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arXiv:2610.04288

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

Youngrong Lim

quant-phSubmitted Oct 3, 2026• Announced Oct 6, 2026• 0 citations
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Abstract

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.

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5+12 pages, 1 figure

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We show that there exist sets of three mutually orthogonal ddd-dimensional maximally entangled states which cannot be perfectly distinguished using one-way local operations and classical communication (LOCC) for arbitrarily large values of ddd. This contrasts with several well-known families of maximally entangled states, for which any three states can be perfectly distinguished. We then show that two-way LOCC is sufficient to distinguish these examples. We also show that any three mutually orthogonal ddd-dimensional maximally entangled states can be perfectly distinguished using measurements with a positive partial transpose (PPT) and can be distinguished with one-way LOCC with high probability. These results circle around the question of whether there exist three maximally entangled states which cannot be distinguished using the full power of LOCC; we discuss possible approaches to answer this question.
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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.
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.
We explicitly exhibit a set of four ququad-ququad orthogonal maximally entangled states that cannot be perfectly distinguished by means of local operations and classical communication. Before our work, it was unknown whether there is a set of d locally indistinguishable d⊗d orthogonal maximally entangled states for some positive integer d. We further show that a 2⊗2 maximally entangled state can be used to locally distinguish this set of states without being consumed, thus demonstrate a novel phenomenon of entanglement discrimination catalysis. Based on this set of states, we construct a new set K consisting of four locally indistinguishable states such that K(⊗m) (with 4(m) members) is locally distinguishable for some m greater than one. As an immediate application, we construct a noisy quantum channel with one sender and two receivers whose local zero-error classical capacity can achieve the full dimension of the input space but only with a multi-shot protocol.