Detecting the local indistinguishability of maximally entangled states

TL;DR

This paper develops a new criterion for detecting local indistinguishability of maximally entangled states, enabling systematic construction of such states in various dimensions and revealing their properties under local protocols.

Contribution

It introduces a computable criterion incorporating asymmetry of local protocols and constructs explicit sets of locally indistinguishable maximally entangled states for all dimensions d ≥ 4.

Findings

Systematic construction of locally indistinguishable maximally entangled states for all d ≥ 4.

Explicit small sets of states with ratio approaching 3/4 of d.

Existence of d-1 maximally entangled states in even d ≥ 6 that are locally indistinguishable.

Abstract

By incorporating the asymmetry of local protocols, i.e., some party has to start with a nontrivial measurement, into an operational method of detecting the local indistinguishability proposed by Horodecki {\it et al.} [Phys.Rev.Lett. 90 047902 (2003)], we derive a computable criterion to efficiently detect the local indistinguishability of maximally entangled states. Locally indistinguishable sets of $d$ maximally entangled states in a $d\otimes d$ system are systematically constructed for all $d\ge 4$ as an application. Furthermore, by exploiting the fact that local protocols are necessarily separable, we explicitly construct small sets of $k$ locally indistinguishable maximally entangled states with the ratio $k/d$ approaching 3/4. In particular, in a $d\otimes d$ system with even $d\ge 6$, there always exist $d-1$ maximally entangled states that are locally indistinguishable by separable measurements.