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arXiv:2102.07400 · Physical Review A

Simple criterion for local distinguishability of generalized Bell states in prime dimension

T. Hashimoto, M. Horibe, A. Hayashi

Published May 25, 2021• 10 citations
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Abstract

Local distinguishability of sets of generalized Bell states (GBSs) is investigated. We first clarify the conditions such that a set of GBSs can be locally transformed to a certain type of GBS set that is easily distinguishable within local operations and one-way classical communication. We then show that if the space dimension $d$ is a prime, these conditions are necessary and sufficient for sets of $d$ GBSs in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ to be locally distinguishable. Thus we obtain a simple computable criterion for local distinguishability of sets of $d$ GBSs in prime dimension $d$.

DOI
10.1103/physreva.103.052429

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Local distinguishability of sets of generalized Bell states (GBSs) is investigated. We first clarify the conditions such that a set of GBSs can be locally transformed to a certain type of GBS set that is easily distinguishable within local operations and one-way classical communication. We then show that if the space dimension ddd is a prime, these conditions are necessary and sufficient for sets of ddd GBSs in Cd\ensuremath⨂Cd{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}Cd\ensuremath⨂Cd to be locally distinguishable. Thus we obtain a simple computable criterion for local distinguishability of sets of ddd GBSs in prime dimension ddd.
In general, for a bipartite quantum system Cd⊗Cd\mathbb{C}^{d}\otimes\mathbb{C}^{d}Cd⊗Cd and an integer kkk such that 4≤k≤d4\leq k\le d4≤k≤d,there are few necessary and sufficient conditions for local discrimination of sets of kkk generalized Bell states (GBSs) and it is difficult to locally distinguish kkk-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\mathbb{C}^{4}\otimes\mathbb{C}^{4}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\mathbb{C}^{5}\otimes\mathbb{C}^{5}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=5d=5d=5 of the problem proposed by Wang et al. (Phys Rev A 99:022307, 2019).
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 order to study a large number of generalized Bell states (GBSs) efficiently, it is important to classify them by local unitary (LU) equivalence. For example, there are C254(=12650) sets of four GBSs in C5⊗C5, which greatly exceed the C164(=1820) sets of four GBSs in C4⊗C4, and it is inconceivable to explore these sets directly without prior classification. In this paper, we consider the classification of GBS sets in C5⊗C5. First, two simple and useful characterizations of LU-equivalence of GBS sets are given. Then, one characterization is used to divide all 12 650 sets of four GBSs into eight equivalence classes including six locally distinguishable classes and two classes that cannot be discriminated by one-way local operations and classical communication. The second characterization is powerful and can be used to write a program, by using which all C255(=53130) sets of five GBSs in C5⊗C5 are divided into 21 equivalence classes including 9 locally distinguishable classes and 12 locally indistinguishable classes.
In this paper, we study the distinguishability of arbitrary dimensional generalized Bell states under one-way local operations and classical communication (LOCC). We introduce an admissible solutions set and a nonadmissible solutions set for each set of generalized Bell states. Any element that exists in the admissible solution set but not in the nonadmissible one implies the one-way LOCC distinguishablity of a given set. This criterion is very convenient in the following two aspects: both solution sets can be calculated easily and, once the criterion can be applied successfully for a given set, it immediately presents an explicit distinguishing strategy. Furthermore, Fan's results in Phys. Rev. Lett. 92, 177905 (2004) can also be deduced as special cases of our results. For the case no such element exists, we disprove a conjecture proposed by Yang et al. in Quantum Inf. Process. 17, 29 (2018) by presenting a counterexample.
Abstract In this paper the local distinguishability of generalized Bell states in arbitrary dimension is investigated. We firstly study the decomposition of a basis which consists of d 2 number of generalized Pauli matrices. We discover that this basis is equal to the union of D number of different sets, where D = 2 ϕ ( d ) ∑ t ∈ Z d gcd ( t , d ) = 1 ∑ i = 2 ⌊ d t ⌋ ϕ ( i ) + 1 and ϕ is Euler ϕ -function. Then we define the generator of the matrices in this decomposition, and exhibit an algorithm to calculate generators of a given set of matrices. This algorithm shows that generators of a given set can be calculated simply and efficiently. Secondly, we show that a set L of GBSs can be distinguished by one-way LOCC if the cardinality of G L is less than Dϕ ( d ), where G L is a set of generators of all the elements in difference set of a set L of GBSs. The previous results in [2004 Phys. Rev. Lett. 92 177905; 2019 Phys. Rev. A 99 022307; 2021 Quantum Inf. Process. 20 52] can be covered by our result. Finally, for the uncovered cases in [2021 Quantum Inf. Process. 20 52], we give a new result to partly solve that problem.