# Discriminating quantum states: The multiple chernoff distance

## Abstract

We consider the problem of testing multiple quantum hypotheses {p n 1 , . . . , p n r }, where an arbitrary prior distribution is given and each of the r hypotheses is n copies of a quantum state. It is known that the minimal average error probability Pe decays exponentially to zero, that is, Pe = exp{an+o(n)}. However, this error exponent I is generally unknown, except for the case that r = 2. In this paper, we solve the long-standing open problem of identifying the above error exponent, by proving Nussbaum and SzkoÅa's conjecture that I = mini=j C(pi,pj ). The right-hand side of this equality is called the multiple quantum Chernoff distance, and C(pi,pj ) := max0â‰sâ‰1{âlogTr ps i p 1âs j } has been previously identified as the optimal error exponent for testing two hypotheses, p n i versus p n j . The main ingredient of our proof is a new upper bound for the average error probability, for testing an ensemble of finite-dimensional, but otherwise general, quantum states. This upper bound, up to a states-dependent factor, matches the multiple-state generalization of Nussbaum and SzkoÅa's lower bound. Specialized to the case r = 2, we give an alternative proof to the achievability of the binary-hypothesis Chernoff distance, which was originally proved by Audenaert et al.