Distinguishability of locally diagonal orthogonally invariant quantum states

TL;DR

This paper investigates the distinguishability of a broad class of quantum states under various measurement constraints, providing bounds and simplifications that enhance understanding of quantum state discrimination.

Contribution

It demonstrates that optimal measurements for LDOI states can be restricted to LDOI, reducing computational complexity and establishing conditions where LOCC, PPT, and separable measurements coincide.

Findings

Optimal PPT and separable measurements for LDOI states are also LDOI.

LOCC supremum can be approached by LDOI LOCC POVMs, reducing problem dimension.

The gap between PPT and LOCC distinguishability is at most (n-2)/(2n^2).

Abstract

We study the distinguishability of quantum states under local operations with classical communication (LOCC), separable, and positive-partial-transpose (PPT) measurements, focusing on locally diagonal orthogonally invariant (LDOI) states -- those invariant under local diagonal orthogonal twirling. This class includes many important families such as Werner states, isotropic states, X-states, and Dicke states. We show that optimal PPT and separable measurements for distinguishing LDOI states can always be taken to be LDOI, and the LOCC supremum can be approached by LDOI LOCC POVMs, enabling a dimensional reduction from $n^4$ to $O(n^2)$ in the associated optimization problems. We establish efficiently computable bounds on the distinguishability of orthonormal LDOI bases and prove that for a broad class of such bases -- including all two-qubit cases -- the LOCC supremum equals the PPT and separable optima. More generally, we show the gap between PPT and LOCC distinguishability is at most $(n-2)/(2n^2)$ for local dimension $n$.