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Physical Review A

One-way local distinguishability of generalized Bell states in arbitrary dimension

Yan-Ling Wang, Mao-Sheng Li, Zong-Xing Xiong

Published Feb 8, 2019• 24 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.99.022307

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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.
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.
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\mathcal{S}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\mathcal{S}S is disjoint with or completely contained in some MCS, then the set S\mathcal{S}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.
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 $2\ensuremath{\lceil}\sqrt{d}\ensuremath{\rceil}+\ensuremath{\lceil}\frac{\sqrt{d}}{4}\ensuremath{\rceil}+1$.
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 ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ to be locally distinguishable. Thus we obtain a simple computable criterion for local distinguishability of sets of ddd GBSs in prime dimension ddd.
In this paper, we mainly consider the local indistinguishability of the set of mutually orthogonal bipartite generalized Bell states (GBSs). We construct small sets of GBSs with cardinality smaller than ddd which are not distinguished by one-way local operations and classical communication (1-LOCC) in d⊗dd\otimes dd⊗d. The constructions, based on linear system and Vandermonde matrix, is simple and effective. The results give a unified upper bound for the minimum cardinality of 1-LOCC indistinguishable set of GBSs, and greatly improve previous results in [Zhang \emph{et al.}, Phys. Rev. A 91, 012329 (2015); Wang \emph{et al.}, Quantum Inf. Process. 15, 1661 (2016)]. The case that ddd is odd of the results also shows that the set of 4 GBSs in 5⊗55\otimes 55⊗5 in [Fan, Phys. Rev. A 75, 014305 (2007)] is indeed a 1-LOCC indistinguishable set which can not be distinguished by Fan's method.