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arXiv:2111.11275 · Physical Review A

Local discrimination of generalized Bell states via commutativity

Mao-Sheng Li, Fei Shi, Yan-Ling Wang

Published Mar 30, 2022• 9 citations
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Abstract

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 $\mathcal{S}$ of generalized Bell states. In fact, we get an efficient criterion. That is, if the difference set $\mathrm{\ensuremath{\Delta}}\mathcal{S}$ of $\mathcal{S}$ is disjoint with or completely contained in some MCS, then the set $\mathcal{S}$ 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.

DOI
10.1103/physreva.105.032455

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arXiv:2109.07390•Quantum•Jul 14, 2022•5 citations
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 \ensuremathΔS\mathrm{\ensuremath{\Delta}}\mathcal{S}\ensuremathΔS 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 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.
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.
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.
Qudit lattice states, as the generalization of qubit lattice states, are the maximally entangled states determined by qudit lattice unitaries in a ${p}^{r}\ensuremath{\bigotimes}{p}^{r}$ quantum system with ppp being a prime and rrr being an integer. Based on the partitions of qudit lattice unitaries into commuting sets, we present a sufficient condition for local discrimination of qudit lattice states, in which the commutativity plays an efficient role. It turns out that any set of lll qudit lattice states with $2\ensuremath{\le}l\ensuremath{\le}{p}^{r}$, including $k\ensuremath{\le}l$ mutually commuting qudit lattice unitaries and satisfying $l(l\ensuremath{-}1)\ensuremath{-}(k+1)(k\ensuremath{-}2)\ensuremath{\le}2{p}^{r}$, can be locally distinguished, not only extending Fan's result [H. Fan, Phys. Rev. Lett. 92, 177905 (2004)] to the prime power quantum system but also involving the local discrimination of a larger number of maximally entangled states.
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.