Minimum guesswork discrimination between quantum states
aa r X i v : . [ qu a n t - ph ] O c t Minimum guesswork discriminationbetween quantum states
Weien Chen ∗ , Yongzhi Cao , Hanpin Wang , and Yuan Feng † Institute of Software, School of EECS, Peking University, China Key Laboratory of HCST, Ministry of Education, China Centre for QCIS, University of Technology, Sydney, Australia
July 4, 2018
Abstract
Error probability is a popular and well-studied optimization cri-terion in discriminating non-orthogonal quantum states. It capturesthe threat from an adversary who can only query the actual stateonce. However, when the adversary is able to use a brute-force strat-egy to query the state, discrimination measurement with minimumerror probability does not necessarily minimize the number of queriesto get the actual state. In light of this, we take Massey’s guesswork asthe underlying optimization criterion and study the problem of min-imum guesswork discrimination. We show that this problem can bereduced to a semidefinite programming problem. Necessary and suf-ficient conditions when a measurement achieves minimum guessworkare presented. We also reveal the relation between minimum guess-work and minimum error probability. We show that the two criteriagenerally disagree with each other, except for the special case withtwo states. Both upper and lower information-theoretic bounds onminimum guesswork are given. For geometrically uniform quantumstates, we provide sufficient conditions when a measurement achievesminimum guesswork. Moreover, we give the necessary and sufficientcondition under which making no measurement at all would be theoptimal strategy.
Keywords: quantum state discrimination, error probability, guesswork,brute-force strategy, information flow ∗ [email protected] † [email protected] Introduction
Since Helstrom’s pioneering work on quantum binary decision problem [1],quantum state discrimination has been extensively studied [2, 3, 1, 4, 5, 6, 7,8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. The problem is usually described asa protocol between two parties, conventionally named Alice and Bob. Aliceselects a quantum state ρ from a set { ρ i } according to a probability distri-bution { p i } and gives it to Bob. We assume that Bob knows both the setof possible states and their associated probabilities. His aim is to identifythe actual prepared state. To this end, Bob performs some quantum mea-surement on ρ in order to extract information about the index i . This givesrise to an optimization problem with regard to Bob’s choice of measurement.A number of criteria have been considered to concretize the meaning of thisoptimality [1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20], amongwhich error probability and the Shannon mutual information are two repre-sentatives. While the former has led to a research line known as minimumerror discrimination (MED) [14, 15, 16, 17, 18, 19], the later corresponds tothe study of accessible information [21, 22, 7, 8, 9]. Interestingly, as an alter-native to MED, an unambiguous (error-less) scheme of state discriminationhas been proposed, by allowing certain fraction of inconclusive measurementoutcomes [10, 11, 12].We point out that quantum state discrimination can be seen as a spe-cial case of quantitative information flow (QIF) analysis, which has been anactive topic in security community during the last decades [23, 24, 25, 26,27, 28, 29, 30, 31, 32]. In QIF analysis, the aim is to quantify the amountof information leaked by a covert channel from a high-level entity, whose se-cret information (e.g., a password) is mathematically described as a randomvariable X with alphabet { x i } and the associated probability distribution { p ( x i ) } , to a low-level entity, whose partial information about X is describedas another random variable Y with alphabet { y j } . The correlation between X and Y is determined by the channel matrix { p ( y j | x i ) } of the covert chan-nel. To put quantum state discrimination in the context of QIF analysis, wemay view Alice as the high-level entity and Bob the low-level entity. Theonly restriction is that the correlation between these two entities are ruledby quantum mechanics: Alice encodes her classical secret messages { x i } intoquantum states { ρ x i } ; Bob performs a measurement on Alice’s prepared state ρ x to get information about X and stores his measurement outcome in Y ;the channel matrix is then given by the Born rule [33], p ( y j | x i ) = Tr( ρ x i π y j ),where π y j is the measurement operator corresponding to the outcome y j .In the literature of QIF analysis, researchers have proposed different fig-ures of merit to quantify how successfully a low-level entity Bob can identify2he secret value of X given knowledge about Y , according to different adver-sarial strategies which Bob may adopt. In particular, it is well-known that er-ror probability, guesswork, and the Shannon entropy deal with one-shot strat-egy, brute-force strategy, and subset membership strategy, respectively, andthus play important and complementary roles in QIF analysis [34, 35, 36]. Inthe quantum setting, it is clear that one-shot strategy and subset membershipstrategy have been considered. Error probability and the Shannon entropyhave been widely studied in quantum information theory, and led a largeamount of research on, besides MED and accessible information discussedabove, quantum source coding [5, 37], quantum channel capacity [38, 39, 40],etc. However, to the best of our knowledge, no work has addressed brute-force strategy in the context of quantum state discrimination.The above observation motivates us to consider Massey’s guesswork [41]as the optimization criterion in quantum state discrimination. We name thenew problem minimum guesswork discrimination (MGD). In contrast to theMED scenario where Bob has only one chance to ask Alice “is ρ x = ρ x i ” forsome x i chosen based on his measurement outcome, in this study Bob carriesout multiple such queries until hitting Alice’s prepared state ρ x . Guesswork,the new criterion, quantifies the expected number of queries that Bob needsto make. We hope our preliminary step towards the study of brute-forcestrategy in the quantum setting will initiate a sibling direction of MED.The rest of this paper is organized as follows. In Section 2, we first reviewthe classical guessing problem, then extend it to the quantum setting. Weshow that MGD can be reduced to a semidefinite programming (SDP) prob-lem, and present necessary and sufficient conditions which must be satisfiedby the optimal measurement to achieve minimum guesswork. Section 3 is de-voted to the relation between the minimum error criterion and the minimumguesswork criterion. We provide both upper and lower information-theoreticbounds on minimum guesswork in Section 4, and sufficient conditions when ameasurement achieves minimum guesswork for geometrically uniform statesin Section 5. In Section 6, we answer the question “when would making nomeasurement at all be the optimal strategy?” by a sufficient and necessarycondition on the quantum ensemble. We discuss several other interesting is-sues worthy of consideration in Section 7 and conclude this work in Section 8. We first review the classical guessing problem. Then its quantum variantcan be simply formalized by instantiating the classical problem with quan-tum information and mechanics. Suppose that Alice has a discrete random3ariable X with alphabet X = { x i : 1 ≤ i ≤ n } and the associated probabil-ity distribution { Pr( X = x i ) , p ( x i ) : 1 ≤ i ≤ n } . Bob, who knows both thealphabet and the distribution, aims to identify the true value of X by keepingon asking questions of the form “is X = x i ?” until getting the answer “yes”.How many guesses is he expected to make? Massey [41] observed that Bob’soptimal strategy for minimizing his work is to arrange his queries accordingto the non-increasing order of probabilities p ( x i )’s. Formally, the guesswork of Bob is given by G(X) , n X i =1 σ ( i ) p ( x i ) , (1)where σ is a permutation on the index set { , · · · , n } such that p ( x i ) ≥ p ( x j )implies σ ( i ) ≤ σ ( j ). Recall that a permutation on set S is just an one-to-one mapping from S to itself. Here, σ represents formally Bob’s guessingstrategy: he guesses x i in his σ ( i )th query. The guesswork G( X ) quantifiesthe expected number of queries that Bob needs to guess the actual value of X when applying the strategy σ . The constraint on σ ensures its optimalityin that any other permutation yields greater or equal guesswork.The above definition of guesswork has been generalized to conditionalversion [42], which is more appealing in practice. In addition to Alice’salphabet X and the prior distribution on it, Bob may possess some extraknowledge (or side information) of X . In general, we assume a channelexisting between Alice and Bob with input set X and output set Y = { y j :1 ≤ j ≤ m } . The probabilistic behavior of this channel is characterized inthe standard way, i.e., by conditional probabilities { p ( y j | x i ) : 1 ≤ i ≤ n, ≤ j ≤ m } . Consequently, given some fixed input random variable X of Alice,we can derive an output random variable Y on Bob’s side with the associateddistribution obtained by p ( y j ) = n X i =1 p ( x i ) p ( y j | x i ) , for each 1 ≤ j ≤ m . As usual, we denote the joint distribution of X and Y by { p ( x i , y j ) } .Now, instead of consulting the prior distribution of X which leads to theguesswork G( X ), Bob applies an optimal guessing strategy on each poste-rior distribution { p ( x i | y j ) : 1 ≤ i ≤ n } when y j is observed. We denoteeach corresponding posterior guesswork by G( X | Y = y j ). Bob’s conditional4uesswork is then given byG( X | Y ) , m X j =1 p ( y j )G( X | Y = y j )= m X j =1 p ( y j ) n X i =1 σ j ( i ) p ( x i | y j ) , (2)where each σ j is a permutation on { , · · · , n } such that p ( x i | y j ) ≥ p ( x i ′ | y j )implies σ j ( i ) ≤ σ j ( i ′ ). In [42], Arikan showed that extra knowledge alwaysreduces (at least preserves) guesswork, i.e., the inequality G( X | Y ) ≤ G( X )holds.We now define the quantum guessing problem by extending the abovescenario to the quantum setting. Alice selects from her alphabet a secretmessage x i with probability p ( x i ) and encodes it into a (possibly mixed)quantum state ρ x i , which is accessible to Bob. Alice’s operation gives riseto an ensemble of quantum states E = { ( p ( x i ) , ρ x i ) } living in a finite di-mensional Hilbert space H . We call this E a quantum encoding of X . Inorder to identify Alice’s secret message, Bob performs on the quantum statea positive operator-valued measure (POVM) Π = { π y j : 1 ≤ j ≤ m } , whichcomprises m positive semidefinite (PSD) operators satisfying the complete-ness condition P mj =1 π y j = I H , where I H is the identity matrix in H . Theprobability p ( y j | x i ) that Bob obtains the j th measurement outcome whenAlice chooses the i th message is given by Tr( ρ x i π y j ). Note that the randomvariable Y is completely determined by the ensemble E and the POVM Π.Hence, minimizing over all possible POVMs leads to the following definitionof minimum guesswork : G opt ( E ) , min Π ∈M G( X | Y ) , (3)where M is the set of all POVMs. We name this minimization problem minimum guesswork discrimination (MGD).For convenience of the following reasoning, we introduce some notations.Let P be the set of all non-zero PSD operators and P be the set of all non-zero rank-one PSD operators. A complete POVM is a POVM comprisingonly rank-one PSD operators. The set of all complete POVMs is denotedby M c . Given π ∈ P , the random variable X π which takes value from thealphabet of X is defined byPr( X π = x i ) , p ( x i )Tr( ρ x i π ) P ni =1 p ( x i )Tr( ρ x i π ) . X π describes Bob’s posterior distribution over Alice’s messageswhen he obtains the outcome indicated by the measurement operator π . Withthis notation, the guesswork G( X | Y ) can be rewritten as P mj =1 p ( y j )G( X π yj )for some E and Π as given in the preceding paragraph. Sometimes, we writeG( X | Π) instead of G( X | Y ) to indicate the specific POVM adopted by Bob.In the following, we give several alternative characterizations of G opt ( E ).The first one states that the optimal POVM achieving G opt ( E ) can always betaken as a complete measurement. Proposition 1.
Let E be a quantum encoding of a random variable X , and G opt ( E ) be defined in Eq.(3). It holds that G opt ( E ) = min Π ∈M c G( X | Y ) . Proof.
Since G( X | Y ) can be rewritten as P mj =1 p ( y j )G( X π yj ) for a POVMΠ = { π y j } , it is sufficient to prove that for any π y j ∈ Π the guesswork G( X | Y )cannot be increased by splitting π y j into two measurement operators π y j ′ and π y j ′′ such that π y j = π y j ′ + π y j ′′ . Formally, we have the following inference: p ( y j )G( X π yj ) = p ( y j ) n X i =1 σ j ( i ) p ( x i )Tr( ρ x i π y j ) p ( y j )= n X i =1 σ j ( i ) p ( x i )Tr( ρ x i π y j ′ )+ n X i =1 σ j ( i ) p ( x i )Tr( ρ x i π y j ′′ ) ≥ n X i =1 σ j ′ ( i ) p ( x i )Tr( ρ x i π y j ′ )+ n X i =1 σ j ′′ ( i ) p ( x i )Tr( ρ x i π y j ′′ )= p ( y j ′ )G( X π yj ′ ) + p ( y j ′′ )G( X π yj ′′ ) , (4)where p ( y j ) = P ni =1 p ( x i )Tr( ρ x i π y j ) and the inequality is due to the fact thatthe permutation σ j may not necessarily be optimal with respect to X π yj ′ or X π yj ′′ . Therefore, from an optimal POVM Π which achieves G opt ( E ), one canalways derive a complete POVM which also achieves G opt ( E ).Another alternative characterization of G opt ( E ) is suggested by the min-imum error criterion used in MED. As shown in Remark 1 in Section 3,6o minimize error probability it suffices to consider POVMs Π’s which sat-isfy | Π | = n when E comprises n states. Here, | Π | denotes the number ofmeasurement operators in Π. Similarly, in MGD, the following simple obser-vation shows an n ! upper bound on | Π | in minimizing guesswork. Suppose π and π are two measurement operators in Π. If the two posterior distri-butions induced by π and π achieve their minimum guesswork G( X π ) andG( X π ) by the same optimal permutation σ , then merging π and π intoa single measurement operator π + π would not change the overall perfor-mance of Π. Since there are exactly n ! different permutations on the indexset { , , · · · , n } , we can reduce | Π | to n ! for an optimal POVM Π. Proposition 2.
Let E be a quantum encoding of a random variable X , and G opt ( E ) be defined in Eq.(3). It holds that G opt ( E ) = min Π ∈M n ! G( X | Y ) , where M n ! is the set of all POVMs consisting of exactly n ! measurementoperators. Based on Eldar et al.’s analogous results in MED [43], we reduce MGD toan SDP problem, which has numerical solutions within any desired accuracyin mathematics, and derive necessary and sufficient conditions satisfied bythe optimal POVM to achieve minimum guesswork. We present the resultsbelow and omit the proof which is simply an imitation of the reasoning in [43].
Proposition 3.
Let E be a quantum encoding of a random variable X , and G opt ( E ) be defined in Eq.(3). It holds that G opt ( E ) = max A Tr( A ) , where A ranges over all Hermitian operators in H satisfying A ≤ P ni =1 σ ( i ) p ( x i ) ρ x i for any permutation σ on { , · · · , n } . Proposition 4.
Let E be a quantum encoding of a random variable X , and G opt ( E ) be defined in Eq.(3). A POVM { π y , π y , · · · , π y n ! } achieves G opt ( E ) if and only if for any permutation σ on { , · · · , n } it holds that n X i =1 n ! X j =1 σ j ( i ) p ( x i ) ρ x i π y j ≤ n X i =1 σ ( i ) p ( x i ) ρ x i .
7t is worth noting that Proposition 4 can also be proved directly usingthe technique introduced in [44].
Given two random variables X and Y , the unconditional and conditionalerror probabilities of guessing the value of X are defined respectively byP err ( X ) , − max ≤ i ≤ n p ( x i ) , (5)and P err ( X | Y ) , m X j =1 p ( y j )P err ( X | Y = y j )= 1 − m X j =1 p ( y j ) max ≤ i ≤ n p ( x i | y j ) . (6)The underlying observation here is that to minimize the error probability,Bob should always guess the most probable message according to his prioror posterior distributions. In addition, if X and Y are correlated by somequantum encoding E and some POVM Π as in the preceding section, Bob’serror probability can be written asP err ( X | Y ) = 1 − m X j =1 max ≤ i ≤ n p ( x i )Tr( ρ x i π y j ) , because of the fact that p ( y j | x i ) = Tr( ρ x i π y j ). The definition of minimumerror probability in MED is then given byP opterr ( E ) , min Π ∈M P err ( X | Y ) . (7) Remark . It is worth noting that in the literature of MED, error probabilityis usually formulated byP ′ err ( X | Y ) , − n X i =1 p ( x i )Tr( ρ x i π y i ) , n measurement op-erators. It is easy to show that this simplified characterization is equivalentto P err ( X | Y ) in the sense thatmin Π ∈M n P ′ err ( X | Y ) = min Π ∈M P err ( X | Y ) , where M n is the set of all POVMs consisting of exactly n measurementoperators.Now, we are ready to investigate the relation between G opt ( E ) and P opterr ( E ).To this end, we start by examining the two notions G( X ) and P err ( X ) inclassic setting. Lemma 1 ([45], Lemma 2.4) . Let G( X ) and P err ( X ) be defined in Eq.(1)and Eq.(5), respectively. It holds that G( X ) ≤ n · P err ( X ) + 1 . (8)If we assume without loss of generality that p ( x ) = max i p ( x i ), thenEq.(8) achieves equality when p ( x i ) = (1 − p ( x )) / ( n −
1) for 2 ≤ i ≤ n . Itis straightforward to prove the conditional counterpart of Eq.(8). Corollary 1.
Let G( X | Y ) and P err ( X | Y ) be defined in Eq.(2) and Eq.(6),respectively. It holds that G( X | Y ) ≤ n · P err ( X | Y ) + 1 . Interestingly, guesswork can also be lower bounded in terms of error prob-ability.
Lemma 2.
Let G( X ) and P err ( X ) be defined in Eq.(1) and Eq.(5), respec-tively. It holds that G( X ) ≥ − P err ( X )) + 12 , (9) with the equality achieved when k probabilities in { p ( x i ) } are equal to /k forsome integer k (1 ≤ k ≤ n ) and the other probabilities all equal to zero. roof. Without loss of generality, we assume that p ( x ) ≥ · · · ≥ p ( x n ).Given p ( x ) fixed, in order to minimize G( X ) we need to require as manyprobabilities p ( x i )’s being equal to p ( x ) as possible. It follows thatG( X ) ≥ k X i =1 i · p ( x ) + ( k + 1) · (1 − p ( x ) · k )= − p ( x )2 k − p ( x )2 k + k + 1 ≥ p ( x ) + 12= 12(1 − P err ( X )) + 12 , where k = j p ( x ) k . The only non-trivial part of the above reasoning is thesecond inequality. To prove it, let t = 1 /p ( x ) − k . Then, it is equivalent toprove that − k k + t ) − k k + t ) + k + 1 ≥ k + t , which can be further simplified to t ≥ t . Since 0 ≤ t <
1, we conclude theproof. When t = 0, thus j p ( x ) k = p ( x ) , the equality is achieved.Again, we derive the conditional counterpart of Eq.(9). Corollary 2.
Let G( X | Y ) and P err ( X | Y ) be defined in Eq.(2) and Eq.(6),respectively. It holds that G( X | Y ) ≥ − P err ( X | Y )) + 12 . (10) Proof.
We have the following inference:G( X | Y ) = m X j =1 p ( y j )G( X | Y = y j ) ≥ m X j =1 p ( y j )( 12(1 − P err ( X | Y = y j )) + 12 ) ≥ − P mj =1 p ( y j )P err ( X | Y = y j )) + 12= 12(1 − P err ( X | Y )) + 12 , where the second inequality is from Jensen’s inequality.10e observe that there is an even stronger connection between guessworkand error probability for the special case where n = 2. Lemma 3.
If the alphabet of X comprises exactly two elements, i.e., n = 2 ,then it holds that(i) G( X ) = P err ( X ) + 1 ;(ii) G( X | Y ) = P err ( X | Y ) + 1 . We omit the proof of this lemma, because it is easy from the definitions ofguesswork and error probability. Based on the preceding results, we obtainthe relation between G opt ( E ) and P opterr ( E ) as follows. Theorem 1.
Let E be a quantum encoding of a random variable X , and G opt ( E ) and P opterr ( E ) be defined in Eq.(3) and Eq.(7), respectively. It holdsthat − P opterr ( E )) + 12 ≤ G opt ( E ) ≤ n opterr ( E ) + 1 (11) and if n = 2 , G opt ( E ) = P opterr ( E ) + 1 . (12) Proof.
We prove the left inequality in Eq.(11). Let Π be an optimal POVMachieving G opt ( E ) and Y the corresponding random variable induced by E and Π. Applying Eq.(10), we have the following inference:G opt ( E ) = G( X | Y ) ≥ − P err ( X | Y )) + 12 ≥ − P opterr ( E )) + 12 . Eq.(12) and the right inequality in Eq.(11) follows from similar reasoning butusing optimal POVM achieving P opterr ( E ) instead.This theorem states that the minimum error criterion coincides with theminimum guesswork criterion in the context of two state discrimination.However, regarding to general cases, the two criteria may not agree with eachother, though there still exists a weak correlation between them as shown inEq.(11). The following example shows their disagreement when three statesare presented. 11 xample 1. We consider the so-called trine ensemble [46, 47], which consistsof three equiprobable pure qubit states (living in a 2-dimensional Hilbertspace), given by | φ x i = | i , | φ x i = − | i + √ | i , | φ x i = − | i − √ | i . For the trine ensemble, the 3-component POVM, denoted by Π E , with eachoperator given by π y i = 23 ρ x i = 23 | φ x i ih φ x i | , achieves the minimum error with P err ( X | Π E ) = 1 /
3. This POVM is knownas the square-root measurement [46]. On the other hand, in Section 5 we willshow that the POVM Π G = { / | ψ x k ih ψ x k | : 1 ≤ k ≤ } with | ψ x k i = cos( 2 kπ − π
12 ) | i + sin( 2 kπ − π
12 ) | i achieves the minimum guesswork with G( X | Π G ) = 2 − √ /
3. It is also easyto verify that P err ( X | Π G ) = 23 − √ > P err ( X | Π E ) , G( X | Π E ) = 32 > G( X | Π G ) , where we write Π E or Π G for Y to avoid confusion. Indeed, by observing theproof of the optimality of Π G in Section 5, we can conclude that for the trineensemble there does not exist any POVM which can achieve both minimumerror and minimum guesswork. As a matter of fact, the optimization problem in MED is usually hard tosolve analytically. Closed-form result or optimal measurement is only knownfor some special quantum systems, e.g., the case with exactly two states [1],12quiprobable symmetric states [16], or multiply symmetric states [17]. Webelieve that it is also the case for MGD, because of the analogy between thesetwo problems. In light of this, upper/lower bounds on minimum guessworkare as desirable as those on minimum error [48, 49, 50, 51, 52]. In what fol-lows, we start by reviewing some existing upper/lower bounds on guessworkin classic setting. We then combine them with the celebrated
Holevo bound and the less well-known subentropy bound on accessible information, resultingin upper/lower bounds on minimum guesswork in the quantum setting.In [41], Massey proved the following lower bound on guesswork G( X ). Lemma 4 ([41]) . Let G( X ) be defined in Eq.(1). Provided H( X ) ≥ , itholds that G( X ) ≥ · H( X ) + 1 . Note that H( X ) , − P ni =1 p ( x i ) log p ( x i ) is the Shannon entropy. Thelogarithm is taken with base 2. Let us also recall the conditional Shannonentropy: H( X | Y ) , m X j =1 p ( y j )H( X | Y = y j ) , = − m X j =1 p ( y j ) n X i =1 p ( x i | y j ) log p ( x i | y j ) . Based on Massey’s result, we derive a similar lower bound on conditionalguesswork.
Corollary 3.
Let G( X | Y ) be defined in Eq.(2). Provided H( X | Y = y j ) ≥ for each y j , it holds that G( X | Y ) ≥ · H( X | Y ) + 1 . (13)13 roof. We have the following inference:G( X | Y ) = m X j =1 p ( y j )G( X | Y = y j ) ≥ m X j =1 p ( y j )( 14 · H( X | Y = y j ) + 1) ≥ · P mj =1 p ( y j )H( X | Y = y j ) + 1= 14 · H( X | Y ) + 1 , where the second inequality is from Jensen’s inequality.Upper bound on guesswork in terms of the Shannon entropy also exists. Lemma 5 ([53]) . Let G( X ) be defined in Eq.(1). It holds that G( X ) ≤ n −
12 log n H( X ) + 1 . (14)Again, we present the conditional counterpart of Eq.(14). Corollary 4.
Let G( X | Y ) be defined in Eq.(2). It holds that G( X | Y ) ≤ n −
12 log n H( X | Y ) + 1 . (15) Proof. G( X | Y ) = m X j =1 p ( y j )G( X | Y = y j ) ≤ m X j =1 p ( y j )( n −
12 log n H( X | Y = y j ) + 1)= n −
12 log n H( X | Y ) + 1 . E be aquantum encoding of a random variable X , and Y be the random variableinduced by E and some POVM Π as described in Section 2. The accessibleinformation I acc ( E ) of the ensemble E is defined as the maximum mutualinformation between X and Y obtainable via varying the POVM Π:I acc ( E ) , max Π ∈M I( X : Y ) , where I( X : Y ) = H( X ) − H( X | Y ). Although accessible information isdifficult to characterize analytically, various upper and lower bounds havebeen found. In a celebrated paper [21], Holevo bounded I acc ( E ) as follows:I acc ( E ) ≤ χ ( E ) , S( n X i =1 p ( x i ) ρ x i ) − n X i =1 p ( x i )S( ρ x i ) , (16)where the von Neumann entropy S( ρ ) of quantum state ρ is given by S( ρ ) , − Tr( ρ log ρ ) which is a natural extension of the Shannon entropy. The quan-tity χ ( E ) is usually referred to as the Holevo information of the ensemble E .Interestingly, accessible information can also be lower bounded by a quantitywhich has an analogous form to the Holevo information. With the notion of subentropy which is defined byQ( ρ ) , − X k Y l = k λ k λ k − λ l λ k log λ k , with λ k ’s being the eigenvalues of the state ρ , Jozsa, Robb and Wootters [22]obtained the following inequality:I acc ( E ) ≥ Λ( E ) , Q( n X i =1 p ( x i ) ρ x i ) − n X i =1 p ( x i )Q( ρ x i ) . (17)Combining Eq.(13) and Eq.(16), we obtain the following lower bound onminimum guesswork. Theorem 2.
Let E be a quantum encoding of a random variable X and G opt ( E ) be defined in Eq.(3). Provided H( X π ) ≥ for any π ∈ P , it holdsthat G opt ( E ) ≥ · H( X ) − χ ( E ) + 1 . (18)15 roof. Since I acc ( E ) = H( X ) − min Π ∈M H( X | Y ) ≤ χ ( E ) , we have that min Π ∈M H( X | Y ) ≥ H( X ) − χ ( E ) . (19)Let Π be an optimal POVM achieving G opt ( E ). Due to the guarantee thatH( X π ) ≥ π ∈ P , we are safe to apply Eq.(13):G( X | Y ) ≥ · H( X | Y ) + 1 , where Y is the random variable induced by E and Π. Then, Eq.(19) andEq.(13) together imply thatG opt ( E ) ≥ · H( X ) − χ ( E ) + 1 , as required.From the above proof, we see that the equality in Eq.(18) holds if and onlyif there exists an optimal POVM Π achieving G opt ( E ) such that G( X | Y ) = · H( X | Y ) + 1 and H( X | Y ) = H( X ) − χ ( E ). To satisfy the first equation,we have to require G( X π ) = · H( X π ) + 1 for each π ∈ Π (see the proof ofCorollary 3). Consequently, the alphabet of X must be countably infiniteand each X π must obey the geometric distribution { , , · · · } [41]. A trivialcase is when X π are identical, e.g.,Pr( X π = x i ) = 12 i , i = 1 , , · · · , for each π ∈ Π. In this case, we have that H( X ) = H( X | Y ) = G( X | Y ) = 2,which further implies that χ ( E ) = 0 as required by the second equation.Therefore, Alice’s choice of E has to be a trivial encoding with all the statesbeing identical! Indeed, this special system satisfies the “no-measurement”condition discussed in Section 6. That is, Bob cannot decrease his priorguesswork G( X ) by applying any measurement.Nonetheless, there also exists non-trivial quantum encoding E satisfyingthe equality condition of Eq.(18). We give an example below. Let X be arandom variable with the associated distribution given by p ( x ) = p ( x ) = 38 ,p ( x i ) = 12 i , i = 3 , , · · · . E of X : ρ x = 23 | ih | + 13 | ih | ,ρ x = 13 | ih | + 23 | ih | ,ρ x i = 12 | ih | + 12 | ih | , i = 3 , , · · · . It is straightforward to verify that H( X π ) ≥ π ∈ P . Then, with thePOVM Π = { π y = | ih | , π y = | ih |} , we see that both X π y and X π y obeythe geometric distribution { , , · · · } , and thus H( X | Y ) = G( X | Y ) = 2.On the other hand, we can calculate that H( X ) = 13 / − / χ ( E ) = 5 / − /
4. Hence, it holds that H( X | Y ) = H( X ) − χ ( E ) asrequired by the equality condition of Eq.(18). Remark . We note that in order to apply Eq.(18) we need to verify inadvance the precondition H( X π ) ≥ π . This con-straint, which limits the applicability of the bound, originates from the onein Lemma 4. Intuitively, it can only be fulfilled by quantum states whosespanning spaces overlap to a certain extent. We give an example where thisconstraint holds. Consider a 5-dimensional Hilbert space H with an orthog-onal basis {| i , · · · , | i} . Let E = { (1 / , ρ x i ) : 1 ≤ i ≤ } be a quantumencoding of some random variable X , with ρ x i defined by ρ x i = 14 ( I − | i ih i | ) , where I is the identity operator on H . For any π ∈ P , we can calculate thatH( X π ) = − P i =1 p i log p i with p i = 14 (1 − h i | π | i i Tr( π ) ) . It is easy to verify that H( X π ) ≥ Theorem 3.
Let E be a quantum encoding of a random variable X and G opt ( E ) be defined in Eq.(3). It holds that G opt ( E ) ≤ n −
12 log n (H( X ) − Λ( E )) + 1 . (20)17 roof. Since I acc ( E ) = H( X ) − min Π ∈M H( X | Y ) ≥ Λ( E ) , we have that min Π ∈M H( X | Y ) ≤ H( X ) − Λ( E ) . Consequently, there must exist some Π and Y such that H( X | Y ) ≤ H( X ) − Λ( E ). Applying Eq.(15), we obtain the following inference:G opt ( E ) ≤ G( X | Y ) ≤ n −
12 log n H( X | Y ) + 1 ≤ n −
12 log n (H( X ) − Λ( E )) + 1 . (21)Let us examine when the inequality in Eq.(20) is saturated. First, it isrequired that I acc ( E ) = Λ( E ). Otherwise, there must exist a random variable Y induced by E and some POVM Π such that H( X | Y ) < H( X ) − Λ( E ), whichfurther implies that G opt ( E ) is strictly less than the RHS term in Eq.(20).According to the discussion in [22], the lower bound Λ( E ) on the accessibleinformation of E can only be achieved by the so-called Scrooge ensemble ortrivially by an ensemble with all the states being identical. It was showedthat the amount of information that we can obtain from the Scrooge ensembleby performing some complete POVM is independent of the choice of POVM.Second, for any complete POVM Π and the corresponding random vari-able Y induced by E and Π, we require that G( X | Y ) = n −
12 log n H( X | Y ) + 1 tosaturate the second inequality in Eq.(21). It then follows from the proof ofCorollary 4 that G( X π ) = n −
12 log n H( X π ) + 1 for any π ∈ P . On the other hand,the equality in Eq.(14) holds if and only if X obeys either the uniform distri-bution, i.e., p ( x i ) = 1 /n for each 1 ≤ i ≤ n , or trivially a point distribution,i.e., p ( x i ) = 1 for some 1 ≤ i ≤ n [53]. As a consequence, all the quantumstates in E must be either mutually orthogonal or identical. Since mutu-ally orthogonal states cannot form a Scrooge ensemble, we conclude that theequality in Eq.(20) holds if and only if all the states in E are identical. In this section, we consider MGD for a special type of symmetric quan-tum states, the geometrically uniform states [13]. We provide sufficient con-ditions when some geometrically uniform measurement achieves minimum18uesswork. With this technique, we are able to prove the optimality of thePOVM Π G , as given in Example 1, for the trine ensemble.Let G = { U i : 1 ≤ i ≤ n } be a finite group which contains n unitaryoperators. The identity element of G and the inverse of U i is denoted by I and U † i , respectively. An ensemble of states E = { ( p ( x i ) , ρ x i ) : 1 ≤ i ≤ n } is called geometrically uniform if ρ x i ’s can be generated by a state ρ anda group G such that ρ x i = U i ρ U † i . It is also assumed that p ( x i ) = 1 /n , asrequired by symmetry. Similarly, we can define such symmetry for quantummeasurements. A POVM Π = { π y i : 1 ≤ i ≤ n } is called geometricallyuniform if the measurement operators are generated by some π and G suchthat π y i = U i π U † i . In [18], Eldar, Megretski, and Verghese showed thatthe square-root measurement of a geometrically uniform ensemble is alsogeometrically uniform, and that there always exists a geometrically uniformmeasurement which achieves the minimum error for this ensemble.We state our sufficient conditions as follows. Given a unitary operator V and a POVM Π = { π y i : 1 ≤ i ≤ n } , we use V Π V † to denote the new POVM { V π y i V † : 1 ≤ i ≤ n } . Theorem 4.
Let E = { ( p ( x i ) , ρ x i ) : 1 ≤ i ≤ n } be a quantum encoding of arandom variable X such that E is geometrically uniform being generated bya state ρ and a group G = { U i : 1 ≤ i ≤ n } . Given a geometrically uniformPOVM Π = { π y i : 1 ≤ i ≤ n } , which is generated by π and G , if there existsa unitary operator V such that(i) V U i = U i V , for each ≤ i ≤ n , and(ii) G( X V π V † ) = min π ∈P G( X π ) ,then V Π V † achieves the minimum guesswork for E , i.e., G opt ( E ) = G( X | V Π V † ) .Proof. The proof of this theorem consists of two steps. First, we show thatthe minimum of G( X π ) over all π ∈ P can always be obtained in P ; thatis, min π ∈P G( X π ) = min π ∈P G( X π ). To see this, consider Eq.(4) used in theproof of Proposition 1. For any π y , π y , π y ∈ P such that π y = π y + π y ,we have proved that p ( y )G( X π y ) ≥ p ( y )G( X π y ) + p ( y )G( X π y ) , where p ( y j ) = P ni =1 p ( x i )Tr( ρ x i π y j ). Since p ( y ) = p ( y ) + p ( y ), it musthold either G( X π y ) ≥ G( X π y ) or G( X π y ) ≥ G( X π y ). Consequently, onlyrank-one PSD operators need to be considered in minimizing G( X π ).Second, recall that G( X | Y ) can be written as P nj =1 p ( y j )G( X π yj ). Pro-vided the conditions (i) and (ii) are satisfied, we prove the optimality of the19OVM V Π V † by showing that G( X V π yj V † ) = G( X V π V † ) for each 1 ≤ j ≤ n .As the identity matrix I is contained in G , without loss of generality weassume that U = I and thus π y = π . It holds thatG( X V π yj V † ) = P ni =1 σ j ( i )Tr( ρ x i V π y j V † ) P ni =1 Tr( ρ x i V π y j V † )= P ni =1 σ j ( i )Tr( U i ρ U † i V U j π U † j V † ) P ni =1 Tr( U i ρ U † i V U j π U † j V † )= P ni =1 σ j ( i )Tr( U † j U i ρ U † i U j V π V † ) P ni =1 Tr( U † j U i ρ U † i U j V π V † )= P ni =1 σ ( i )Tr( U i ρ U † i V π y V † ) P ni =1 Tr( U i ρ U † i V π y V † )= G( X V π y V † ) = G( X V π V † ) , where σ j is an optimal permutation with respect to X V π yj V † for 1 ≤ j ≤ n .In this reasoning, the third equation follows from the condition (i) and thefourth is due to that { U † j U i : 1 ≤ i ≤ n } = G for each 1 ≤ j ≤ n . Then, thecondition (ii) implies the optimality of V Π V † .It is worth noting that the above theorem just reduces the general min-imization problem into another restricted one, i.e., the term min π ∈P G( X π )in the condition (ii), which in itself is not easy to solve.Nevertheless, this theorem suffices to prove that the POVM Π G in Exam-ple 1 is optimal for the trine ensemble. It can be easily verified that the trineensemble and the POVM Π E are geometrically uniform. The three statesof the ensemble can be obtained from one another by a rotation R y (4 π/ y -axis of the Bloch sphere, with R y ( θ ) defined as R y ( θ ) = (cid:20) cos θ − sin θ sin θ cos θ (cid:21) , where 0 ≤ θ < π . The POVM Π G can be obtained from Π E in a similar way:Π G = R y ( π/ E R y ( π/ † . Since rotations about the same axis commute,i.e., R y ( α ) R y ( β ) = R y ( β ) R y ( α ), the condition (i) of Theorem 4 is satisfied.It remains to verify that the measurement operator R y ( π/ | ih | R y ( π/ † actually achieves min π ∈P G( X π ). Let π ′ = | w ih w | be an arbitrary rank-onePSD operator on a 2-dimensional Hilbert space, | w i can be given as | w i = cos α | i + e iβ sin α | i , ≤ α, β < π . To calculate the value of G( X π ′ ), we need the followingprobabilities: Pr( X π ′ = x ) = 13 (1 + cos 2 α ) , Pr( X π ′ = x ) = 16 (2 − cos 2 α − √ α cos β ) , Pr( X π ′ = x ) = 16 (2 − cos 2 α + √ α cos β ) . There are two cases of the value of G( X π ′ ): ( − √ sin 2 α cos β, if √ α ≤ sin 2 α cos β, − cos 2 α − √ sin 2 α cos β, otherwise.For the first case, G( X π ′ ) achieves the minimum value 2 −√ / α = π/ β = 0. For the second case, G( X π ′ ) achieves the same minimum valuewith α = π/
12 and β = 0. Since π ′ = R y ( π/ | ih | R y ( π/ † when α = π/ β = 0, it follows that Π G is optimal for the trine ensemble. So far, we have focused on the adversarial viewpoint taken by Bob aiming tominimize his guesswork. In this section, let us take the protective viewpointof Alice, whose task is to choose an optimal encoding ensemble to maximizeBob’s minimum guesswork. Similar max-min problem has been addressed inthe context of MED [54]. Formally, we are concerned with the following goal:max E G opt ( E ) . Since it always holds that G( X | Y ) ≤ G( X ) [42], we simply know thatmax E G opt ( E ) = max E min Π G( X | Y ) ≤ G( X ) . In what follows, we show that for any classical information source X , Alicecan always find a quantum encoding E which renders Bob’s measurementuseless in the sense that no measurement can reduce his prior guesswork,i.e., max E G opt ( E ) = G( X ). 21e start from the proof of G( X | Y ) ≤ G( X ):G( X | Y ) = m X j =1 p ( y j ) n X i =1 σ j ( i ) p ( x i | y j ) ≤ m X j =1 p ( y j ) n X i =1 σ ( i ) p ( x i | y j )= n X i =1 σ ( i ) m X j =1 p ( x i , y j )= n X i =1 σ ( i ) p ( x i ) = G( X ) , where the inequality is due to the fact that substituting σ for σ j can onlyincrease (at best preserve) the posterior guesswork G( X | Y = y j ). Moreover,the equality is achieved if and only if each prior optimal permutation σ isoptimal with respect to any posterior distribution { p ( x i | y j ) : 1 ≤ i ≤ n } .(Since it may be the case that p ( x i ) = p ( x j ) for some 1 ≤ i = j ≤ n , theremay exist more than one optimal permutations achieving the prior minimumguesswork G( X ).) Now, suppose that E = { ( p ( x i ) , ρ x i ) } is a quantum encod-ing of X such that G opt ( E ) = G( X ). As we need to range over all POVMs inobtaining G opt ( E ), the aforementioned equality condition can be restated as ∀ ≤ i, j ≤ n, π ∈ P .p ( x i ) ≥ p ( x j ) ⇒ p ( x i )Tr( ρ x i π ) ≥ p ( x j )Tr( ρ x j π ) , which is equivalent to ∀ ≤ i, j ≤ n. p ( x i ) ≥ p ( x j ) ⇒ p ( x i ) ρ x i ≥ p ( x j ) ρ x j . We formalize our result as the following theorem.
Theorem 5.
Let E = { ( p ( x i ) , ρ x i ) : 1 ≤ i ≤ n } be a quantum encoding of arandom variable X . Let G( X ) and G opt ( E ) be defined in Eq.(1) and Eq.(3),respectively. Then G opt ( E ) = G( X ) holds if and only if for any ≤ i, j ≤ n the following condition holds: p ( x i ) ≥ p ( x j ) ⇒ p ( x i ) ρ x i ≥ p ( x j ) ρ x j . Consequently, it holds that max E G opt ( E ) = G( X ) for any random variable X . A direct corollary of the theorem is that, for a uniformly distributedvariable X , a quantum encoding E achieving G opt ( E ) = G( X ) must be formedby identical states. 22 Discussion
By Proposition 2, we have shown that the optimal POVM Π achievingminimum guesswork can always be taken as an n !-component measurementwhen the ensemble E comprises n states. Still, better upper bound on | Π | may be possible, especially for cases where the quantum states have certaintype of symmetry. Recall that in Example 1 we have found the optimalPOVM Π G , which consists of three elements, for the trine ensemble.In Section 4, we combined information-theoretic bounds on guessworkG( X | Y ) in classic setting and bounds on accessible information I acc ( E ) in thequantum setting to generate bounds on minimum guesswork G opt ( E ). Thereare alternative approaches to deriving bounds on minimum guesswork. Recallthat in Section 3 we bounded G opt ( E ) in terms of P opterr ( E ) in both directions(see Eq.(11)). Thus, we can substitute existing bounds on minimum errorprobability [48, 49, 50, 51, 52] for P opterr ( E ) in Eq.(11) to generate bounds onG opt ( E ). Also, we can relate G opt ( E ) to the inconclusive probability p inc ofan unambiguous discrimination scheme (if applicable) in the following way:G opt ( E ) ≤ (1 − p inc ) + n + 12 p inc = 1 + n − p inc , where n is the number of states in E . This inequality follows from the factthat G( X ) ≤ ( n + 1) / p inc [55, 56] for p inc in the above inequality, we obtain an upper boundon G opt ( E ). Comparing the performance of various bounds yielded by theseapproaches is a subject for future work. In this work, with the concern of brute-force adversarial strategy, we re-examined the problem of quantum state discrimination by adopting guess-work [41, 42], rather than the widely accepted and well-studied error proba-bility [1], as the optimization criterion. The new problem, named quantumguesswork discrimination, can thus be viewed as a sibling of minimum errordiscrimination. Following the approach in [43], we reduced the new op-timization problem to a semidefinite programming problem. Necessary andsufficient conditions which must be satisfied by the optimal POVM to achieveminimum guesswork were also presented. Then we investigated the relationbetween minimum guesswork and minimum error probability, showing thatthe former can be bounded in terms of the latter in both directions. Ad-ditionally, the general disagreement between the two criteria was illustrated23y the trine ensemble. Combining Massey’s and McEliece and Yu’s classicalbounds on guesswork [41, 53] with two elegant bounds on accessible informa-tion [21, 22], we obtained both upper and lower bounds on minimum guess-work. Other approaches to deriving bounds were also discussed in Section 7.For geometrically uniform quantum states [13], we gave sufficient conditionsfor a geometrically uniform POVM to achieve minimum guesswork. Usingthis result, we proved the optimality of the POVM Π G (see Example 1) forthe trine ensemble. Furthermore, inspired by a similar result in minimumerror discrimination [54], we provided the necessary and sufficient conditionunder which making no measurement at all would be the optimal strategy. Acknowledgements
The work is supported by the Australian Research Council (Grant Nos.DP130102764 and FT100100218) and the National Natural Science Foun-dation of China (Grant Nos. 61170299 and 61370053). Y. Feng is alsosupported by the Overseas Team Program of the Academy of Mathematicsand Systems Science, CAS and the CAS/SAFEA International PartnershipProgram for Creative Research Team.
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