On the optimal error exponents for classical and quantum antidistinguishability

The concept of antidistinguishability of quantum states has been studied to investigate foundational questions in quantum mechanics. It is also called quantum state elimination, because the goal of such a protocol is to guess which state, among finitely many chosen at random, the system is not prepared in (that is, it can be thought of as the first step in a process of elimination). Antidistinguishability has been used to investigate the reality of quantum states, ruling out $\psi$-epistemic ontological models of quantum mechanics [Pusey et al., Nat. Phys., 8(6):475-478, 2012]. Thus, due to the established importance of antidistinguishability in quantum mechanics, exploring it further is warranted. In this paper, we provide a comprehensive study of the optimal error exponent -- the rate at which the optimal error probability vanishes to zero asymptotically -- for classical and quantum antidistinguishability. We derive an exact expression for the optimal error exponent in the classical case and show that it is given by the classical Chernoff--Hellinger divergence. Our work thus provides this multi-variate divergence with a meaningful operational interpretation as the optimal error exponent for antidistinguishing a set of probability measures. We provide several bounds on the optimal error exponent in the quantum case: a lower bound given by the best pairwise Chernoff divergence of the states, an upper bound in terms of max-relative entropy, and lower and upper bounds in terms of minimal and maximal quantum Chernoff--Hellinger divergences. It remains an open problem to obtain an explicit expression for the optimal error exponent for quantum antidistinguishability.