A fundamental problem in quantum information processing is the discrimination among a set of orthogonal quantum states of a composite system under local operations and classical communication (LOCC). Corresponding to the LOCC indistinguishable sets of four ququad-ququad orthogonal maximally entangled states constructed by Yu et al. [Phys. Rev. Lett. 109, 020506 (2012)], the maximum commutative sets were introduced as detectors for the local distinguishability of the set of generalized Bell states (GBSs) for which the detectors are sufficient to determine the LOCC distinguishability. In this work, we show how to determine all the detectors for a given GBS set. We also construct several four-GBS sets without detectors, most of which are one-way LOCC indistinguishable and only one of which is one-way LOCC distinguishable, indicating that the detectors are not necessary for LOCC distinguishability. Furthermore, we show that for four-GBS sets in the quantum system ${\mathbb{C}}^{6}\ensuremath{\bigotimes}{\mathbb{C}}^{6}$, the detectors are almost necessary for one-way LOCC distinguishability, except for one set in the sense of local unitary equivalence. The problem of one-way LOCC discrimination of four-GBS sets in ${\mathbb{C}}^{6}\ensuremath{\bigotimes}{\mathbb{C}}^{6}$ is completely resolved.
A fundamental problem in quantum information processing is the discrimination among a set of orthogonal quantum states of a composite system under local operations and classical communication (LOCC). Corresponding to the LOCC indistinguishable sets of four ququad-ququad orthogonal maximally entangled states constructed by Yu et al. [Phys. Rev. Lett. 109, 020506 (2012)], the maximum commutative sets were introduced as detectors for the local distinguishability of the set of generalized Bell states (GBSs) for which the detectors are sufficient to determine the LOCC distinguishability. In this work, we show how to determine all the detectors for a given GBS set. We also construct several four-GBS sets without detectors, most of which are one-way LOCC indistinguishable and only one of which is one-way LOCC distinguishable, indicating that the detectors are not necessary for LOCC distinguishability. Furthermore, we show that for four-GBS sets in the quantum system C6\ensuremath⨂C6, the detectors are almost necessary for one-way LOCC distinguishability, except for one set in the sense of local unitary equivalence. The problem of one-way LOCC discrimination of four-GBS sets in C6\ensuremath⨂C6 is completely resolved.
Local unitary classification of sets of generalized Bell states in Cd⊗Cd
Two sets of quantum entangled states that are equivalent under local unitary transformations may exhibit identical effectiveness and versatility in various quantum information processing tasks. Consequently, classification under local unitary transformations has become a fundamental issue in the theory of quantum entanglement. The primary objective of this work is to establish a practical LU-classification for all sets of l(≥2) generalized Bell states (GBSs), high-dimensional generalizations of Bell states, in a bipartite system Cd⊗Cd with d≥3 . Based on this classification, we determine the minimal cardinality of indistinguishable GBS sets in C6⊗C6 under one-way local operations and classical communication (one-way LOCC). We first propose two classification methods based on LU-equivalence for all sets of l GBSs (l-GBS sets). We then establish LU-classification for all 2-GBS, 3-GBS, 4-GBS and 5-GBS sets in C6⊗C6 . Since LU-equivalent sets share identical local distinguishability, it suffices to examine representative GBS sets from equivalent classes. Notably, we identify a one-way LOCC indistinguishable 4-GBS set among these representatives, thereby resolving the case of d=6 for the problem of determining the minimum cardinality of one-way LOCC indistinguishable GBS sets in (Yuan et al. in Quantum Inf Process. 18:145, 2019) or (Zhang et al. in Phys Rev A 91:012329, 2015).
Abstract It is an interesting topic how much entanglement resources are necessary to distinguish locally indistinguishable orthogonal quantum states by local operations and classical communication (LOCC). Using only one ebit of entanglement some locally indistinguishable orthogonal product bases can be distinguished by LOCC [2019 Phys. Rev. A 99 012343]. However, it is still an open question whether one ebit of entanglement is sufficient for distinguishing a locally indistinguishable set of orthogonal maximally entangled states. We focus on the generalized Bell states (GBSs), and firstly find that any four GBSs in a 4 ⊗ 4 quantum system are always distinguished by one-way LOCC using one ebit of entanglement. Then we consider a general case, i.e., d ⊗ d quantum system ( d is even), and show that the set L of any GBSs can be distinguished by one-way LOCC using one ebit of entanglement if S A \ S L is nonempty, where S A is the admissible solutions set of L and S L is the genuinely nonadmissible solutions set of L . This criterion is very simple and convenient since only some congruence equations are calculated. Finally, we find a curious and counterintuitive case, i.e., in a 3 ⊗ 3 quantum system there exist four GBSs which cannot be distinguished by one-way LOCC using one ebit of entanglement, although the dimension of the subsystem is less than four. These results can help us better understand the role of entanglement as a resource in local state discrimination problems.
In general, for a bipartite quantum system Cd⊗Cd and an integer k such that 4≤k≤d ,there are few necessary and sufficient conditions for local discrimination of sets of k generalized Bell states (GBSs) and it is difficult to locally distinguish k -GBS sets.The purpose of this paper is to completely solve the problem of local discrimination of GBS sets in some bipartite quantum systems.Firstly three practical and effective sufficient conditions are given,Fan , s and Wang et al. , s results [Phys Rev Lett 92, 177905 (2004); Phys Rev A 99, 022307 (2019)] can be deduced as special cases of these conditions.Secondly in C4⊗C4 , a necessary and sufficient condition for local discrimination of GBS sets is provided, and a list of all locally indistinguishable 4-GBS sets is provided,and then the problem of local discrimination of GBS sets is completely C5⊗C5 , a concise necessary and sufficient condition for one-way local discrimination of GBS sets is obtained,which gives an affirmative answer to the case d=5 of the problem proposed by Wang et al.
In general, for a bipartite quantum system Cd⊗Cd and an integer k such that 4≤k≤d,there are few necessary and sufficient conditions for local discrimination of sets of k generalized Bell states (GBSs) and it is difficult to locally distinguish k-GBS this http URL this paper, we consider the local discrimination of GBS sets and the purpose is to completely solve the problem of local discrimination of GBS sets in some bipartite quantum systems,specifically, we show some necessary and sufficient conditions for local discrimination of GBS sets by which the local discrimination of GBS sets can be quickly determined.Firstly some sufficient conditions are given, these sufficient conditions are practical and effective.Fan,s and Wang et al.,s results (Phys Rev Lett 92:177905, 2004: Phys Rev A 99:022307, 2019) can be deduced as special cases of these conditions.Secondly in C4⊗C4, a necessary and sufficient condition for local discrimination of GBS sets is provided,all locally indistinguishable 4-GBS sets are found,and then we can quickly determine the local discriminability of an arbitrary GBS this http URL C5⊗C5, a concise necessary and sufficient condition for one-way local discrimination of GBS sets is obtained,which gives an affirmative answer to the case d=5 of the problem proposed by Wang et al. (Phys Rev A 99:022307, 2019).
In this paper, we investigate the indistinguishability of generalized Bell states (GBSs) by one-way local operations and classical communication (LOCC). We first introduce two sets, i.e., the difference set and the testing set, and show that the GBSs cannot always be distinguished by one-way LOCC if the testing set is a subset of the difference set of GBSs. Then we give a unified upper bound about indistinguishable GBSs by one-way LOCC. The newly derived bound is smaller than those presented in Zhang et al. [Phys. Rev. A 91, 012329 (2015)] and Wang et al. [Quantum Inf. Process. 15, 1661 (2016).]. More precisely, the minimum number of one-way locally indistinguishable GBSs is not more than $2\ensuremath{\lceil}\sqrt{d}\ensuremath{\rceil}+\ensuremath{\lceil}\frac{\sqrt{d}}{4}\ensuremath{\rceil}+1$.
We study the distinguishability of generalized Bell states under local operations and classical communication. We introduce the concept of a maximally commutative set (MCS), a subset of generalized Pauli matrices whose elements are mutually commutative, and there is no other generalized Pauli matrix that commutes with all the elements of this set. We find that MCS can be considered a detector for the local distinguishability of a set S of generalized Bell states. In fact, we get an efficient criterion. That is, if the difference set $\mathrm{\ensuremath{\Delta}}\mathcal{S}$ of S is disjoint with or completely contained in some MCS, then the set S is locally distinguishable. Furthermore, we give a useful characterization of MCS for arbitrary dimensions, which provides great convenience for detecting the local discrimination of generalized Bell states. Our method can be generalized to more general settings which contain the lattice qudit basis. The results of Fan [Phys. Rev. Lett. 92, 177905 (2004)], Tian et al. [Phys. Rev. A 92, 042320 (2015)], and a recent work Yuan et al. [arXiv:2109.07390] can be deduced as special cases of our result.