Small sets of locally indistinguishable orthogonal maximally entangled states

TL;DR

This paper proves the existence of small sets of orthogonal maximally entangled states in bipartite quantum systems that cannot be distinguished by LOCC, extending previous results to cases where the number of states is less than the local dimension.

Contribution

It provides the first constructive proof that such indistinguishable sets exist for fewer than d states, also applicable to PPT measurements.

Findings

Existence of indistinguishable sets for k < d states

Constructive proof using semidefinite programming

Applicable to PPT measurement class

Abstract

We study the problem of distinguishing quantum states using local operations and classical communication (LOCC). A question of fundamental interest is whether there exist sets of $k \leq d$ orthogonal maximally entangled states in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ that are not perfectly distinguishable by LOCC. A recent result by Yu, Duan, and Ying [Phys. Rev. Lett. 109 020506 (2012) -- arXiv:1107.3224 [quant-ph]] gives an affirmative answer for the case $k = d$. We give, for the first time, a proof that such sets of states indeed exist even in the case $k < d$. Our result is constructive and holds for an even wider class of operations known as positive-partial-transpose measurements (PPT). The proof uses the characterization of the PPT-distinguishability problem as a semidefinite program.