Tight bounds for antidistinguishability and circulant sets of pure quantum states
Nathaniel Johnston, Vincent M. Russo, Jamie Sikora · Quantum · 2025
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 ( n − 2 ) / ( 2 n − 2 ) then the set must be antidistinguishable, and we show that this bound is tight when n ≤ 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 .