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

arXiv:1509.08450 · Physical Review A

Framework for distinguishability of orthogonal bipartite states by one-way local operations and classical communication

Tanmay Singal

Published Mar 7, 2016• 6 citations
arXivHTMLPDF

Abstract

In the topic of perfect local distinguishability of orthogonal multipartite quantum states, most results obtained so far pertain to bipartite systems whose subsystems are of specific dimensions. In contrast very few results for bipartite systems whose subsystems are of arbitrary dimensions, are known. This is because a rich variety of (algebraic or geometric) structure is exhibited by different sets of orthogonal states owing to which it is difficult to associate some common property underlying them all, i.e., a common property that would play a crucial role in the local distinguishability of these states. In this Rapid Communication, I propose a framework for the distinguishability by one-way local operations and classical communication (1-LOCC) of sets of orthogonal bipartite states in a ${d}_{A}\ensuremath{\bigotimes}{d}_{B}$ bipartite system, where ${d}_{A},{d}_{B}$ are the dimensions of both subsytems, labeled as $A$ and $B$. I show that if the $i\mathrm{th}$ party (where $i=A,B$) can initiate a 1-LOCC protocol to perfectly distinguish among a set of orthogonal bipartite states, then the information of the existence of such a 1-LOCC protocol lies in a subspace of ${d}_{i}\ifmmode\times\else\texttimes\fi{}{d}_{i}$ Hermitian matrices, denoted by ${\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$, and that the method to extract this information (of the existence of this 1-LOCC protocol) from ${\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$ depends on the value of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$. In this way one can give sweeping results for the 1-LOCC (in)distinguishability of all sets of orthogonal bipartite states corresponding to certain values of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$. Thus I propose that the value of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$ gives the common underlying property based on which sweeping results for the 1-LOCC (in)distinguishability of orthogonal bipartite quantum states can be made.

DOI
10.1103/physreva.93.030301

Related papers

Structure for Distinguishability of Orthogonal Bipartite States by One-Way LOCC

Tanmay Singal

In the topic of perfect local distinguishability of orthogonal multipartite quantum states, most results obtained so far pertain to bipartite systems whose subsystems are of specific dimensions. In contrast very few results for bipartite systems whose subsystems are of arbitrary dimensions, are known. This is because a rich variety of (algebraic or geometric) structure is exhibited by different sets of orthogonal states owing to which it is difficult to associate some common property underlying them all, i.e., a common property that would play a crucial role in the local distinguishability of these states. In this paper, I propose a framework for the distinguishability by one-way LOCC ($1$-LOCC) of sets of orthogonal bipartite states in a $d_A \otimes d_B$ bipartite system, where $d_A, d_B$ are the dimensions of both subsytems, labelled as $A$ and $B$. I show that if the $i$-th party (where $i=A,B$) can initiate a $1$-LOCC protocol to perfectly distinguish among a set of orthogonal bipartite states, then the information of the existence of such a $1$-LOCC protocol lies in a subspace of $d_i \times d_i$ hermitian matrices, denoted by $\Tb^{(i)}$, and that the method to extract this information (of the existence of this $1$-LOCC protocol) from $\Tb^{(i)}$ depends on the value of $dim \Tb^{(i)}$. In this way one can give sweeping results for the $1$-LOCC (in)distinguishability of all sets of orthogonal bipartite states corresponding to certain values of $dim \Tb^{(i)}$. Thus I propose that the value of $dim \Tb^{(i)}$ gives the common underlying property based on which sweeping results for the $1$-LOCC (in)distinguishability of orthogonal bipartite quantum states can be made.

•Sep 28, 2015•1 citation

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 Multipartite Orthogonal Quantum States: Generalized and Simplified

Ian George, Mohammad A. Alhejji

In a seminal work [PRL85.4972], Walgate, Short, Hardy, and Vedral prove in finite dimensions that for every pair of pure multipartite orthogonal quantum states, there exists a one-way local operations and classical communication (LOCC) protocol that perfectly distinguishes the pair. We extend this result to infinite dimensions with a simpler proof. For states on $\mathbb{C}^{d_A \times d_A} \otimes \mathbb{C}^{d_B \times d_B}$, we strengthen this existence result by constructing an $O(d_A^2 d_B^2)$-time algorithm that specifies such a perfect one-way LOCC protocol. Finally, we establish the equivalence between Walgate et al.'s result and the fact that the one-shot environment-assisted classical capacity of every quantum channel is at least 1 bit per channel use, thereby clarifying the literature on these notions. At the core of all of these results is the fact that every operator with vanishing trace admits a basis where its diagonal entries are all zero.

quant-pharXiv:2601.20074•Jan 29, 2026•0 citations

An algorithm for one-way LOCC distinguishability of orthogonal bipartite states

Meng-Meng Si, Zhihui Li, Chen‐Ming Bai

Local distinguishability of quantum states is a basic problem in quantum information theory. For an arbitrary group of entangled states in d ⊗ d bipartite system, Singal presented a subspace related to these states, and then he determined that the group of states can be distinguished by one-way local operations and classical communication(1-LOCC) if this subspace contains a maximally Abelian subspace. In this paper according to Singal's results, we present an algorithm to distinguish orthogonal bipartite states in d ⊗ d system based on 1-LOCC; we gives a sufficient condition to judge whether generalized orthogonal quantum states in 4 ⊗ 4 can be distinguished. Finally, we show an corresponding example to verify the conclusion.

IEEE International Conference on Nanotechnology (IEEE-NANO)•Jul 1, 2017•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

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
In the topic of perfect local distinguishability of orthogonal multipartite quantum states, most results obtained so far pertain to bipartite systems whose subsystems are of specific dimensions. In contrast very few results for bipartite systems whose subsystems are of arbitrary dimensions, are known. This is because a rich variety of (algebraic or geometric) structure is exhibited by different sets of orthogonal states owing to which it is difficult to associate some common property underlying them all, i.e., a common property that would play a crucial role in the local distinguishability of these states. In this Rapid Communication, I propose a framework for the distinguishability by one-way local operations and classical communication (1-LOCC) of sets of orthogonal bipartite states in a dA\ensuremath⨂dB{d}_{A}\ensuremath{\bigotimes}{d}_{B}dA​\ensuremath⨂dB​ bipartite system, where dA,dB{d}_{A},{d}_{B}dA​,dB​ are the dimensions of both subsytems, labeled as AAA and BBB. I show that if the ithi\mathrm{th}ith party (where i=A,Bi=A,Bi=A,B) can initiate a 1-LOCC protocol to perfectly distinguish among a set of orthogonal bipartite states, then the information of the existence of such a 1-LOCC protocol lies in a subspace of di\ifmmode×\else\texttimes\fidi{d}_{i}\ifmmode\times\else\texttimes\fi{}{d}_{i}di​\ifmmode×\else\texttimes\fidi​ Hermitian matrices, denoted by T\ensuremath⊥(i){\mathcal{T}}_{\ensuremath{\perp}}^{(i)}T\ensuremath⊥(i)​, and that the method to extract this information (of the existence of this 1-LOCC protocol) from T\ensuremath⊥(i){\mathcal{T}}_{\ensuremath{\perp}}^{(i)}T\ensuremath⊥(i)​ depends on the value of dimT\ensuremath⊥(i)\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}dimT\ensuremath⊥(i)​. In this way one can give sweeping results for the 1-LOCC (in)distinguishability of all sets of orthogonal bipartite states corresponding to certain values of dimT\ensuremath⊥(i)\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}dimT\ensuremath⊥(i)​. Thus I propose that the value of dimT\ensuremath⊥(i)\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}dimT\ensuremath⊥(i)​ gives the common underlying property based on which sweeping results for the 1-LOCC (in)distinguishability of orthogonal bipartite quantum states can be made.
In the topic of perfect local distinguishability of orthogonal multipartite quantum states, most results obtained so far pertain to bipartite systems whose subsystems are of specific dimensions. In contrast very few results for bipartite systems whose subsystems are of arbitrary dimensions, are known. This is because a rich variety of (algebraic or geometric) structure is exhibited by different sets of orthogonal states owing to which it is difficult to associate some common property underlying them all, i.e., a common property that would play a crucial role in the local distinguishability of these states. In this paper, I propose a framework for the distinguishability by one-way LOCC (111-LOCC) of sets of orthogonal bipartite states in a dA⊗dBd_A \otimes d_BdA​⊗dB​ bipartite system, where dA,dBd_A, d_BdA​,dB​ are the dimensions of both subsytems, labelled as AAA and BBB. I show that if the iii-th party (where i=A,Bi=A,Bi=A,B) can initiate a 111-LOCC protocol to perfectly distinguish among a set of orthogonal bipartite states, then the information of the existence of such a 111-LOCC protocol lies in a subspace of di×did_i \times d_idi​×di​ hermitian matrices, denoted by $\Tb^{(i)}$, and that the method to extract this information (of the existence of this 111-LOCC protocol) from $\Tb^{(i)}$ depends on the value of $dim \Tb^{(i)}$. In this way one can give sweeping results for the 111-LOCC (in)distinguishability of all sets of orthogonal bipartite states corresponding to certain values of $dim \Tb^{(i)}$. Thus I propose that the value of $dim \Tb^{(i)}$ gives the common underlying property based on which sweeping results for the 111-LOCC (in)distinguishability of orthogonal bipartite quantum states can be made.
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 a seminal work [PRL85.4972], Walgate, Short, Hardy, and Vedral prove in finite dimensions that for every pair of pure multipartite orthogonal quantum states, there exists a one-way local operations and classical communication (LOCC) protocol that perfectly distinguishes the pair. We extend this result to infinite dimensions with a simpler proof. For states on CdA×dA⊗CdB×dB\mathbb{C}^{d_A \times d_A} \otimes \mathbb{C}^{d_B \times d_B}CdA​×dA​⊗CdB​×dB​, we strengthen this existence result by constructing an O(dA2dB2)O(d_A^2 d_B^2)O(dA2​dB2​)-time algorithm that specifies such a perfect one-way LOCC protocol. Finally, we establish the equivalence between Walgate et al.'s result and the fact that the one-shot environment-assisted classical capacity of every quantum channel is at least 1 bit per channel use, thereby clarifying the literature on these notions. At the core of all of these results is the fact that every operator with vanishing trace admits a basis where its diagonal entries are all zero.
Local distinguishability of quantum states is a basic problem in quantum information theory. For an arbitrary group of entangled states in d ⊗ d bipartite system, Singal presented a subspace related to these states, and then he determined that the group of states can be distinguished by one-way local operations and classical communication(1-LOCC) if this subspace contains a maximally Abelian subspace. In this paper according to Singal's results, we present an algorithm to distinguish orthogonal bipartite states in d ⊗ d system based on 1-LOCC; we gives a sufficient condition to judge whether generalized orthogonal quantum states in 4 ⊗ 4 can be distinguished. Finally, we show an corresponding example to verify the conclusion.
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.
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".