A new quantum version of fdivergence
Abstract
This paper proposes and studies new quantum version of $f$divergences, a class of convex functionals of a pair of probability distributions including KullbackLeibler divergence, Rnyitype relative entropy and so on. There are several quantum versions so far, including the one by Petz. We introduce another quantum version ($\mathrm{D}_{f}^{\max}$, 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 $\mathrm{D}_{f}^{\max}$ is given either if $f$ is operator convex, or if one of the state is a pure state. Also, concise representation of $\mathrm{D}_{f}^{\max}$ 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 $\mathrm{D}_{f}^{\max}$, we show: Suppose $f$ is operator convex. Then the\ maximum $f\,$ divergence of the probability distributions of a measurement under the state $\rho$ and $\sigma$ is strictly less than $\mathrm{D}_{f}^{\max}\left( \rho\Vert\sigma\right) $. This statement may seem intuitively trivial, but when $f$ is not operator convex, this is not always true. A counter example is $f\left( \lambda\right) =\left\vert 1\lambda\right\vert $, which corresponds to total variation distance. We mostly work on finite dimensional Hilbert space, but some results are extended to infinite dimensional case.
 Publication:

arXiv eprints
 Pub Date:
 November 2013
 arXiv:
 arXiv:1311.4722
 Bibcode:
 2013arXiv1311.4722M
 Keywords:

 Quantum Physics
 EPrint:
 The proof of dual representation of the former version was misstated. An alternative proof is presented