Non-commutativity and Local Indistinguishability of Quantum States

We study the local indistinguishability problem of quantum states. By introducing an easily calculated quantity, non-commutativity, we present an criterion which is both necessary and sufficient for the local indistinguishability of a complete set of pure orthogonal product states. A constructive distinguishing procedure to obtain the concrete local measurements and classical communications is given. The non-commutativity of ensembles can be also used to characterize the quantumness for classical-quantum or quantum-classical correlated states.

We study the local indistinguishability problem of quantum states. By introducing an easily calculated quantity, non-commutativity, we present an criterion which is both necessary and sufficient for the local indistinguishability of a complete set of pure orthogonal product states. A constructive distinguishing procedure to obtain the concrete local measurements and classical communications is given. The non-commutativity of ensembles can be also used to characterize the quantumness for classical-quantum or quantum-classical correlated states. N onlocality is one of the most important features in quantum mechanics. Quantum entanglement was firstly introduced to characterize and quantify the nonlocality, as it acts as a crucial resource in many quantum information processing tasks 1 . On the other hand, there are also quantum information and computation processing tasks without quantum entanglement, like the quantum computation with one qubit (DQC1) 2 . Other types of quantum correlations such as quantum discord 3 are introduced to capture this kind of quantum advantage. The quantum discord is considered more general than entanglement and captures the quantum correlations that entanglement fails to capture 4 .
A well known quantum phenomenon that exhibits quantum nonlocality is the local indistinguishability for some quantum states. State discrimination or distinguishing is essentially primitive for many quantum information tasks, such as quantum cryptography 5 and quantum algorithms 6 . Moreover, with the remarkable experimental advances in preparation and measurement of quantum states 7,8 , it becomes essential to have a theory to assess the performance of quantum state discrimination protocols. One basic problem of state discrimination is judging the local distinguishability for a set of pure orthogonal product states (POPS). Consider a set of bipartite POPS, each with a prior probability, which constitutes an ensemble of a density operator r AB . Although a set of POPS can always be distinguished globally, it may not be distinguished locally by local operations and classical communications (LOCC). This is called quantum nonlocality without entanglement 9 . In 10 the relation between quantum discord and local indistinguishability has been investigated. However, contrary to one's intuitive appeal, the quantum discord of r AB is not an indicator of local indistinguishability for a set of states that constitute the pure-state decomposition of r AB 4,10 . It shows no relation between zero quantum correlation and the local distinguishability for a set of POPS. The local indistinguishability problem of a set of POPS remains open, although many research have been done so far [9][10][11][12][13][14]18 .
It is natural to ask what kind of quantity or quantumness accounts for the quantum nonlocality. In this article, instead of quantum correlation measures, such as quantum discord, quantum deficit, etc., we investigate local indistinguishability for a set of POPS from the point of quantumness of ensemble. We first introduce an easily calculated quantity, non-commutativity, to quantify the quantumness of a quantum ensemble. Based on the noncommutativity, we present a necessary and sufficient criterion for the local indistinguishability. We also give a constructive distinguishing procedure to judge the local indistinguishability for any given set of POPS by using the criterion. Moreover, by proving the uniqueness of the expression of for semi-classical quantum correlated states, we show that our definition for quantumness of ensembles can be used to characterize the quantumness for semi-classical states. orthogonal, the ensemble becomes classical since different states of classical information can be thought of merely as orthogonal quantum states 1 . Non complete overlap of the two states, x 5 jAEyjwaej, 0 , x , 1, induces quantumness for the ensemble. In fact the ensemble is most 'quantum' when x~1 . ffiffi ffi 2 p , i.e. when the two states have an angle 45u between them 15 . Taking the above observation into account, we introduce noncommutativity to characterize the quantumness for a quantum ensemble. Let A 1 , A 2 , …, A n be a set of operators. We define the total non-commutativity for this set, where [A, B] 5 AB 2 BA, jjAjj is the trace norm of the operator A, The non-commutativity is a natural measure of 'quantumness' for a quantum ensemble e 5 {p i , r i }, where r i are density operators with probability p i . Denote A i 5 p i r i , then N(A 1 , A 2 , …, A n ) is the measure of quantumness for the ensemble e. Here for the problem of local distinguishability of states r 1 , r 2 , …, r n , the prior probabilities p i are irrelevant. One only needs to concern the quantity N(e) 5 N(r 1 , r 2 , …, r n ) to judge the local distinguishability of the set of states r 1 , r 2 , …, r n . The prior probabilities p i do make sense when one concerns the quantumness of a state given by the ensemble e 5 {p i , r i } (see section ''The quantumness of semi-classical states'').
The non-commutativity N has the following properties, which make it a well defined measure for quantumness of an ensemble: A set of quantum states corresponds to a quantum ensemble. Nevertheless, one density operator may have many quantum ensemble decompositions. For instance, consider the density operator in a 3 fl 3 system, r~1 9 I~1 9 where jy 1 ae 5 j1aej1ae,  9 . Hence that r having zero quantum correlation is not an indicator for the local distinguishability of the states in r's pure-state decomposition. On the other hand, if we change the probability of jy i ae, r is no longer an identity and can have nonzero quantum correlation, but the nine states jy i ae still remain locally indistinguishable. Usually, the local indistinguishability of a set of pure states has no simple relations with the properties of the related density operator 4 .
To study the local indistinguishability, and its relations with the quantumness of a quantum ensemble in terms of non-commutativity, in the following we call an ensemble e 5 {p i , jy i ae} classical if N(e) 5 0, and quantum if N(e) . 0. If N(e) 5 0, from the properties of noncommutativity, {jy i ae} must be either mutually orthogonal or identical and the states form a set of classical signals from informatics point of view. If N(e) . 0, among {jy i ae} there must be at least one pair of states that are neither orthogonal nor identical. Consider a set of bipartite POPS {jy i ae 5 ja i ae fl jb i ae}, where jy i ae are all mutually orthogonal, ja i ae and jb i ae are associated to partite A and B respectively. Obviously, this set forms a classical ensemble e 5 {p i , jy i ae} since N(e) 5 0, where p i is the probability with respect to jy i ae. Correspondingly one has ensembles e A 5 {p i , ja i ae} and e B 5 {p i , jb i ae} respectively. If N(e A ) 5 N(e B ) 5 0, we call e a classical-classical ensemble. If N(e A ) 5 0, N(e B ) . 0, we call e a classical-quantum ensemble. Analogously we can define quantumclassical and quantum-quantum ensembles.
Local distinguishability for a set quantum states. A set of states is said to be reliably distinguished locally or distinguished locally if all the states in the set can be distinguished by finite rounds of LOCC protocols. If there are at least two states in a set which can not be distinguished by finite rounds of LOCC protocols, we say that the set can not be distinguished locally. The arguably more operational ''asymptotic local operations and classical communications discrimination problem 16,17 '' is not under our consideration.
Theorem 1 If a set of bipartite pure orthogonal product states cannot be locally distinguished, the ensemble composed of these states must be a quantum-quantum ensemble.
Proof. Consider a set of bipartite POPS, not necessarily complete, {jy i ae 5 ja i ae fl jb i ae} with AEy i jy j ae 5 0, ;i ? j. Suppose these states form a classical-quantum or classical-classical ensemble e 5 {p i , ja i ae fl jb i ae} with non zero p i . Since N({ja i ae}) 5 0, {ja i ae} must be either mutually orthogonal or identical. To distinguish {jy i ae} locally, we can first take projective measurement on A side to distinguish those orthogonal states in {ja i ae}. For those identical states ja i ae, for example, ja 1 ae 5 ja 2 ae, we can take a projective measurement on B to distinguish state ja 1 ae from state ja 2 ae, since jb 1 ae and jb 2 ae must be mutually orthogonal to ensure that AEy 1 jy 2 ae 5 0. In this way we can distinguish the states in the set e reliably. The analysis is similar if e is a quantumclassical ensemble. Therefore if a set of bipartite POPS forms a classical-classical, quantum-classical or classical-quantum ensemble, the set can be locally distinguished. And if the set of POPS states cannot be locally distinguished, the ensemble composed of these states must be a quantum-quantum one. % Since it is impossible to form a quantum-quantum ensemble for a set of POPS in a 2 fl 2 system, Theorem 1 is both necessary and sufficient for 2 fl 2 systems. For higher dimensional cases, there are quantum-quantum ensembles whose states can be locally distinguished. In fact, for 2 fl n systems, any set of POPS is locally distinguishable 11 . Theorem 1 reveals a relation between local distinguishability and quantumness of a set of states. In the following, we give a necessary and sufficient criterion for local distinguishability for all POPS from the view of quantumness of ensembles.
We say that two sets {ja i ae} and {ja9 j ae} are orthogonal if and only if AEa i ja9 j ae 5 0, ;i, j. If a set of states cannot be divided into subsets such that those subsets are mutually orthogonal, then we call the set a single set. Consider a set of states e 5 {ja 1 ae 5 j0ae, ja 2 ae 5 j0 1 1ae, ja 3 ae 5 j2ae}. This set can be divided into two single subsets e 1 5 {ja 1 ae, ja 2 ae} and e 2 5 {ja 3 ae} which are orthogonal and each subset cannot be split further. We call this partition the direct sum decomposition and denote it as e 5 e 1 › e 2 . A single set means that the set cannot be decomposed into such direct sums. If all the subsets are single sets in a direct sum decomposition, we say that the decomposition is a single set decomposition. It can verified that the single set decomposition of a non-single set is unique. For a single set, adding some states in the vector space spanned by itself will keep the set a single one. Nevertheless, taking away some states from a single set, the set could www.nature.com/scientificreports SCIENTIFIC REPORTS | 4 : 6336 | DOI: 10.1038/srep06336 become a non-single one. In the following, by a set of states' decomposition we mean the single set decomposition.
Lemma For a set of states e 5 {ja i ae}, m 5 dim(span{ja i ae}), the following statements are equivalent: (a) e is a single set. (b) There are m linear independent states a i k j i f g m k~1 in e satisfying the following relations: (c) A nondestructive projective measurement, a measurement which keeps the quantum state unchanged 18   gare single sets. % From the view of accessible information, the more quantum an ensemble is, the less information one can get from the ensemble 1,15 . Hence the quantumness of the two parts of a complete set of POPS must be ''large'' enough, at least larger than N a i k j i f g ð Þ and N b j l È É À Á , so that one can only get limited information and the states in the set cannot be locally distinguished. Therefore the local quantumness of an ensemble determines the local indistinguishability.
Theorem 2 gives a constructive distinguishing procedure to judge the local distinguishability for a complete set of POPS. Let {jy i ae 5 ja i ae fl jb i ae} be a complete set of POPS, corresponding to two sets e A 5 {ja i ae} and e B 5 {jb i ae}. To verify the inequality (3), one needs to find all the subsets involved. This can be done in the following way.
1~y2 j i, y 3 j i f g 11 . Each {jy s ae 5 ja s ae fl jb s ae} ij corresponds to two new subsets e Aij~a s j i f g and e Bij~b s j i f g.
We repeat the above process n rounds for the new subsets until each of those new subsets has only one element or both A and B parts cannot be decomposed further. At last we have subset {jy k ae 5 ja k ae fl jb k ae} st,…,ij (here st, …, ij are the measurement outcomes) corresponding to two new sets e A st,...,ij~a k j i f g and e B st,...,ij~b k j i f g, where both e Ast,...,ij and e Bst,...,ij are single sets. If all the final single subsets have one element, the set {jy i ae} can be locally distinguished.
The quantumness of semi-classical states. The non-commutativity defined by (1) can be also used to describe the quantumness for bipartite semi-classical states. For a given semi-classical state, r qc~X i p i r i 6 i j i i h j, the corresponding ensemble e 5 {p i , r i } is fixed (see Methods). Denote X i 5 p i r i . We have the noncommutativity describing the quantumness for the state r qc , It is easy to check that N(r qc ) satisfies the following properties, which actually makes it a candidate for quantum-correlation measures for semi-classical states 4 : (a) it is positive, (b) it is zero for classicalclassical correlated states, (c) it is invariance under local-unitary transformations, and (d) it is non-increasing when an ancillary system is introduced. This quantity N(r qc ) describes the quantumness of semi-classical states. It is different from those quantum correlation measures, such as quantum discord, quantum deficit, etc., which are based on the measurements and their estimations involve extremely complicated optimization process. Interestingly, while the quantity N(r qc ) can be easily computed, it shows similar behavior to other quantum correlation measures for semi-classical states. Hence, instead of those quantum correlation measures, one can use the non-commutativity to characterize the quantum correlations of semi-classical states r qc . Moreover, noncommutativity provides a tool to explore the relation between the ensemble quantumness and those quantum correlation measures. Fig. 1 shows the comparison between quantum correlations and ensemble quantumness (non-commutativity) for a 3 fl 3 system for which it is extremely difficult to calculate quantum correlations analytically.

Discussion
We have proposed non-commutativity as a quantumness measure for an ensemble. It has been shown that the local ensemble quantumness, instead of quantum correlation measure, like quantum discord, quantum deficit, etc., which is a function of a density operator, accounts for the local indistinguishability for a complete set of POPS. It implies that the quantumness of local ensembles must satisfy certain conditions so that the states in the ensemble cannot be locally distinguished. A constructive distinguishing procedure to obtain the concrete local measurements and classical communications has been presented to judge the local indistinguishability for a complete set of POPS. Our approach to judge the local indistinguishability can also be directly extended to distinguish a complete set of multipartite POPS or a non-complete set of POPS, when the local operations are restricted within nondestructive projective measurements.
Due to that one semi-classical state corresponds to one quantum ensemble, the non-commutativity has been shown to be able to characterize the ensemble quantumness for classical-quantum or quantum-classical systems.

Methods
where y i,j 5 AEw i jyaeAEyjw j ae and y Ã i,j is the conjugate of y i,j . . Therefore for two pure states jyae and jwae, jj[jwae, jyae]jj is 0 when jAEwjyaej 5 0 or 1, and 1 when Proof of Lemma 1. (a) ) (b) First choose any vector in e, say, a i1 j i. Then choose the second vector a i2 j i in e such that it is not equal to or orthogonal to a i1 j i, i.e., N a i1 j i, a i2 j i ð Þ w0. The existence of a i2 j iis guaranteed by the fact that e is a single set. Next choose the third vector a i3 j iin e such that it is independent of a i1 j i, a i2 j i f g , and also not orthogonal to a i1 j i, a i2 j i f g , i.e., N a i1 j i, a i2 j i ð Þ vN a i1 j i, a i2 j i, a i3 j i ð Þ . Or else, if a i3 j idoes not exist, then e can be direct sum decomposed into two parts. Continuing with the above process, we can finally get m linear independent states a ik j i f g m k~1 in e satisfying inequality (2).
(b) ) (c) A nondestructive projective measurement is described by an observable, M, an Hermitian operator such that all ja i ae in e are the eigenvectors of M. If N a i1 j i, a i2 j i ð Þ w0, a i1 j i and a i2 j i are neither identical nor orthogonal. Hence the corresponding eigenvalues of a i1 j i and a i2 j i must be equal, since the eigenvectors corresponding to different eigenvalues are orthogonal. If N a i1 j i, a i2 j i ð Þ vN a i1 j i, a i2 j i, a i3 j i ð Þ , a i3 j i is at least not orthogonal to one of a i1 j i and a i2 j i. For instance, suppose N a i1 j i, a i3 j i ð Þ w0, then the eigenvalues corresponding to a i1 j i and a i3 j i are equal. Therefore we have that if (2) is satisfied, those m linear independent a ik j is have the same eigenvalues of M. Therefore M becomes an identity (apart of real factor) and it cannot be used to distinguish the states in e.
(c) ) (a) Suppose the set is not a single one and has a decomposition e 5 e 1 › e 2 . Then one can use a corresponding nondestructive projective measurement M 5 aI 1 › bI 2 , where a ? b, I 1 , I 2 are identity operators, to distinguish the states in e 1 from e 2 . Hence if M cannot do any in distinguishing the states in e, e must be a single set, that is, statement (a) holds.
for s ? t, which means that both {ja i ae} and {jb i ae} are the eigenvectors of Q kl , andq i kl are the corresponding eigenvalues. Note that the set {q i s i } of state s can be divided into degenerate part (in which all the states are identical) and non-degenerate part (in which all the states are mutually different). For non-degenerate part, taking over all k, l, one will find that the intersection of the eigenspaces belonging to the eigenvaluesq i kl must be one dimensional, namely, ja i ae 5 jb i ae and then p i r i 5 q i s i . For degenerate part, without loss of generality, assume q 1 s 1 5 q 2 s 2 , then r 5 s means p 1 r 1 fl ja 1 aeAEa 1 j 1 p 2 r 2 fl ja 2 aeAEa 2 j 5 q 1 s 1 fl (jb 1 aeAEb 1 j 1 jb 2 aeAEb 2 j). Therefore p 1 r 1 5 p 2 r 2 5 q 1 s 1 5 q 2 s 2 . Finally we get p i r i 5 q i s i , ;i.