Upper bounds on the error probabilities and asymptotic error exponents in quantum multiple state discrimination

Abstract

We consider the multiple hypothesis testing problem for symmetric quantum state discrimination between r given states σ_1,...,σ_r. By splitting up the overall test into multiple binary tests in various ways we obtain a number of upper bounds on the optimal error probability in terms of the binary error probabilities. These upper bounds allow us to deduce various bounds on the asymptotic error rate, for which it has been hypothesised that it is given by the multi-hypothesis quantum Chernoff bound (or Chernoff divergence) C(σ_1,...,σ_r), as recently introduced by Nussbaum and Szkoła in analogy with Salikhov's classical multi-hypothesis Chernoff bound. This quantity is defined as the minimum of the pairwise binary Chernoff divergences min_{j<k}C(σ_j,σ_k). It was known already that the optimal asymptotic rate must lie between C/3 and C, and that for certain classes of sets of states the bound is actually achieved. It was known to be achieved, in particular, when the state pair that is closest together in Chernoff divergence is more than 6 times closer than the next closest pair. Our results improve on this in two ways. Firstly, we show that the optimal asymptotic rate must lie between C/2 and C. Secondly, we show that the Chernoff bound is already achieved when the closest state pair is more than 2 times closer than the next closest pair. We also show that the Chernoff bound is achieved when at least $r-2$ of the states are pure, improving on a previous result by Nussbaum and Szkoła. Finally, we indicate a number of potential pathways along which a proof (or disproof) may eventually be found that the multi-hypothesis quantum Chernoff bound is always achieved.