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 ($\mathrm {D}_{f}^{\max }$, 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 $f\left( \lambda \right) =\left| 1-\lambda \right|$, which corresponds to total variation distance. We mostly work on finite dimensional Hilbert space, but some results are extended to infinite dimensional case.

Appendix 1: Some Backgrounds from Matrix Analysis

Proposition 5

(Theorem V.2.3 of [22]) Let f be a continuous function on $[0,\infty )$. Then, if f is operator convex and $f\left( 0\right) \le 0$, for any positive operator X and an operator C such that $\left\| C\right\| \le 1$, $f\left( C^{\dagger }XC\right) \le C^{\dagger }f\left( X\right) C$.

Proposition 6

(Equation (2.43) of [23]) Let f be a operator convex function defined on $[0,\infty )$. Let $\varLambda ^{\dagger }$ be a unital positive map. Then

$$ f\left( \varLambda ^{\dagger }\left( A\right) \right) \le \varLambda ^{\dagger }\left( f\left( A\right) \right) $$

holds for any $A\ge 0$.

Proposition 7

(Proposition 8.4 of [1]) Let f be a continuous operator convex function on $[0,\infty )$. Then, if

$$ a:=\lim _{\varepsilon \downarrow 0}\varepsilon f\left( 1/\varepsilon \right) =\lim _{x\rightarrow \infty }f\left( \lambda \right) /\lambda <\infty , $$

there is a real number a and a positive Borel measure $\mu$ such that

$$ f\left( \lambda \right) =f\left( 0\right) +a\lambda +\int _{\left( 0,\infty \right) }\psi _{t}\left( \lambda \right) \mathrm {d}\mu \left( t\right) ,\,\,\,\,\psi _{t}\left( \lambda \right) :=-\frac{\lambda }{\lambda +t}, $$

and $\int _{\left( 0,\infty \right) }\frac{\mathrm {d}\mu \left( t\right) }{1+t}<\infty$. Since $\psi _{t}$ is operator monotone decreasing, this means that $f\left( \lambda \right)$ is sum of linear function and operator monotone decreasing function.

Proposition 8

(Lemma 5.2 of [1]) If f is a complex-valued function on finitely many points $\left\{ x_{i};i\in I\right\} \subset [0,\infty )$, then for any pairwise different positive numbers $\left\{ t_{i};i\in I\right\}$ there exist complex numbers $\left\{ c_{i};i\in I\right\}$ such that $f\left( x_{i}\right) =\sum _{j\in I}\frac{c_{j}}{x_{i}+t_{j}}$ , $i\in I$.

Proposition 9

(Exercise 1.3.5 of [23]) Let X, Y be a positive definite matrices. Then,

$$\begin{aligned} \left[ \begin{array} [c]{cc} X &{} C\\ C^{\dagger } &{} Y \end{array} \right] \ge 0 \end{aligned}$$

(83)

implies

$$\begin{aligned} X\ge CY^{-1}C^{\dagger },\,\,\,Y\ge C^{\dagger }X^{-1}C. \end{aligned}$$

(84)