Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz quasi-entropies. Many well known distinguishability measures of quantum states are given by, or derived from, f-divergences; s
pecial examples include the quantum relative entropy, the Renyi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under the dual of Schwarz maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Holder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.
We show that the quantum $alpha$-relative entropies with parameter $alphain (0,1)$ can be represented as generalized cutoff rates in the sense of [I. Csiszar, IEEE Trans. Inf. Theory 41, 26-34, (1995)], which provides a direct operational interpretat
ion to the quantum $alpha$-relative entropies. We also show that various generalizations of the Holevo capacity, defined in terms of the $alpha$-relative entropies, coincide for the parameter range $alphain (0,2]$, and show an upper bound on the one-shot epsilon-capacity of a classical-quantum channel in terms of these capacities.
We apply the recent results of F. Hiai, M. Mosonyi and T. Ogawa [arXiv:0707.2020, to appear in J. Math. Phys.] to the asymptotic hypothesis testing problem of locally faithful shift-invariant quasi-free states on a CAR algebra. We use a multivariate
extension of Szegos theorem to show the existence of the mean Chernoff and Hoeffding bounds and the mean relative entropy, and show that these quantities arise as the optimal error exponents in suitable settings.
