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.

翻译：量子态的反区分概念已被研究用于探究量子力学的基础问题。该方法亦被称为量子态消除，因为此类协议的核心目标是推测系统未被制备为有限随机选择态中的哪一个（即可视为消除过程的第一步）。反区分已被用于检验量子态实在性，排除了量子力学中$\psi$-本体论模型的可能性[Pusey等，Nat. Phys., 8(6):475-478, 2012]。鉴于反区分在量子力学中的既定重要性，对其进行深入探索具有充分必要性。本文对经典与量子反区分的最优误差指数（即最优错误概率渐近趋于零的速率）进行了系统性研究。我们推导出经典情况下最优误差指数的精确表达式，证明其由经典Chernoff-Hellinger散度给出，从而为该多变量散度赋予了具有操作意义的解释——作为反区分一组概率测度的最优误差指数。针对量子情形，我们给出最优误差指数的若干界：下界由态的最佳成对Chernoff散度构成，上界以最大相对熵表示，以及基于最小与最大量子Chernoff-Hellinger散度建立的上下界。量子反区分最优误差指数的显式表达式仍有待解决。