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 multivariate classical Chernoff divergence. Our work thus provides this divergence with a meaningful operational interpretation as the optimal error exponent for antidistinguishing a set of probability measures. For the quantum case, we provide several bounds on the optimal error exponent: a lower bound given by the best pairwise Chernoff divergence of the states, a single-letter semi-definite programming upper bound, and lower and upper bounds in terms of minimal and maximal multivariate quantum Chernoff divergences. It remains an open problem to obtain an explicit expression for the optimal error exponent for quantum antidistinguishability.

翻译：量子态反区分概念已被研究用于探讨量子力学的基础问题。该概念亦称为量子态消除，因为此类协议的目标是：在随机选取的有限量子态中，推断系统未处于哪个态（即可视为消除过程的第一步）。反区分已被用于探究量子态的实在性，推翻了量子力学中ψ-实在论的本体论模型[Pusey等人，《自然·物理学》，第8卷第6期第475-478页，2012年]。鉴于反区分在量子力学中的既定重要性，对其进行深入研究具有必要性。本文对经典与量子反区分的优化误差指数（即最优错误概率渐近趋于零的速率）进行了全面研究。我们推导了经典情形下优化误差指数的精确表达式，并证明其等于多元经典Chernoff散度。该工作为这一散度赋予了具有操作意义的解释——即经典反区分概率测度集的优化误差指数。对于量子情形，我们给出了优化误差指数的若干界：基于态对最佳Chernoff散度的下界、单字母半定规划上界，以及基于最小与最大多元量子Chernoff散度的上下界。量子反区分优化误差指数的显式表达式仍有待探究。