About cookies on this site Our websites require some cookies to function properly (required). In addition, other cookies may be used with your consent to analyze site usage, improve the user experience and for advertising. For more information, please review your options. By visiting our website, you agree to our processing of information as described in IBM’sprivacy statement. To provide a smooth navigation, your cookie preferences will be shared across the IBM web domains listed here.
Publication
Annals of Statistics
Paper
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.