Necessary Condition for Local Distinguishability of Maximally Entangled States: Beyond Orthogonality Preservation

TL;DR

This paper introduces a new necessary condition for the local distinguishability of maximally entangled states in bipartite systems, which is stronger than the existing orthogonality preservation criterion and is shown to be both necessary and sufficient in certain cases.

Contribution

The authors derive a simple, stronger necessary condition for local distinguishability of MES, extending understanding beyond orthogonality preservation, and demonstrate its sufficiency for specific state sets.

Findings

The new condition is stronger than OP for MES distinguishability.

It is necessary for all sets of pairwise orthogonal MES in d7d7 systems.

It is sufficient for distinguishing four generalized Bell states in d7d7 systems.

Abstract

The (im)possibility of local distinguishability of orthogonal multipartite quantum states still remains an intriguing question. Beyond $\mathbb{C}^{3}\otimes\mathbb{C}^{3}$, the problem remains unsolved even for maximally entangled states (MES). So far, the only known condition for the local distinguishability of states is the well-known orthogonality preservation (OP). Using an upper bound on the locally accessible information for bipartite states, we derive a very simple necessary condition for any set of pairwise orthogonal MES in $\mathbb{C}^{d}\otimes \mathbb{C}^{d}$ to be perfectly locally distinguishable. This condition is seen to be stronger than the OP condition. This is particularly so for any set of $d$ number of pairwise orthogonal MES in $\mathbb{C}^{d}\otimes \mathbb{C}^{d}$. When testing this condition for the local distinguishability of all sets of four generalized Bell states in $\mathbb{C}^{4}\otimes \mathbb{C}^{4}$, we find that it is not only necessary but also sufficient to determine their local distinguishability. This demonstrates that the aforementioned upper-bound may play a significant role in the general scenario of local distinguishability of bipartite states.