# A new quantum version of f-divergence. (arXiv:1311.4722v4 [quant-ph] UPDATED)

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 ($\mathrm{D}_{f}^{\max}$, 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.