Tight bounds for antidistinguishability and circulant sets of pure quantum states

TL;DR

This paper establishes a connection between antidistinguishability of pure quantum states and Gram matrix properties, providing explicit criteria and bounds for circulant sets, with implications for quantum resource theories.

Contribution

It introduces a novel equivalence between antidistinguishability and Gram matrix incoherence, and provides explicit, non-semidefinite programming criteria for circulant sets.

Findings

Antidistinguishability is equivalent to Gram matrix $(n-1)$-incoherence.

Explicit formula for circulant Gram matrices determines antidistinguishability.

Bounds on inner products for antidistinguishability are tight for certain $n$.

Abstract

A set of pure quantum states is said to be antidistinguishable if upon sampling one at random, there exists a measurement to perfectly determine some state that was not sampled. We show that antidistinguishability of a set of $n$ pure states is equivalent to a property of its Gram matrix called $(n-1)$-incoherence, thus establishing a connection with quantum resource theories that lets us apply a wide variety of new tools to antidistinguishability. As a particular application of our result, we present an explicit formula (not involving any semidefinite programming) that determines whether or not a set with a circulant Gram matrix is antidistinguishable. We also show that if all inner products are smaller than $\sqrt{(n-2)/(2n-2)}$ then the set must be antidistinguishable, and we show that this bound is tight when $n \leq 4$. We also give a simpler proof that if all the inner products are strictly larger than $(n-2)/(n-1)$, then the set cannot be antidistinguishable, and we show that this bound is tight for all $n$.