Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Local discrimination of qudit lattice states via commutativity

Guojing Tian, Sixia Yu, Fei Gao, Qiaoyan Wen, C. H. Oh

Published Oct 19, 2015• 23 citations
Publisher

Abstract

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 $p$ being a prime and $r$ 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 $l$ 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.

DOI
10.1103/physreva.92.042320

Related papers

Local discrimination of generalized Bell states via commutativity

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

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.

arXiv:2111.11275•Physical Review A•Mar 30, 2022•9 citations

Local discrimination of lattice states via complete graph

Qianqian Liu, Ying‐Hui Yang, Pei-Ying Chen +1 more

Abstract In this paper, we use graph theory to study the distinguishability of lattice states under local operations and classical communication (LOCC) in C p 2 ⨂ C p 2 . Firstly, we present that for the basis of lattice unitary matrices, there are (p 2 + 1)(p + 1) distinct maximal commuting sets in p 2-dimensional system. Secondly, for any set  of lattice states, we can obtain a graph G which consists of some k-complete subgraphs. Using the graph G, we show that the set  can be distinguished by one-way LOCC if n ( p + 1 ) − ∑ k = 1 p + 1 s k ( k − 1 ) < ( p 2 + 1 ) ( p + 1 ) , where n is the number of vertices of a graph G and s k is the number of k-complete subgraphs. Moreover, we also proved that this conclusion is necessary and sufficient in C 2 2 ⨂ C 2 2 . For C p 2 ⨂ C p 2 system, our result can cover the results in (2015, Phys. Rev. A 92, 042 320); (2020, Sci. China-Phys. Mech. Astron. 63, 280 312).

Physica Scripta•Sep 20, 2023•2 citations

Detectors for local discrimination of sets of generalized Bell states

Caihong Wang, Jiangtao Yuan, Ying‐Hui Yang +3 more

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.

arXiv:2504.11747•Physical Review A•Apr 4, 2025•3 citations

Local discrimination of four or more maximally entangled states

Guojing Tian, Sixia Yu, Fei Gao +2 more

In the $4\ensuremath{\bigotimes}4$ quantum system, four qubit lattice states are found to be either locally indistinguishable or distinguishable by using one-way classical communication only, presenting a curious and counterintuitive case in which two-way classical communication has no advantage. Based on these results, we explicitly construct sets of four, five, and even more maximally entangled states that can be locally distinguished only with the help of two-way classical communication, solving partially the open problem described by Nathanson [Phys. Rev. A 88, 062316 (2013)] and highlighting in more detail the difference between local operations with and without two-way classical communication in distinguishing more than three states.

Physical Review A•May 15, 2015•33 citations

Necessary and sufficient conditions for local discrimination of generalized Bell states: finding out all locally indistinguishable sets of generalized Bell states

Jiangtao Yuan, Ying‐Hui Yang, Caihong Wang

In general, for a bipartite quantum system $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ and an integer $k$ such that $4\leq k\le d$,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 $\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, 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 $\mathbb{C}^{5}\otimes\mathbb{C}^{5}$, 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).

•Sep 15, 2021•1 citation

Local entanglement is not necessary for perfect discrimination between unitary operations acting on two qudits by local operations and classical communication

Lvzhou Li, Daowen Qiu

Recently, the problem of discriminating multipartite unitary operations by local operations and classical communication (LOCC) has attracted significant attention. The latest work in the literature on this problem showed that two multipartite unitary operations can always be perfectly distinguished by LOCC when a finite number of runs are allowable. However, in these schemes, local entanglement (an entangled state held by one party) was required, which seems to imply that local entanglement is necessary for perfect discrimination between unitary operations by LOCC. In this paper, we show that a perfect discrimination between two unitary operations acting on a two-qudit system can always be achieved without exploiting any entanglement. As a result, we conclude that local entanglement is not necessary for perfect discrimination between unitary operations acting on a two-qudit system by LOCC. Our result, in general, is based on the assumption made by X. F. Zhou, Y. S. Zhang, and G. C. Guo [Phys. Rev. Lett. 99, 170401 (2007)] that for any unitary operation $U$, we can obtain its reverse transformation ${U}^{\ifmmode\dagger\else\textdagger\fi{}}$ by exchanging the input and output ports of the whole setup. Thus, we further consider whether we can avoid using this assumption, and we find that for unitary operations acting on two qubits, our result can be obtained without using this assumption. However, in higher dimension how to do that seems to be unclear.

arXiv:0706.0941•Physical Review A•Mar 24, 2008•18 citations
Qudit lattice states, as the generalization of qubit lattice states, are the maximally entangled states determined by qudit lattice unitaries in a pr\ensuremath⨂pr{p}^{r}\ensuremath{\bigotimes}{p}^{r}pr\ensuremath⨂pr 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≤l\ensuremath≤pr2\ensuremath{\le}l\ensuremath{\le}{p}^{r}2\ensuremath≤l\ensuremath≤pr, including k\ensuremath≤lk\ensuremath{\le}lk\ensuremath≤l mutually commuting qudit lattice unitaries and satisfying l(l\ensuremath−1)\ensuremath−(k+1)(k\ensuremath−2)\ensuremath≤2prl(l\ensuremath{-}1)\ensuremath{-}(k+1)(k\ensuremath{-}2)\ensuremath{\le}2{p}^{r}l(l\ensuremath−1)\ensuremath−(k+1)(k\ensuremath−2)\ensuremath≤2pr, 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.
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.
Abstract In this paper, we use graph theory to study the distinguishability of lattice states under local operations and classical communication (LOCC) in C p 2 ⨂ C p 2 . Firstly, we present that for the basis of lattice unitary matrices, there are (p 2 + 1)(p + 1) distinct maximal commuting sets in p 2-dimensional system. Secondly, for any set  of lattice states, we can obtain a graph G which consists of some k-complete subgraphs. Using the graph G, we show that the set  can be distinguished by one-way LOCC if n ( p + 1 ) − ∑ k = 1 p + 1 s k ( k − 1 ) < ( p 2 + 1 ) ( p + 1 ) , where n is the number of vertices of a graph G and s k is the number of k-complete subgraphs. Moreover, we also proved that this conclusion is necessary and sufficient in C 2 2 ⨂ C 2 2 . For C p 2 ⨂ C p 2 system, our result can cover the results in (2015, Phys. Rev. A 92, 042 320); (2020, Sci. China-Phys. Mech. Astron. 63, 280 312).
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.
In the $4\ensuremath{\bigotimes}4$ quantum system, four qubit lattice states are found to be either locally indistinguishable or distinguishable by using one-way classical communication only, presenting a curious and counterintuitive case in which two-way classical communication has no advantage. Based on these results, we explicitly construct sets of four, five, and even more maximally entangled states that can be locally distinguished only with the help of two-way classical communication, solving partially the open problem described by Nathanson [Phys. Rev. A 88, 062316 (2013)] and highlighting in more detail the difference between local operations with and without two-way classical communication in distinguishing more than three states.
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).
Recently, the problem of discriminating multipartite unitary operations by local operations and classical communication (LOCC) has attracted significant attention. The latest work in the literature on this problem showed that two multipartite unitary operations can always be perfectly distinguished by LOCC when a finite number of runs are allowable. However, in these schemes, local entanglement (an entangled state held by one party) was required, which seems to imply that local entanglement is necessary for perfect discrimination between unitary operations by LOCC. In this paper, we show that a perfect discrimination between two unitary operations acting on a two-qudit system can always be achieved without exploiting any entanglement. As a result, we conclude that local entanglement is not necessary for perfect discrimination between unitary operations acting on a two-qudit system by LOCC. Our result, in general, is based on the assumption made by X. F. Zhou, Y. S. Zhang, and G. C. Guo [Phys. Rev. Lett. 99, 170401 (2007)] that for any unitary operation UUU, we can obtain its reverse transformation ${U}^{\ifmmode\dagger\else\textdagger\fi{}}$ by exchanging the input and output ports of the whole setup. Thus, we further consider whether we can avoid using this assumption, and we find that for unitary operations acting on two qubits, our result can be obtained without using this assumption. However, in higher dimension how to do that seems to be unclear.