2013/11/19 by Keiji Matsumoto, Matsumoto, Keiji · 10 citations
Decision Sciences · Mathematics · Physics and Astronomy · #Mathematical Inequalities and Applications #Multi-Criteria Decision Making #Statistical Mechanics and Entropy #quant-ph
paper · pdf · doi:10.48550/arxiv.1311.4722
The proof of dual representation of the former version was misstated. An alternative proof is presented
arxiv created 2018/02/06 · arxiv updated 2018/02/07
This paper proposes and studies new quantum version of f-divergences, a class of convex functionals of a pair of probability distributions including Kullback-Leibler divergence, Rnyi-type relative entropy and so on. There are several quantum versions so far, including the one by Petz. We introduce another quantum version (Dfmax, 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 Dfmax is given either if f is operator convex, or if one of the state is a pure state. Also, concise representation of Dfmax 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 Dfmax, we show: Suppose f is operator convex. Then the maximum f - divergence of the probability distributions of a measurement under the state ρ and σ is strictly less than Dfmax( ρ\Vertσ) . This statement may seem intuitively trivial, but when f is not operator convex, this is not always true. A counter example is f( λ) =\vert 1-λ\vert , which corresponds to total variation distance. We mostly work on finite dimensional Hilbert space, but some results are extended to infinite dimensional case.