Tight bounds for antidistinguishability and circulant sets of pure quantum states

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 (n1)-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 (n2)/(2n2) then the set must be antidistinguishable, and we show that this bound is tight when n4. We also give a simpler proof that if all the inner products are strictly larger than (n2)/(n1), then the set cannot be antidistinguishable, and we show that this bound is tight for all n.