A New Quantum Version of f-Divergence

Abstract

This paper proposes and studies new quantum version of f-divergences, a class of functionals of a pair of probability distributions including Kullback–Leibler divergence, Renyi-type relative entropy and so on. distance. There are several quantum versions so far, including the one by Petz (Rev Math Phys 23:691–747, 2011, [1]). We introduce another quantum version, below), defined as the solution to an optimization problem, or the minimum classical f-divergence necessary to generate a given pair of quantum states. It turns out to be the largest quantum f-divergence. The closed formula of is given either if f is operator convex, or if one of the state is a pure state. Also, concise representation of as a pointwise supremum of linear functionals is given and used for the clarification of various properties of the quality. Using the closed formula of we show: Suppose f is operator convex. Then the maximum f-divergence of the probability distributions of a measurement under the state and is strictly less than (formula presented). This statement may seem intuitively trivial, but when f is not operator convex, this is not always true. A counter example is (formula presented), which corresponds to total variation distance. We mostly work on finite dimensional Hilbert space, but some results are extended to infinite dimensional case.