Inner products of pure states and their antidistinguishability

We study the antidistinguishability problem, which is a fundamental task in quantum computing. A set of $d$ quantum states is said to be antidistinguishable if there exists a $d$-outcome positive-operator-valued measure that can perfectly identify which state was not measured. We revisit a conjecture by Havlíçek and Barrett which states that if a set of $d$ pure states has small pairwise inner products, then the set must be antidistinguishable. We develop a certificate of antidistinguishability via semidefinite programming duality and use it to provide a counterexample to this conjecture when $d=4$. Our work thus opens up again the investigation into which sets of pure states are antidistinguishable.