  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.
