√

1 + 3 cos2 2θ. The polarisingbeamsplitters in our experiment transmit horizontal po-larisation, |H〉 and reflect vertical polarisation |V 〉. ThusQWP2 (at 45◦) and HWP4 (at α/2) are used to performa unitary transformation from the |λ0〉, |λ1〉 basis to the|H〉, |V 〉 basis. It is easily verified by comparison of thematrix representations in the |H〉, |V 〉 basis that up to anoverall phase factor

Λ1/2(α/2)Λ1/4(45◦) = |H〉〈λ0| − |V 〉〈λ1|. (29)

Following the same reasoning as above we can see that theinterferometer in our set-up transforms the input polari-sation state |Ψ〉 as follows

|Ψ〉 → ( − eiφ(1 − λ1)1/2|HL〉〈λ1| + (1 − λ0)1/2|VL〉〈λ0|+ λ

1/20 |HU 〉〈λ0| + eiφλ

1/21 |VU 〉〈λ1|

)|Ψ〉. (30)

The two desired outcomes in each arm are then realised bysetting HWP7 to β/2 and QWP3 to 45◦, where cos 2β =(3 cos 2θ − 1)/

√1 + 3 cos2 2θ. Note that the phase of the

interferometer, which is set to π radians in our apparatus,determines the polarisation of the output state in eacharm of the interferometer, and therefore does affect theseresults. The input state is therefore now transformed to

|Ψ〉 → (1/

√2|HB〉((1 − λ1)1/2〈λ1| + i(1 − λ0)1/2〈λ0|)

+1/√

2|VB〉(i(1 − λ1)1/2〈λ1| + (1 − λ0)1/2〈λ0|)+|HA〉(λ1/2

0 cosβ〈λ0| − λ1/21 sinβ〈λ1|)

+|VA〉(λ1/20 sin β〈λ0| + λ

1/21 cosβ〈λ1|)

)|Ψ〉. (31)

After PBS4-5, the |HB〉, |VB〉 components can reach pho-todetectors PD1, PD2 respectively, while the |HA〉, |VA〉components reach PD? and PD0. Finally, a little algebraconfirms that the action of the entire apparatus on theinput state may be expressed as follows

|Ψ〉 → |PD1〉(1/√

3(tan θ〈R| + e−2πi/3〈L|)|Ψ〉)

+|PD2〉(1/√

3(tan θ〈R| + e2πi/3〈L|)|Ψ〉)

+|PD?〉(1 − tan2 θ)1/2〈R|Ψ〉+|PD0〉(1/

√3(tan θ〈R| + 〈L|)|Ψ〉). (32)

S. Croke et al.: Maximum confidence measurements and their optical implementation 595

where the photodetector state |PDi〉 represents the detec-tion of a photon at PDi. Thus

P (ω1|Ψ) = (3 cos2 θ)−1|〈φ1|Ψ〉|2P (ω2|Ψ) = (3 cos2 θ)−1|〈φ2|Ψ〉|2P (ω?|Ψ) = (1 − tan2 θ)|〈R|Ψ〉|2P (ω0|Ψ) = (3 cos2 θ)−1|〈φ0|Ψ〉|2, (33)

as required, and the apparatus realises the desired mea-surement.

3.4 Experimental details

The maximum confidence measurement outlined abovewas performed for 10 values of the parameter θ, rang-ing from 0◦ to 45◦ in 5◦ steps. The apparatus used inour experiment is shown in Figure 3. The light source wasan 810 nm laser diode (Laser 2000 PPMT model) run inCW mode, with a full width half maximum bandwidthof 0.88 nm and maximum power output of 20 mW. Thiswas coupled into a single mode fibre to act as a spatialfilter and ensure a well-controlled Gaussian spatial modealong the beam path. The output beam from the fibrewas collimated using an aspheric lens. This beam was ini-tially sent through a zero-order half waveplate (HWP1)followed by a polarising beamsplitter (PBS1) to provideclean horizontal input polarisation and to act as a variableattenuator.

Three waveplates were used to prepare each of theinput states in turn. HWP2 was oriented at θ/2 − 45◦,QWP1 at −45◦ for all input states, and HWP3 at −δ/4where δ = 0, 120◦,−120◦ for |Ψ0〉, |Ψ1〉 and |Ψ2〉 respec-tively. Again by comparing the matrix representation inthe |H〉, |V 〉 basis, it may be shown that

Λ1/2(−δ/4)Λ1/4(−45◦)Λ1/2(θ/2 − 45◦) =(cos θ|R〉 + eiδ sin θ|L〉)〈H | + (sin θ|R〉 − eiδ cos θ|L〉)〈V |,

(34)and therefore that this combination produces the desiredinput state when acting on the input |H〉. Thus θ wasset for each of the input states using HWP2, while thephase was set using HWP3. This arrangement allowed theinput to be switched easily between the three input states|Ψ0,1,2〉 by moving only one waveplate (HWP3).

In the measurement section of our apparatus, mir-ror M1 was mounted on a precision translation stage(Melles Griot Nanomax-TS 17MAX303), allowing the rel-ative phase between the arms of the interferometer tobe accurately varied. In order to minimise phase fluctu-ations, the interferometer was shielded from air currentsby a box, and, when operating in Mach-Zender configura-tion, the visibility of the interference fringes were regularlymeasured to be better than 99% over a period of at least10 minutes. It was also ascertained that the phase couldbe adjusted over a range greater than 2π radians withoutaffecting the quality of the interference fringes. The phasecould be most accurately controlled by adjusting the hor-izontal translation axis approximately parallel to the face

PD0

LD

SMF

HWP1 PBS1

HWP2QWP1HWP3

QWP2HWP4

HWP5

HWP6

PBS2

PBS3

HWP7 QWP3

PBS4PBS5

PD? PD1

PD2

M2

M1Phaseadjust

Source

State preparation

Measurement

Fig. 3. Optical network to realise the maximum confidencemeasurement. LD = Laser diode, SMF = single mode fibre,PBS1–5 = polarising beamsplitters, HWP1–7 = zero-order halfwaveplates, QWP1–3 = zero-order quarter waveplate, PD0–2,PD? = amplified photodiodes [27].

of M1 (to give much smaller changes in the position ofM1 for a given rotation of the adjustment knob than waspossible using the translation axis perpendicular to themirror face); therefore an extension was made for this ad-juster to allow the phase to be set from outside the boxcontaining the interferometer, minimising any disturbanceto the interferometer during phase adjustments.

The photodetectors used were amplified photodiodes(Thorlabs PDA520-EC) whose linearity and relative cal-ibration were measured over the relevant range of inci-dent power to be within 0.1% and 2.5% respectively. Theoutput voltages of the four photodetectors were displayedand recorded using a digital oscilloscope (LeCroy Wave-pro 7100). HWP1 was rotated to attenuate the power inthe transmitted arm of PBS1 to keep the photodetectorswithin their linear response range. All the waveplates usedwere calibrated to within 0.1◦ and were measured to pre-serve the purity of the polarisation to better than 1:2000.The waveplates were held in precision mounts that allowedthe angle of the optic axes to be set to within 0.1◦ witha repeatability of 0.1◦. The extinction ratios of the beam-splitters were measured to be approximately 1:200.

596 The European Physical Journal D

0 15 30 450

0.2

0.4

0.6

0.8

1.0P(ωj|ρ0)

θ (o)0 15 30 45

0

0.2

0.4

0.6

0.8

1.0P(ωj|ρ1)

θ (o)

0 15 30 450

0.2

0.4

0.6

0.8

1.0P(ωj|ρ2)

θ (o)

P(ρj|ωj)

15 30 45

0.2

0.4

0.6

00 θ (o)

Fig. 4. (Colour online) Graphs show experimental results for the normalised voltages at each detector (registering outputresult ωj) as a function of θ for input state: |Ψ0〉, |Ψ1〉 (top, l-r); |Ψ2〉, and the confidence figure of merit calculated from theseresults (bottom l-r). Also shown are the theoretical predictions without error (grey dashed lines), and those of a non-ideal modelwhich takes into account the errors introduced at beamsplitters PBS2 and PBS3 (coloured solid lines). Points correspond tothe experimental results, with red triangles, black squares, green circles, blue diamonds corresponding to PD0, PD?, PD1, PD2respectively. In the last plot red diamonds, green squares and blue triangles correspond to states |Ψ0〉, |Ψ1〉, |Ψ2〉 respectively,and the lines indicate the confidence figure of merit for both the maximum confidence (dotted) and minimum error (dashed)measurement strategies. The data show a clear improvement of the state discrimination over the minimum error case. The shadedregions indicate the range of values consistent with the non-ideal model, which is explained fully in the text. Experimental errors,due to fluctuations in the measured voltages, are smaller than the size of the data points.

For the intended measurement, the probability of ob-taining outcome ωj , j = 0, 1, 2, given input state ρi, i =0, 1, 2 may be expressed

P (ωj |ρi) = (3 cos2 θ)−1|〈φj |Ψi〉|2

= (1 − cos 2θ)(

16

+12δij

)(35)

where δij is the Kroeneker delta. Thus the two results forwhich j �= i have the same probability of occurrence, asexpected from the symmetry of the measurement. Whenthe apparatus was set up to perform the measurement,this property was used to set the phase of the interfer-ometer. In practice, at each value of θ, the state |Ψ1〉 wasinput, and the phase of the interferometer was set by ad-justing the position of M1 to minimise the difference be-tween the outputs at detectors PD0 and PD2. Once thephase was set, it was possible to cycle quickly through thethree input states by rotating HWP3. The output voltageswere recorded for 10 seconds for each input state. These

data were then averaged, and then normalised by divid-ing by the total voltage recorded at all detectors for eachinput state, so that the data could be interpreted as theprobability that input state ρi gives result ωj . The mea-surement outcomes depend only on the second order fieldcorrelation functions of the input light, as we look at theoutputs of individual detectors, and not correlations be-tween them. Thus there is no difference in the results fora one photon input or for classical light. What we are dis-criminating is the relative (complex) amplitudes of beamsin two input modes — the two orthogonal polarisations.In the case of weak classical light, this can be interpretedas the relative probability amplitudes of a single photondistributed between the two modes.

4 Results

The results of our experiment [27] are shown in Figure 4.The first three graphs show the normalised voltages at

S. Croke et al.: Maximum confidence measurements and their optical implementation 597

each detector for each input state |Ψ0〉, |Ψ1〉, |Ψ2〉. The the-oretical probabilities of obtaining outcome ωj, j = 0, 1, 2given input state ρi were given above in equation (35).The probability of obtaining outcome ω?, as discussedpreviously, is the same for all three input states in eachset, and equal to cos 2θ. These theoretical predictionsare shown alongside our experimental results. These datawere then used to calculate the confidence figure of merit,shown in the fourth graph in Figure 4. In the experimentalimplementation the confidence figure of merit representsthe proportion of the voltage recorded at detector PDi,i = 0, 1, 2, which was due to the corresponding input stateρi. Again the theoretical value of 2/3 is also shown on thisgraph. For comparison purposes we have also shown theconfidence achieved by the theoretically optimal minimumerror measurement. Experimental errors are due to fluctu-ations in the voltages over the 10 seconds for which theywere recorded, and are smaller than the size of the datapoints.

Modelling of the errors due to the different componentsof our apparatus showed that the largest errors were asso-ciated with the beamsplitters used in the interferometer,PBS2 and PBS3. Ideally, these beamsplitters transmit allincident |H〉 and reflect all incident |V 〉. In modelling theerror, we assumed that 0.5% of the intensity of the inci-dent light leaks into the wrong output port. This level oferror is based on our calibration data, and the polarisingbeamsplitters PBS2 and PBS3 are thus modelled by thefollowing operator:

PBS = |HL〉(ie−iχ√

0.005〈HL| +√

0.995〈HU |)+|VL〉(

√0.995〈VL| + ieiχ

√0.005〈VU |)

+|HU 〉(√

0.995〈HL| + ieiχ√

0.995〈HU |)+|VU 〉(ie−iχ

√0.005〈VL| +

√0.995〈VU |) (36)

where χ is the phase introduced between the intended anderroneous components. No phase information was avail-able, and this was therefore left as a parameter in ourmodel. The form of the above operator is chosen so thatfor the input in each arm the reflected and transmittedcomponents have the same phase, both for the intendedand erroneous components.

It was found that the predictions of the non-idealmodel for the probabilities of obtaining each outcome gavebest agreement with our data for χ = π/2, correspondingto the case in which all the transmission and reflection co-efficients are real. These predictions are shown alongsideour results in Figure 4. However there is little change inthe predictions of the non-ideal model for these probabil-ities for π/3 ≤ χ ≤ 2π/3. The predictions for the confi-dence figure of merit do vary within this range however,and the full range of values for which the non-ideal modelis consistent with our raw data are shown. The remainingdifferences between the model and our experimental re-sults are due to second order effects such as errors in thewaveplates and in setting the phase of the interferometer.Residual errors may also be due to non-pure input statesarising from the bandwidth of the input being greater thanthat of the detectors.

The results show a clear improvement in the confidencefigure of merit over the corresponding minimum error mea-surement. This is most evident in the range 10◦ ≤ θ ≤ 30◦.For θ larger than this range, this figure of merit is compa-rable for the two strategies. For θ smaller than this range,the input states are very close together, and distinguish-ing between them is more difficult. Experimental errorsare therefore more significant in this region, as can beseen from our non-ideal theory plot.

5 Conclusion

In the problem of discriminating between non-orthogonalquantum states, the maximum confidence measurementallows us to be as confident as possible that if we in-fer from the measurement result that a given state waspresent, that state was indeed present. This confidenceis maximised for all states in a given set, independent ofthe frequency with which any given result occurs. There-fore maximising the confidence measure does not define aunique strategy. Moreover, it is sometimes necessary to in-clude an inconclusive outcome in order to form a completemeasurement. The maximum confidence measurement hasthe advantage that it is possible to write down an analyticsolution for the optimal POM elements for an arbitrary setof states. This is not usually possible for other strategies.

We have demonstrated a maximum confidence mea-surement experimentally for a set of three equiprobablesymmetric polarisation states. Our results show an im-provement in the confidence figure of merit over the opti-mal minimum error measurement for the same set.

This work was supported by the EPSRC, and by the Syn-ergy fund of the Universities of Glasgow and Strathclyde.PJM acknowledges support by the National Science Founda-tion through its Information Technology Research (ITR) pro-gram, grant number PHY-0219460, via a collaboration withthe University of Oregon. IAW acknowledges support by theDARPA QuIST programme via a collaboration with Prince-ton University. We thank E. Riis for helpful discussions. IAWthanks R. Kosut and H. Rabitz for introducing him to the ideasof optimal detection measurements and for useful conversationson quantum state discrimination. SC and SMB would like tothank J. Jeffers, E. Andersson and C. Gilson with whom wehave enjoyed working on state discrimination problems.

References

1. C.H. Bennett, G. Brassard, Proceedings of the IEEEConference on Computers, Systems and Signal Processing(Bangalore, India, 1984), p. 175

2. C.H. Bennett, Phys. Rev. Lett. 68, 3121 (1992)3. C.A. Fuchs, Phys. Rev. Lett. 79, 1162 (1997)4. For a review of quantum state discrimination see A.

Chefles, Contemp. Phys. 41, 401 (2000)5. C.W. Helstrom, Quantum Detection and Estimation

Theory (Academic, New York, 1976)6. A.S. Holevo, J. Multivariate Anal. 3, 337 (1973)

598 The European Physical Journal D

7. H.P. Yuen, R.S. Kennedy, M. Lax, IEEE Trans. Inf. TheoryIT-21, 125 (1975)

8. I.D. Ivanovic, Phys. Lett. A 123, 257 (1987); D. Dieks,Phys. Lett. A 126, 303 (1988); A. Peres, Phys. Lett. A128, 19 (1988)

9. A. Peres, D. Terno, J. Phys. A 31, 7105 (1998)10. A. Chefles, Phys. Lett. A 239, 339 (1998)11. S. Croke, E. Andersson, S.M. Barnett, C.R. Gilson, J.

Jeffers, Phys. Rev. Lett. 96, 070401 (2006)12. E.B. Davies, IEEE Trans. Inf. Theory IT-24, 596 (1978)13. M. Sasaki, S.M. Barnett, R. Josza, M. Osaki, O. Hirota,

Phys. Rev. A 59, 3325 (1999)14. S.M. Barnett, C.R. Gilson, M. Sasaki, J. Phys. A 34, 6755

(2001)15. K. Hunter, E. Andersson, C.R. Gilson, S.M. Barnett, J.

Phys. A 36, 4159 (2003)16. R.L. Kosut, I.A. Walmsley, Y.C. Eldar, H. Rabitz, e-print

arXiv:quant-ph/0403150

17. U. Herzog, J.A. Bergou, Phys. Rev. A 65, 050305(R)(2002)

18. Y. Sun, J.A. Bergou, M. Hillery, Phys. Rev. A 66, 032315(2002)

19. J.A. Bergou, U. Herzog, M. Hillery, Phys. Rev. Lett. 90,257901 (2003)

20. For a review see S.M. Barnett, Quantum Inf. Comput. 4,450 (2004)

21. M. Mohseni, A.M. Steinberg, J.A. Bergou, Phys. Rev. Lett.93, 200403 (2004)

22. S.M. Barnett, E. Riis, J. Mod. Opt. 44, 1061 (1997)

23. R.B.M. Clarke, V.M. Kendon, A. Chefles, S.M. Barnett,E. Riis, M. Sasaki, Phys. Rev. A 64, 012303 (2001)

24. B. Huttner, A. Muller, J.D. Gautier, H. Zbinden, N. Gisin,Phys. Rev. A 54, 3783 (1996)

25. R.B.M. Clarke, A. Chefles, S.M. Barnett, E. Riis, Phys.Rev. A 63, 040305 (2001)

26. J. Mizuno, M. Fujiwara, M. Akiba, S.M. Barnett, M.Sasaki, Phys. Rev. A 65, 012315 (2002)

27. P.J. Mosley, S. Croke, I.A. Walmsley, S.M. Barnett, Phys.Rev. Lett. 97, 193601 (2006)

28. G.M. D’Ariano, M.F. Sacchi, J. Kahn, Phys. Rev. A 72,032310 (2005)

29. A. Peres, Quantum Theory: Concepts and Methods(Kluwer Academic Publishers, Dordrecht, 1993)

30. G. Jaeger, A. Shimony, Phys. Lett. A 197, 83 (1995)31. M. Ban, K. Kurokawa, R. Momose, O. Hirota, Int. J.

Theor. Phys. 36, 1269 (1997)32. S.M. Barnett, Phys. Rev. A 64, 030303 (2001)33. E. Andersson, S.M. Barnett, C.R. Gilson, K. Hunter, Phys.

Rev. A 65, 044307 (2002)34. Y.C. Eldar, G.D. Forney, IEEE Trans. Inf. Theory IT-47,

858 (2001)35. P. Hausladen, W.K. Wooters, J. Mod. Opt 41, 2385 (1994)36. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford,

1987)37. E. Hecht, Optics, 2nd edn. (Addison-Wesley, Reading, MA,

1987)38. S.E. Ahnert, M.C. Payne, Phys. Rev. A 71, 012330 (2005)