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 distinguishability of four multipartite orthogonal product states

Guang‐Bao Xu, Zi-Yan Hao, Hua-Kun Wang, Yu‐Guang Yang, Dong‐Huan Jiang

Published Sep 30, 2025• 7 citations
Publisher

Abstract

In this paper, we give the local distinguishability of four orthogonal product states (OPSs) on multipartite and high-dimension systems. To characterize the different orthogonal relationships between four multipartite OPSs, a new concept, i.e., the vector of the numbers of pairwise orthogonal relations, is proposed. We classify four tripartite OPSs, any two of which are orthogonal only on one subsystem, into three categories by using the vectors of the numbers of pairwise orthogonal relations. We prove that any four tripartite OPSs belonging to two of the three categories can be perfectly distinguished by local operations and classical communication (LOCC). Meanwhile, we find an interesting phenomenon that two sets of four tripartite OPSs belonging to the remaining one of the three categories have different local distinguishability even if they have the same orthogonal graph. This means that the orthogonal graph is not a sufficient condition for determining the local distinguishability of a set of OPSs. Additionally, for the remaining category, we specify the conditions under which four tripartite OPSs can be locally distinguished and the conditions under which four tripartite OPSs cannot be perfectly distinguished by LOCC. On the other hand, we prove that any four tripartite OPSs with seven or more pairwise orthogonal relations can be perfectly distinguished by LOCC. Furthermore, we prove that any four OPSs with six or more pairwise orthogonal relations can be perfectly distinguished by LOCC on four-, five-, and six-partite systems.

DOI
10.1103/ht8g-43kp

Related papers

Local distinguishability of five quadripartite orthogonal product states

Guang-Bao Xu, Jie Zhan, Yu-Guang Yang +1 more

The local distinguishability of five bipartite and tripartite orthogonal product states (OPSs) has been effectively resolved, while that of five quadripartite OPSs has not yet been effectively addressed. In this paper, we give the local distinguishability of five quadripartite OPSs, where any two of these five states are orthogonal on exactly one subsystem. Based on the vectors of the numbers of pairwise orthogonality relations, we classify different sets of five quadripartite OPSs into nine categories, which correspond to 371 distinct orthogonality graphs. We show that the sets of five quadripartite OPSs corresponding to 355 of the 371 cases can be perfectly distinguished by local operations and classical communication (LOCC), whereas the sets of five quadripartite OPSs for the remaining 16 cases cannot exactly distinguished by LOCC. Our results enrich the theory of nonlocality for sets of OPSs.

quant-pharXiv:2610.29622•Oct 22, 2026•0 citations

Local distinguishability of five orthogonal product states on bipartite and tripartite quantum systems

Guang-Bao Xu, Zi-Yan Hao, Hua-Kun Wang +2 more

Local distinguishability of orthogonal quantum states can effectively reduce the consumption of quantum resources and lower economic costs in quantum protocols. Although numerous achievements have been made regarding local distinguishability of orthogonal quantum states, some fundamental issues have not been effectively addressed. For example, the local distinguishability of five orthogonal product states (OPSs) is still unknown up to now. In this paper, we give the properties of local distinguishability of five OPSs on bipartite and tripartite quantum systems. Firstly, to characterize the structure of a set of bipartite OPSs, we propose the concept of the vector of orthogonal relations for a set of bipartite OPSs. Secondly, we classify the structures of five bipartite OPSs into six categories by this concept and prove that five of these six categories can be perfectly distinguished by local operations and classical communication (LOCC). Thirdly we show that the local distinguishability of each case of the sixth category singly. On the other hand, we first divide the structures of five tripartite OPSs into eight categories by the vectors of orthogonal relations of five tripartite OPSs. Then we give the local distinguishability of each category. Our work enriches the research results of quantum nonlocality and will provide a clear understanding of the local distinguishability of five OPSs.

quant-pharXiv:2512.00484•Physical Review A•Dec 2, 2025•0 citations

Local distinguishability of six bipartite orthogonal product states

Guang-Bao Xu, Hua-Kun Wang, Yu-Guang Yang +1 more

It is necessary to investigate the local distinguishability of orthogonal quantum state sets, as their adoption in protocol design helps diminish quantum state transmission and cut operational costs. In this paper, we explore the local distinguishability of six orthogonal product states (OPSs) on any bipartite quantum system. We classify different sets of six bipartite OPSs into eight categories by using the vectors of the numbers of pairwise orthogonality relations, where any two states are orthogonal on only one subsystem within each set. We find that these eight categories contain a total of 78 distinct cases, all but five of which are perfectly distinguishable via local operations and classical communication (LOCC). Furthermore, we discuss the local distinguishability of those five distinct cases in detail. Our work explicitly characterizes the local distinguishability of six bipartite OPSs.

quant-pharXiv:2607.02006•Physical Review A•Jul 3, 2026•0 citations

Local distinguishability of quantum states in multipartite System

Xiaoqian Zhang

In this paper, we mainly study the local distinguishable multipartite quantum states by local operations and classical communication (LOCC) in $m_1\otimes m_2\otimes\ldots\otimes m_n$ , where the quantum system $m_1$ belongs to Alice, $m_2$ belongs to Bob, \ldots and $m_n$ belongs to Susan. We first present the pure tripartite distinguishable orthogonal quantum states by LOCC in $m_1\otimes m_2\otimes m_3$. With the conclusion in $m_1\otimes m_2\otimes m_3$, we prove distinguishability or indistinguishability of some quantum states. At last, we give the $n$-party distinguishable quantum states in $m_1\otimes m_2\otimes\cdots\otimes m_n$. Our study further reveals quantum nonlocality in multipartite high-dimensional.

arXiv:1712.08970•Dec 24, 2017•6 citations

Classifying locally distinguishable sets: No activation across bipartitions

Atanu Bhunia, Saronath Halder, Ritabrata Sengupta

A set of orthogonal quantum states is said to be locally indistinguishable if they cannot be perfectly distinguished by local operations and classical communication (LOCC). Otherwise, the states are locally distinguishable. Interestingly, locally indistinguishable states can have productive applications in quantum information processing protocols. In this sense, locally indistinguishable states are useful. On the other hand, it is usual to consider that locally distinguishable states are useless. Nevertheless, recent works suggest that locally distinguishable states should be given due consideration as in certain situations these states can be converted to locally indistinguishable states under orthogonality-preserving LOCC (OP-LOCC). Such a counterintuitive phenomenon motivates us to ask when the aforesaid conversion is possible and when it is not. In this work, we provide different structures of locally distinguishable product and entangled states which do not allow the aforesaid conversion. We also provide certain structures of locally distinguishable states which allow the aforesaid conversion. In this way, we classify the locally distinguishable sets by introducing hierarchies among them. In a multipartite system, this study becomes more involved as there exist multipartite locally distinguishable sets which cannot be converted to locally indistinguishable sets by OP-LOCC across any bipartition. We say this as ``no activation across bi-partitions".

arXiv:2507.10698•Physical Review A•Jan 30, 2026•2 citations

Local distinguishability of orthogonal quantum states with multiple copies of 2 ⊗ 2 maximally entangled states

Zhichao Zhang, Yan-Qi Song, Tingting Song +3 more

For general quantum systems, many sets of locally indistinguishable orthogonal quantum states have been constructed so far. However, it is interesting how much entanglement resources are sufficient and/or necessary to distinguish these states by local operations and classical communication (LOCC). Here we first present a method to locally distinguish a set of orthogonal product states in $5\ensuremath{\bigotimes}5$ by using two copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. Then we generalize the distinguishing method for a class of orthogonal product states in $d\ensuremath{\bigotimes}d$ ($d$ is odd). Furthermore, for a class of nonlocality of orthogonal product states in $d\ensuremath{\bigotimes}d$ ($d\ensuremath{\ge}5$), we prove that these states can be distinguished by LOCC with two copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. Finally, for some multipartite orthogonal product states, we also present a similar method to locally distinguish these states with multiple copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. These results also reveal the phenomenon of less nonlocality with more entanglement.

Physical Review A•Feb 23, 2018•42 citations
In this paper, we give the local distinguishability of four orthogonal product states (OPSs) on multipartite and high-dimension systems. To characterize the different orthogonal relationships between four multipartite OPSs, a new concept, i.e., the vector of the numbers of pairwise orthogonal relations, is proposed. We classify four tripartite OPSs, any two of which are orthogonal only on one subsystem, into three categories by using the vectors of the numbers of pairwise orthogonal relations. We prove that any four tripartite OPSs belonging to two of the three categories can be perfectly distinguished by local operations and classical communication (LOCC). Meanwhile, we find an interesting phenomenon that two sets of four tripartite OPSs belonging to the remaining one of the three categories have different local distinguishability even if they have the same orthogonal graph. This means that the orthogonal graph is not a sufficient condition for determining the local distinguishability of a set of OPSs. Additionally, for the remaining category, we specify the conditions under which four tripartite OPSs can be locally distinguished and the conditions under which four tripartite OPSs cannot be perfectly distinguished by LOCC. On the other hand, we prove that any four tripartite OPSs with seven or more pairwise orthogonal relations can be perfectly distinguished by LOCC. Furthermore, we prove that any four OPSs with six or more pairwise orthogonal relations can be perfectly distinguished by LOCC on four-, five-, and six-partite systems.
The local distinguishability of five bipartite and tripartite orthogonal product states (OPSs) has been effectively resolved, while that of five quadripartite OPSs has not yet been effectively addressed. In this paper, we give the local distinguishability of five quadripartite OPSs, where any two of these five states are orthogonal on exactly one subsystem. Based on the vectors of the numbers of pairwise orthogonality relations, we classify different sets of five quadripartite OPSs into nine categories, which correspond to 371 distinct orthogonality graphs. We show that the sets of five quadripartite OPSs corresponding to 355 of the 371 cases can be perfectly distinguished by local operations and classical communication (LOCC), whereas the sets of five quadripartite OPSs for the remaining 16 cases cannot exactly distinguished by LOCC. Our results enrich the theory of nonlocality for sets of OPSs.
Local distinguishability of orthogonal quantum states can effectively reduce the consumption of quantum resources and lower economic costs in quantum protocols. Although numerous achievements have been made regarding local distinguishability of orthogonal quantum states, some fundamental issues have not been effectively addressed. For example, the local distinguishability of five orthogonal product states (OPSs) is still unknown up to now. In this paper, we give the properties of local distinguishability of five OPSs on bipartite and tripartite quantum systems. Firstly, to characterize the structure of a set of bipartite OPSs, we propose the concept of the vector of orthogonal relations for a set of bipartite OPSs. Secondly, we classify the structures of five bipartite OPSs into six categories by this concept and prove that five of these six categories can be perfectly distinguished by local operations and classical communication (LOCC). Thirdly we show that the local distinguishability of each case of the sixth category singly. On the other hand, we first divide the structures of five tripartite OPSs into eight categories by the vectors of orthogonal relations of five tripartite OPSs. Then we give the local distinguishability of each category. Our work enriches the research results of quantum nonlocality and will provide a clear understanding of the local distinguishability of five OPSs.
It is necessary to investigate the local distinguishability of orthogonal quantum state sets, as their adoption in protocol design helps diminish quantum state transmission and cut operational costs. In this paper, we explore the local distinguishability of six orthogonal product states (OPSs) on any bipartite quantum system. We classify different sets of six bipartite OPSs into eight categories by using the vectors of the numbers of pairwise orthogonality relations, where any two states are orthogonal on only one subsystem within each set. We find that these eight categories contain a total of 78 distinct cases, all but five of which are perfectly distinguishable via local operations and classical communication (LOCC). Furthermore, we discuss the local distinguishability of those five distinct cases in detail. Our work explicitly characterizes the local distinguishability of six bipartite OPSs.
In this paper, we mainly study the local distinguishable multipartite quantum states by local operations and classical communication (LOCC) in m1⊗m2⊗…⊗mnm_1\otimes m_2\otimes\ldots\otimes m_nm1​⊗m2​⊗…⊗mn​ , where the quantum system m1m_1m1​ belongs to Alice, m2m_2m2​ belongs to Bob, \ldots and mnm_nmn​ belongs to Susan. We first present the pure tripartite distinguishable orthogonal quantum states by LOCC in m1⊗m2⊗m3m_1\otimes m_2\otimes m_3m1​⊗m2​⊗m3​. With the conclusion in m1⊗m2⊗m3m_1\otimes m_2\otimes m_3m1​⊗m2​⊗m3​, we prove distinguishability or indistinguishability of some quantum states. At last, we give the nnn-party distinguishable quantum states in m1⊗m2⊗⋯⊗mnm_1\otimes m_2\otimes\cdots\otimes m_nm1​⊗m2​⊗⋯⊗mn​. Our study further reveals quantum nonlocality in multipartite high-dimensional.
A set of orthogonal quantum states is said to be locally indistinguishable if they cannot be perfectly distinguished by local operations and classical communication (LOCC). Otherwise, the states are locally distinguishable. Interestingly, locally indistinguishable states can have productive applications in quantum information processing protocols. In this sense, locally indistinguishable states are useful. On the other hand, it is usual to consider that locally distinguishable states are useless. Nevertheless, recent works suggest that locally distinguishable states should be given due consideration as in certain situations these states can be converted to locally indistinguishable states under orthogonality-preserving LOCC (OP-LOCC). Such a counterintuitive phenomenon motivates us to ask when the aforesaid conversion is possible and when it is not. In this work, we provide different structures of locally distinguishable product and entangled states which do not allow the aforesaid conversion. We also provide certain structures of locally distinguishable states which allow the aforesaid conversion. In this way, we classify the locally distinguishable sets by introducing hierarchies among them. In a multipartite system, this study becomes more involved as there exist multipartite locally distinguishable sets which cannot be converted to locally indistinguishable sets by OP-LOCC across any bipartition. We say this as ``no activation across bi-partitions".
For general quantum systems, many sets of locally indistinguishable orthogonal quantum states have been constructed so far. However, it is interesting how much entanglement resources are sufficient and/or necessary to distinguish these states by local operations and classical communication (LOCC). Here we first present a method to locally distinguish a set of orthogonal product states in $5\ensuremath{\bigotimes}5$ by using two copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. Then we generalize the distinguishing method for a class of orthogonal product states in $d\ensuremath{\bigotimes}d$ (ddd is odd). Furthermore, for a class of nonlocality of orthogonal product states in $d\ensuremath{\bigotimes}d$ ($d\ensuremath{\ge}5$), we prove that these states can be distinguished by LOCC with two copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. Finally, for some multipartite orthogonal product states, we also present a similar method to locally distinguish these states with multiple copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. These results also reveal the phenomenon of less nonlocality with more entanglement.