Asymptotic error rates in quantum hypothesis testing

We give an overview of some recent results in quantum hypothesis testing mostly presented in [1]. Using both a new trace inequality for pairs of density operators due to Audenaert et al. [2] and a special mapping from pairs of density operators to pairs of probability distributions [3] we derive quantum extensions of the Chernoff distance and the Hoeffding bound. As a byproduct we demonstrate how one part of the quantum Stein's Lemma -the achievability of the quantum relative entropy as the best rate of the probability of error of the type-II- arises from these quantities. Moreover, we discuss the properties of the quantum Chernoff distance as a distance measure on the quantum state space of finite-dimensional complex Hilbert spaces.

[1] K.M.R. Audenaert, M. Nussbaum, A. Szkola, F. Verstraete, "Asymptotic Error Rates in Quantum Hypothesis Testing", Preprint No. 84/2007, MPI MiS Leipzig
[2] K.M.R. Audenaert, J. Calsamiglia, R. Munoz-Tapia, E. bagan, Li. masanes, A. Acin and F. Verstraete, " Discriminating States: The Quantum Chernoff Bound" , quant-ph/0607216
[3] M. Nussbaum, A. Szkola, "The Chernoff lower bound in quantum hypothesis testing", Preprint No. 69/2006, MPI MiS Leipzig