In $2\ensuremath{\bigotimes}2$, more than two orthogonal Bell states with a single copy can never be discriminated with certainty if only local operations and classical communication (LOCC) are allowed. We show here that more than $d$ numbers of pairwise orthogonal maximally entangled states in $d\ensuremath{\bigotimes}d$, which are in canonical form, used by Bennett et al. [Phys. Rev. Lett. 70, 1895 (1993)], can never be discriminated with certainty by LOCC, when single copies of the states are provided. Interestingly we show here that all orthogonal maximally entangled states, which are in canonical form, can be discriminated with certainty by LOCC if and only if two copies of each of the states are provided. We provide here a conjecture regarding the highly nontrivial problem of local distinguishability of any $d$ or fewer numbers of pairwise orthogonal maximally entangled states in $d\ensuremath{\bigotimes}d$ (in the single copy case).In $2\ensuremath{\bigotimes}2$, more than two orthogonal Bell states with a single copy can never be discriminated with certainty if only local operations and classical communication (LOCC) are allowed. We show here that more than $d$ numbers of pairwise orthogonal maximally entangled states in $d\ensuremath{\bigotimes}d$, which are in canonical form, used by Bennett et al. [Phys. Rev. Lett. 70, 1895 (1993)], can never be discriminated with certainty by LOCC, when single copies of the states are provided. Interestingly we show here that all orthogonal maximally entangled states, which are in canonical form, can be discriminated with certainty by LOCC if and only if two copies of each of the states are provided. We provide here a conjecture regarding the highly nontrivial problem of local distinguishability of any $d$ or fewer numbers of pairwise orthogonal maximally entangled states in $d\ensuremath{\bigotimes}d$ (in the single copy case).

arXiv:quant-ph/0205105Physical Review AAug 10, 2004150 citations

Mixture of multiple copies of maximally entangled states is quasipure

Dong Yang, Yixin Chen

Employing the general bilateral controlled-NOT operation and local state discrimination, the mixed state of the form ${\ensuremath{\rho}}_{d}^{(k)}{=(1/d}^{2}){\ensuremath{\sum}}_{m,n=0}^{d\ensuremath{-}1}(|{\ensuremath{\varphi}}_{\mathrm{mn}}〉〈{\ensuremath{\varphi}}_{\mathrm{mn}}|{)}^{\ensuremath{\bigotimes}k}$ is proved to be quasipure, where ${|{\ensuremath{\varphi}}_{\mathrm{mn}}〉}$ is the canonical set of mutually orthogonal maximally entangled states in $d\ifmmode\times\else\texttimes\fi{}d.$ Therefore irreversibility does not occur in the process of distillation for this family of states. Also, the distillable entanglement is calculated explicitly.Employing the general bilateral controlled-NOT operation and local state discrimination, the mixed state of the form ${\ensuremath{\rho}}_{d}^{(k)}{=(1/d}^{2}){\ensuremath{\sum}}_{m,n=0}^{d\ensuremath{-}1}(|{\ensuremath{\varphi}}_{\mathrm{mn}}〉〈{\ensuremath{\varphi}}_{\mathrm{mn}}|{)}^{\ensuremath{\bigotimes}k}$ is proved to be quasipure, where ${|{\ensuremath{\varphi}}_{\mathrm{mn}}〉}$ is the canonical set of mutually orthogonal maximally entangled states in $d\ifmmode\times\else\texttimes\fi{}d.$ Therefore irreversibility does not occur in the process of distillation for this family of states. Also, the distillable entanglement is calculated explicitly.

arXiv:quant-ph/0304194Physical Review AFeb 9, 200410 citations

Polynomial Separation between One-Way and Two-Way LOCC in Discrimination of Maximally Entangled States

Youngrong Lim

One-way local operations and classical communication (LOCC) can fail to distinguish four orthogonal maximally entangled states in every local dimension $d\ge4$ . We show that every set of $k$ pairwise orthogonal maximally entangled states in $\mathbb C^d\otimes\mathbb C^d$ is perfectly distinguishable by an $A\to B\to A$ protocol whenever $d\ge Ck^4[\log(2k)]^ν$ , for universal constants $C,ν>0$ . A single feedback message therefore raises the number of candidates that can always be distinguished from a constant to $Ω\!\left(d^{1/4}/(\log d)^{ν/4}\right)$ , establishing a polynomial separation between one-way and two-way LOCC. In our protocol, Alice preserves exact orthogonality while reducing overlaps between Bob's conditional subspaces. Bob reduces the candidates to a singleton or an orthogonal pair and refines his outcome so that Alice can complete the discrimination. As a corollary, nine copies suffice for perfect two-way LOCC discrimination of any orthogonal maximally entangled set in all sufficiently large local dimensions.One-way local operations and classical communication (LOCC) can fail to distinguish four orthogonal maximally entangled states in every local dimension $d\ge4$. We show that every set of $k$ pairwise orthogonal maximally entangled states in $\mathbb C^d\otimes\mathbb C^d$ is perfectly distinguishable by an $A\to B\to A$ protocol whenever $d\ge Ck^4[\log(2k)]^ν$, for universal constants $C,ν>0$. A single feedback message therefore raises the number of candidates that can always be distinguished from a constant to $Ω\!\left(d^{1/4}/(\log d)^{ν/4}\right)$, establishing a polynomial separation between one-way and two-way LOCC. In our protocol, Alice preserves exact orthogonality while reducing overlaps between Bob's conditional subspaces. Bob reduces the candidates to a singleton or an orthogonal pair and refines his outcome so that Alice can complete the discrimination. As a corollary, nine copies suffice for perfect two-way LOCC discrimination of any orthogonal maximally entangled set in all sufficiently large local dimensions.

quant-pharXiv:2610.04288Oct 6, 20260 citations

d

numbers of pairwise orthogonal maximally entangled states in $d\ensuremath{\bigotimes}d$, which are in canonical form, used by Bennett et al. [Phys. Rev. Lett. 70, 1895 (1993)], can never be discriminated with certainty by LOCC, when single copies of the states are provided. Interestingly we show here that all orthogonal maximally entangled states, which are in canonical form, can be discriminated with certainty by LOCC if and only if two copies of each of the states are provided. We provide here a conjecture regarding the highly nontrivial problem of local distinguishability of any

d

or fewer numbers of pairwise orthogonal maximally entangled states in $d\ensuremath{\bigotimes}d$ (in the single copy case).

One-way local operations and classical communication (LOCC) can fail to distinguish four orthogonal maximally entangled states in every local dimension

d\ge4

. We show that every set of

k

pairwise orthogonal maximally entangled states in

\mathbb C^d\otimes\mathbb C^d

is perfectly distinguishable by an

A\to B\to A

protocol whenever

d\ge Ck^4[\log(2k)]^ν

, for universal constants

C,ν>0

. A single feedback message therefore raises the number of candidates that can always be distinguished from a constant to

Ω\!\left(d^{1/4}/(\log d)^{ν/4}\right)

, establishing a polynomial separation between one-way and two-way LOCC. In our protocol, Alice preserves exact orthogonality while reducing overlaps between Bob's conditional subspaces. Bob reduces the candidates to a singleton or an orthogonal pair and refines his outcome so that Alice can complete the discrimination. As a corollary, nine copies suffice for perfect two-way LOCC discrimination of any orthogonal maximally entangled set in all sufficiently large local dimensions.