Relations between coherence and path information
Emilio Bagan, Janos A. Bergou, Seth S. Cottrell, Mark Hillery
aa r X i v : . [ qu a n t - ph ] D ec Relations between coherence and path information
Emilio Bagan , , J´anos A. Bergou , , Seth S. Cottrell , and Mark Hillery , Department of Physics and Astronomy, Hunter College of the CityUniversity of New York, 695 Park Avenue, New York, NY 10065 USA Graduate Center of the City University of New York, 365 Fifth Avenue, New York, NY 10016 F´ısica Te`orica: Informaci´o i Fen`omens Qu`antics,Universitat Aut`onoma de Barcelona, 08193 Bellaterra (Barcelona), Spain Department of Mathematics, New York City College of Technology, 300 Jay Street, Brooklyn, NY 11201
We find two relations between coherence and path-information in a multi-path interferometer.The first builds on earlier results for the two-path interferometer, which used minimum-error statediscrimination between detector states to provide the path information. For visibility, which wasused in the two-path case, we substitute a recently defined l measure of quantum coherence. Thesecond is an entropic relation in which the path information is characterized by the mutual infor-mation between the detector states and the outcome of the measurement performed on them, andthe coherence measure is one based on relative entropy. PACS numbers: 03.65.Ta,03.65.Yz
Recently a theory of quantum coherence as a resourcefor quantum information processing was proposed alongwith two possible coherence measures, an entropic mea-sure and an l measure [1]. This work has led to arenewed interest in the properties of quantum coher-ence [2, 3]. The l measure is of interest, because it is,in principle, observable. It depends on the magnitudesof off-diagonal density matrix elements, whose real andimaginary parts can be estimated. The entropic measureis the difference between the von Neumann entropies ofa density matrix and a diagonal density matrix formedfrom its diagonal elements. In this paper we focus on theduality relations between these coherence measures andthe which-path information for a particle going throughan interferometer with two or more internal paths.It is well known that a particle going through an inter-ferometer can exhibit wave or particle properties. Theparticle properties are characterized by how much in-formation one has about which path the particle tookthrough the device. The wave properties determine thevisibility of the interference pattern. There is an inverserelation between the particle and wave properties, thestronger one is the weaker is the other. This was studiedfor interferometers with two internal paths in a quanti-tative way by Wootters and Zurek [4]. The relation wasput into an elegant form by Greenberger and YaSin [5] D + V ≤ , (1)where D is a measure of path information and V is thevisibility of the interference pattern. This work was car-ried further by Jaeger, et al . [6], who proposed a possibledefinition for path information for interferometers withmore than two paths. Wootters and Zurek employed apath detector in their analysis but did not derive a re-lation of the form given in Eq. (1). In the subsequentwork [5, 6], which did derive a path-visibility relation,the path information is related not to information from a detector but to the preparation of the particle state, i.e.whether it is more likely to be in one path rather than inthe other. In a seminal study, Englert [7] combined theseapproaches. He introduced detectors into the problemin order to define the path information and derived arelation between this type of path information and thevisibility that took the form of Eq. (1). In his model, asystem of detectors, one in each path, is coupled to thepaths, so that when the particle passes through the in-terferometer correlations are produced between the pathstates and the detector states. Path information is thenrelated to the distinguishability of the detector states. Ifthe detector states are orthogonal, one has perfect pathinformation, but if they are not, then the informationone can obtain about the path is smaller.The first derivation of a path-visibility relation formore than two paths is due to D¨urr [8]. Expressing thedensity matrix of the particle inside the interferometerin a path basis, in which each path corresponds to oneof a set of orthonormal states, his measure of path infor-mation depended on the diagonal elements of the densitymatrix, and his measure of visibility depended on the off-diagonal elements. Some difficulties with the definitionintroduced by D¨urr were pointed out in [9] and this, inturn, following the earlier discussion in [10], led to properdefinitions of the quantities that are free of the difficul-ties [11] and also to some alternative definitions [12].We want to emphasize that in [8–12] an l measure ofcoherence was employed and, hence, the results presentedthere are relations between second moments that, in turn,are closely connected to uncertainty relations. As arguedin [1], a proper operational definition of coherence mustbe related to first moments, i.e., an l measure. Therethey defined the l coherence of a density matrix ρ to be C l ( ρ ) = N X i,j =1 i = j | ρ ij | . (2)Here we shall consider a normalized version of this quan-tity given by X = (1 /N ) C l ( ρ ), which has the propertythat 0 ≤ X ≤ ( N − /N . There is also an entropic mea-sure of coherence that satisfies the criteria in [1], which,as we shall see, also leads to a duality relation. Both def-initions of coherence are basis dependent. We will treatthe case of the l measure first.It is natural to search for a duality relation betweenpath coherence and path information. The first approachto this problem using the l measure and a detector ineach path was taken by Bera, et al. [13]. The discrim-ination of the detector states can be done in a numberof ways, and the fact that the states are not orthogo-nal means it cannot be done perfectly. Bera, et al . usedunambiguous discrimination, in which one never obtainsa wrong answer, but the procedure can sometimes failproviding no information about the detector state. Theyfound that the sum of the path coherence and an upperbound to the probability of successfully discriminatingthe detector states is less than or equal to one. Thisdoes not produce a relation of the form given in Eq. (1).Furthermore, unambiguous discrimination is not possiblewhen the detector states are linearly dependent. WhenEnglert derived his relation, which did take the form ofEq. (1), he used minimum-error state discrimination [7],which is always possible, even if the detector states arelinearly dependent. In this procedure, one always obtainsa result, but it can be wrong, though the probability ofmaking an error is minimized. The probability of suc-cessfully identifying the detector states, P s , quantifies theavailable path information via optimized measurements.In this letter we study the duality in the N -path inter-ferometer between coherence and path information. Ourfirst result is a relation between these two quantities thathas a form similar to Eq. (1), (cid:18) P s − N (cid:19) + X ≤ (cid:18) − N (cid:19) . (3)The reason P s − (1 /N ), rather than just P s , appears isthat it is the measure of how much better we can doby using prior information and detectors than by justguessing. With no prior information about the path andno detectors, we must assume that each path is equallylikely. Then, if we just guess the path our probability ofbeing right is 1 /N , which is the worst case scenario. Ifwe read out the detectors and use prior information, ourprobability of being right is P s . Note that if the detectorstates are orthogonal to each other, the two sides of theinequality are equal, so the inequality is tight. In order to derive (3) we start with a particle enteringan N -port interferometer via a generalized beamsplitterthat puts it in the superposition state | ψ i = N X i =1 √ p i | i i . (4)The orthonormal basis states, | i i , i = 1 , , . . . , N corre-spond to the N possible paths and span the N dimen-sional Hilbert space, H p . Equation (4) represents themost general state of the particle inside the interferome-ter.While in the interferometer, the particle interacts withanother system, called the detector. The detector startsin a global state | η i . The interaction of the particlewith the detector is described by the controlled unitary U ( | i i| η i ) = | i i| η i i , which entangles the path degree offreedom | i i of the particle with the detector state | η i i .After the particle has interacted with the detector, thestate of the entire system is | Ψ i = N X i =1 √ p i | i i| η i i . (5)Tracing out the detector, we find that the particle densitymatrix is given by ρ = Tr det ( | Ψ ih Ψ | ) = N X i,j =1 √ p i p j h η j | η i i | i ih j | , (6)which, in turn, yields for our coherence measure XX = 1 N C l ( ρ ) = 1 N N X i,j =1 i = j √ p i p j |h η j | η i i| . (7)Since path information is encoded in the detectorstates we also need to introduce the detector density ma-trix, ρ det . Tracing out the particle states, we find ρ det = Tr particle ( | Ψ ih Ψ | ) = N X i =1 p i ρ i , (8)where ρ i = | η i ih η i | . In order to obtain which-path infor-mation, we need to discriminate among the states {| η i i} .To this end, we will employ the minimum-error strat-egy. For N states, we have an N -element POVM withelements Π i ≥
0, which satisfy P Ni =1 Π i = I . The proba-bility that if we are given the state | η j i detector i clicksis h η j | Π i | η j i . We identify a click in detector i with thedetection of the state | η i i , so the average probability ofsuccessfully identifying the state is P s = N X i =1 p i h η i | Π i | η i i = N X i =1 p i Tr(Π i ρ i ) . (9)In minimum-error state discrimination, we seek to find aPOVM that maximizes P s . The solution to the problemis known in complete generality for two states [14], butonly in special cases for more than two states. Here weshall employ an upper bound on the success probabilityto obtain our main result.There are several upper bounds on the success prob-ability for minimum-error state discrimination [17–21].However, for our goals we find that another one, whichwe first state and later prove, is more useful. If we have N density matrices, { ρ i | j = 1 , , . . . N } , where ρ i ap-pears with probability p i , then the success probabilityfor minimum-error state discrimination obeys P s ≤ N + 12 N N X i,j =1 k Λ ij k , (10)where Λ ij = p i ρ i − p j ρ j is the Helstrom matrix of thepair of states ρ i , ρ j , and the norm in this inequality isthe trace norm. In the case of pure states, ρ i = | η i ih η i | ,we find, by diagonalizing the operator Λ ij = p i | η i ih η i | − p j | η j ih η j | , that k Λ ij k = 2 s(cid:18) p i + p j (cid:19) − p i p j |h η i | η j i| . (11)This implies that the average probability of success-fully identifying the detector state, entangled with agiven path, is bounded above by Eq. (10), with k Λ ij k given by (11). The quantity X , which describes the co-herence, is given in Eq. (7). This gives us the upperbound for the expression on the left-hand side of Eq. (3), (cid:18) P s − N (cid:19) + X ≤ N N X i,j =1 i = j N X k,l =1 k = l (cid:16) k Λ ij k k Λ kl k + √ p i p j |h η i | η j i|√ p k p l |h η k | η l i| (cid:17) . (12)For fixed i and j , the pair ( k Λ ij k , √ p i p j |h η i | η j i| ) canbe viewed as a bra vector h v ij | of length ( p i + p j ) / k and l ). The term in parentheses inthe r.h.s. of (12) is the scalar product, h v ij | v kl i , of twosuch vectors. Using the Schwarz inequality, the r.h.s. canbe bounded above by1 N N X i,j =1 j = k p i + p j ! = (cid:18) − N (cid:19) , (13)and we recover Eq. (3).We now give the proof of Eq. (10). The success prob-ability of the N -element POVM was introduced in (9).An upper bound for the individual terms in the success probability can be found as p i Tr(Π i ρ i ) = p j Tr(Π i ρ j ) + Tr (Π i Λ ij ) ≤ p j Tr(Π i ρ j ) + max ≤ Π ≤ I Tr (ΠΛ ij )= p j Tr(Π i ρ j ) + Tr (Λ ij, + )= p j Tr(Π i ρ j ) + p i − p j + k Λ ij k , (14)where the subscript “+” stands for positive part of the op-erator, i.e. if P + is the projection onto the space of eigen-vectors of the hermitian operator Λ ij with positive eigen-values, then Λ ij, + = P + Λ ij P + . Similarly, if P − is the pro-jection onto the space of eigenvectors with non-positiveeigenvalues, then we define Λ ij, − = P − Λ ij P − . For theinequality in the second line, we have used the fact thatfor Λ ij , one has Tr(Π i Λ ij ) ≤ max ≤ Π ≤ I Tr(ΠΛ ij ), sincethe maximization is over the set of positive operators lessthan the identity, which contains Π i . The equality in thethird line results from choosing Π to be the projectoronto the positive part of Λ ij and noting that Tr(ΠΛ ij ) ≤ Tr(ΠΛ ij, + ) ≤ Tr(Λ ij, + ) for a positive operator Π withoperator norm less than or equal to 1. The equalityin the last line uses the fact that Tr(Λ ij ) = p i − p j =Tr(Λ ij, + )+Tr(Λ ij, − ), and k Λ ij k = Tr(Λ ij, + ) − Tr(Λ ij, − )from where Tr(Λ ij, + ) = ( p i − p j + k Λ ij k ) / i, j ; i = j of both sides in the in-equality (14), we find( N − P s ≤ − P s + 12 N X j,k =1 j = k k Λ ij k . (15)Noting that k Λ ij k = 0 for i = j , the sum in the lastterm can be extended to include the i = j terms, im-mediately yielding Eq. (10). We want to point out thatfor N = 2, Eq. (10) is actually an equality, it reproducesthe Helstrom bound. Our bound generalizes this for ar-bitrary N . The first term on the r.h.s. of (10) would bethe result of pure guessing, so the second term can beregarded as the gain provided by the measurement thattakes into account the available prior information (prob-ability with which an individual detector states occursand the overlaps of the states).It is also possible to derive a duality relation using theentropic definition of coherence. The relative entropycoherence measure for a density matrix ρ is given by C rel ent ( ρ ) = S ( ρ diag ) − S ( ρ ) , (16)where ρ diag is a diagonal density matrix in the specifiedbasis whose diagonal elements are the same as those of ρ ,and S denotes the von Neumann entropy, with the log-arithms taken base 2. In our case the relevant densitymatrix is given by Eq. (6). This gives us C rel ent ( ρ ) = H ( { p j } ) − S ( ρ ) , (17)where H ( { p j } ) = − P Nj =1 p j log p j is the Shannon en-tropy.For path information we can consider the mutual in-formation between the detector states labeling the pathsand the results of probing them. The detector den-sity matrix was introduced in (8), so ρ i appears witha probability of p i . Let D be a random variable cor-responding to the choice of detector state; it takes thevalue i ∈ { , , . . . N } , corresponding to ρ i , with proba-bility p i . We probe the detector states with a POVM, M = { Π i | i = 1 , . . . N } in order to identify them, andthereby identify the path. Let the random variable cor-responding to the measurement result be M . It takesvalues in the set { , , . . . N } , with i corresponding to thedetection of the state ρ i . The joint distribution for thetwo variables is given by p ( M = i, D = j ) = Tr(Π i ρ j ) p j .Note that this situation is analogous to one in which Alicesends the state ρ i with a probability of p i to Bob, and Bobperforms a state discrimination measurement in order todetermine what state he received. We will quantify thepath information by the mutual information, H ( M : D ).If the random variables are perfectly correlated this is H ( D ) = H ( { p i } ), while if they are uncorrelated this isequal to zero.In this situation we can make use of the Holevobound [22], which states that if Alice sends ρ i with prob-ability p i , and Bob measures the state he receives, then H ( M : D ) ≤ S ( ρ det ) − X i p i S ( ρ i ) . (18)We recall that in our case ρ i = | η i ih η i | , which meansthat the second term above is zero so that H ( M : D ) ≤ S ( ρ det ). Now ρ and ρ det are reduced density matrices ofthe same pure state [recall (6) and (8)], and, therefore, S ( ρ ) = S ( ρ det ). Consequently, we have that C rel ent ( ρ ) + H ( M : D ) ≤ H ( { p i } ) , (19)which is an entropic version of the coherence-path-information duality relation. The relation is tight, be-cause the bound is attained when the detector states areorthogonal.Since the bound (19) holds for any measurement,it also holds for the accessible information, defined asAcc( D ) = max M H ( M : D ), where the maximization isover all POVMs. Thus, we can also write C rel ent ( ρ ) + Acc( D ) ≤ H ( { p i } ) . We also note that our bound based on the l coherencemeasure holds for any deterministic discrimination pro-tocol, for which all outcomes give a conclusive answerabout the identity of the detector states, not just for theoptimal measurement, with minimum discrimination er-ror.In summary, we have derived two relations relating thepath information about a particle inside a multi-path in- terferometer to two recently defined measures of the co-herence of a quantum system. The first of these providesa generalization of the visibility-path-information rela-tion derived by Englert for the two path case. Previousstudies used a number of different quantities as multi-path generalizations of the visibility, but our results heresuggest that the recently defined l and entropic coher-ence measures are strong candidates. Acknowledgment . This publication was made possiblethrough the support of a Grant from the John TempletonFoundation. The opinions expressed in this publicationare those of the authors and do not necessarily reflect theviews of the John Templeton Foundation. Partial finan-cial support by a Grant from PSC-CUNY is also grate-fully acknowledged. The research of EB was addition-ally supported by the Spanish MICINN, through contractFIS2013-40627-P, the Generalitat de Catalunya CIRIT,contract 2014SGR-966, and ERDF: European RegionalDevelopment Fund. EB also thanks Hunter College forthe hospitality extended to him during his research stay. [1] T. 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