An asymptotic error bound for discriminating between several quantum states

Statistics and Modeling for Complex Data

Given a K-tuple of density matrices, we consider the problem of detecting the true state of a quantum system on the basis of measurements performed on N  copies.  We investigate the exponential rate of decay of the averaged error probability of the optimal quantum detector. In the classical case of several  probability measures on a finite sample space,  it is known that the optimal error exponent is given by the worst case binary Chernoff bound between any possible pair from the K distributions, and attained by the MLE. In the quantum case, the analogous worst case binary quantum Chernoff bound may be called the multiple quantum Chernoff bound. Recently it has been shown that this bound is generally unimprovable, and also attainable in the case of K pure states. We extend the attainability result to a larger class of K-tuples of states which are possibly mixed, but are all singular (nonfaithful). For arbitrary finite dimensional states, we construct a detector which attains the multiple quantum Chernoff bound up to a factor 1/3.