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  Fidelity and the communication of quantum information Stephen M. Barnett 1 , Claire R. Gilson 2 and Masahide Sasaki 3 , 4 1 Department of Physics and Applied Physics, University of Strathclyde, Glasgow G4 0NG, Scotland  2 Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland  3 Communications Research Laboratory, Koganei, Tokyo 184-8795, Japan  4 CREST, Japan Science and Technology  We compare and contrast the error probability and fidelity as measures of the quality of thereceiver’s measurement strategy for a quantum communications system. The error probability isa measure of the ability to retrieve  classical   information and the fidelity measures the retrieval of  quantum   information. We present the optimal measurement strategies for maximising the fidelitygiven a source that encodes information on the symmetric qubit-states.PACS numbers:03.67.-a, 03.65.Bz, 89.70.+c I. INTRODUCTION The principles governing the communication of information by a quantum channel are now well-known [1–3]. Thetransmiting party (Alice) selects from a set of signal states  | ψ j   and uses a string of these to encode her message.These states are known to the receiving party (Bob) who also knows the  a priori   probabilities  p j  for selection of eachof the signal states. Bob’s problem is to decide upon an optimal detection strategy. His choice of strategy will dependon the way in which the information he receives is to be used. In mathematical terms, Bob must choose a strategyso as to extremise some function of his measurement outcomes and commonly occurring examples are the minimumerror probability or minimum Bayes cost [1–5] and the accessible information [6–10]. These quantities determine thequality of Bob’s strategy for recovering the classical information associated with Alice’s selection of the transmittedstate.In this paper we will be concerned with a different measure of Bob’s detection strategy. This quantity, which werefer to as the fidelity, determines Bob’s ability to access the  quantum   information contained in Alice’s signal. Thefidelity depends on Bob’s choice of measurement strategy and also on his subsequent selection of a new quantumstate. The extent to which the selected state matches that chosen by Alice will determine Bob’s ability to reconstructthe selected quantum state. We will introduce the fidelity and compare its properties with those of the more familiarerror probability in the following section. At this stage, we can motivate our idea by considering the familiar problemof eavesdropping in quantum key distribution [11]. The error probability and fidelity relate, in this case, to the twoprincipal factors in assessing any eavesdropping strategy. The error probability is simply the probability that theeavesdropper will fail to learn the state selected by Alice, while the fidelity is the probability that the state selectedby the eavesdropper for transmission to Bob will appear to Bob as the state selected by Alice. In this way, errorprobability is related to the security of the classical information encoded by Alice and the fidelity is related to thelikelihood of escaping detection [12].We have not been able to find general criteria for maximising the fidelity. This maximum fidelity was introducedby Fuchs [13] who referred to it as the accessible fidelity. For a special class of qubit-states known as the symmetricstates, however, we have been able to derive the strategy that maximises the fidelity. The measurement part of theoptimal strategy is not unique, but includes the strategy that also minimises the error probability [5]. II. FIDELITY AND ERROR PROBABILITY In a quantum communications channel, Bob’s problem is to distinguish between the set of possible signal states, | ψ j  (  j  = 1 ,...M  ), that Alice may have sent. He does this by performing a measurement the results of which areassociated with the POM elements [1,14] ˆ π k . There is, of course, no particular reason for the number of possiblemeasurement outcomes to equal  M  , the number of possible signal states. The probability that Bob observes the result‘ k  ’ given that Alice selected the state  | ψ j   is P  ( k |  j ) =  ψ j | ˆ π k | ψ j  .  (1)If Bob wishes to determine the signal state then the probability that he will do so correctly is1  P  c  = M   j =1 P  (  j |  j )  p j  = M   j =1  ψ j | ˆ π j | ψ j   p j .  (2)This quantity is a measure of the success of Bob’s strategy at recovering Alice’s (classical) choice of signal state. Theerror probability is simply 1 − P  c : P  e  = 1 − M   j =1  ψ j | ˆ π j | ψ j   p j .  (3)Necessary and sufficient conditions are known for minimising  P  e  (or maximising  P  c ) [1–4] although very few explicitexamples of the required POM elements have been given. Some of these minimum error POMs have recently beenimplemented optically [15–17].The fidelity is more closely related to the retrieval of the quantum information ‘ | ψ j  ’. As a physical picture, considerBob to be operating some relay station in a communications channel. He must measure the signal and then, on thebasis of his measurement, he selects a state to retransmit. The fidelity is then a measure of how well the selectedstate matches the srcinal signal state selected by Alice. We can see this by considering one of the possible sequenceof events. Let us suppose that Alice has sent the signal state | ψ j   and that Bob measurement has given the result ‘ k  ’corresponding to the POM element ˆ π k . He then selects a state,  | φ k  , that depends on the measurement result, forretransmission. The simplest question that we can ask, to assess the retransmitted state, is “is this state | ψ j  ?”. Theprobability that this question will be answered in the affirmative is just the modulus squared overlap of the signalstate and the retransmitted state,  | ψ j | φ k | 2 . The  a priori   probability that the retransmitted state will pass this testis the fidelity F   = M   j =1  k | ψ j | φ k | 2  ψ j | ˆ π k | ψ j   p j .  (4)This quantity determines the quality of the measurement-retransmission strategy adopted by Bob. The strategyadopted by Bob depends on both his choice of measurement (associated with the POM elements ˆ π k ) and the selectionof the associated retransmission states ( | φ k  ). A large value of   F   corresponds to a good strategy while a smaller valueindicates a less good one. The strategy that best extracts the quantum information will be the one that gives themaximum fidelity. The general principles governing the maximum fidelity are unknown to us although the maximumfidelity, the associated measurement and retransmission states have been derived for a special case [13]. We willpresent strategies for maximising the fidelity for a wider set of possible signal states (the symmetric qubit-states) insection IV. III. SYMMETRIC STATES The symmetric states were introduced for the problems of state discrimination by Ban  et. al.  [5]. These states, | ψ j  , are generated from a single state,  | ψ 1  , by the action of a unitary operator ˆ V  : | ψ j  = ˆ V  j − 1 | ψ 1  .  (5)These  M   states are said to be symmetrical if they are  a priori   equally likely to have been selected andˆ V  M  = ˆ I   (6)so that  | ψ j + M   = | ψ j   [18].The minimum error probability occurs [5] if we adopt the so-called square-root measurement [19–21] for which the M   POM elements areˆ π k  = ˆΦ − 1 / 2 | ψ k  ψ k | ˆΦ − 1 / 2 ,  (7)whereˆΦ = M   j =1 | ψ j  ψ j | .  (8)2  The resulting minimum error probability is then P  mine  = 1 −| ψ 1 | ˆΦ − 1 / 2 | ψ 1 | 2 .  (9)In this paper we will obtain the maximum fidelity for any symmetric states of a single qubit. We can represent thesestates in terms of the orthonormal eigenstates,  |±  of the unitary operatorˆ V   = exp  i 2 πM   |−−|  .  (10)This operator clearly satisfies the requirement Eq. (6) for a symmetric set of states. Our  M  , equiprobable symmetricstates are | ψ j  = cos  θ 2  | +  + exp  i 2 πM   (  j − 1)  sin  θ 2  |− ,  (0 ≤ θ  ≤  π 2) .  (11)It is helpful to picture these states on the Bloch sphere (see Fig. 1). ψ  2 ψ  1  µ  1  µ  2  µ  Μ  +−   ψ  Μ  FIG. 1. The symmetric set of states  {| ψ j } . The square root measurement has POM elements that are proportional toprojectors onto the states  {| µ j } . This measurement minimises the average error probability for distinguishing between thestates. Each of the states is represented by a point on the surface of the sphere with the polar coordinates,  θ  and  φ ,corresponding to  θ  and 2 πj/M   respectively in (Eq. 11). The symmetric states lie on a single circle of the Blochsphere at the latitude  π/ 2 − θ . For this set of symmetric states the minimum error probability is obtained by meansof a POM with elementsˆ π j  = 2 M   | µ j  µ j | ,  (12)where | µ j  = 1 √  2  | +  + exp  i 2 πM   (  j − 1)  |−  .  (13)These states correspond to points on the equator of the Bloch sphere at the same longitude ( φ  - coordinate) as thecorresponding signal states  | ψ j   (see Fig. 1). The associated minimum error probability is P  mine  = 1 −  1 M   (1 + sin θ ) .  (14)3  As  θ  varies between 0 and  π/ 2 this error probability varies between 1 − 1 /M   and 1 − 2 /M  . These values correspond toguessing the value ‘  j  ’ when the states all correspond to the single ket | +  and the minimum attainable error probabilityfor symmetric states [4,16] which occurs when the symmetric states lie on the equator of the Bloch sphere.In the following section we establish the maximum fidelity attainable for this ensemble of states. IV. MAXIMUM FIDELITY In seeking to maximise the fidelity it is helpful to write it in the form [13] F   =  k  φ k | ˆ O k | φ k  ,  (15)where ˆ O k  is the Hermitian operatorˆ O k  = 1 M  M   j =1 | ψ j  ψ j | ˆ π k | ψ j  ψ j | .  (16)The selection of the retransmission states  | φ k   is now straightforward. The best state to select will be the eigenstateof  ˆ O k  having the largest eigenvalue and the corresponding maximum fidelity is the sum of the maximum eigenvaluesof the operators ˆ O k  [13].The problem of maximising  F   is now simply one of selecting the POM or POMs that produce the largest eigenvaluesum. Naturally, there are constraints associated with the fact that our POM elements must be Hermitian, positive-semidefinite and must sum to the identity. In seeking the optimal POM, it is sufficient to consider only rank-oneelements correponding to weighted projectors onto pure states [22]. The (rank-one) POM elements can be written inthe formˆ π k  = 2 w k  cos  θ k 2  | +  + e iφ k sin  θ k 2  |−  cos  θ k 2   + | + e − iφ k sin  θ k 2  −|   (17)or, more simply, as the matrixˆ π k  =  w k   1 + cos θ k  e − iφ k sin θ k e iφ k sin θ k  1 − cos θ k  ,  (18)where the basis states  | +   and  |−  correspond to the column vectors (1 , 0) T  and (0 , 1) T  respectively. Here,  w k  is aweight factor bounded by 0  ≤  w k  ≤  1. The requirement that the POM elements should sum to the identity placesrestrictions in the allowed values of the parameters  θ k ,  φ k  and  w k . These take the form:  k w k  = 1 ,  (19)  k w k  cos θ k  = 0 ,  (20)  k w k e iφ k sin θ k  = 0 .  (21)Our first task is to obtain the greater of the two eigenvalues for each of the operators ˆ O k . Evaluating the sum inEq. (16) and writing the resulting operator in matrix form givesˆ O k  =  w k 2   (1 + cos θ )(1 + cos θ cos θ k )  12  sin 2 θ sin θ k (e − iφ k + δ  M, 2 e iφ k ) 12  sin 2 θ sin θ k (e iφ k + δ  M, 2 e − iφ k ) (1 − cos θ )(1 + cos θ cos θ k )  ,  (22)where  δ  M, 2  is the usual Kronecker delta. We see that this matrix has one of two possible forms, one if   M >  2 andone if   M   = 2. It is simplest to deal these two cases separately.4
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