Convexity of quantum $\chi^2$-divergence

TL;DR

This paper proves that all quantum ^2-divergences are convex functions in their arguments, extending classical convexity results to the quantum setting and clarifying their mathematical properties.

Contribution

It establishes the convexity of any quantum ^2-divergence in its arguments, generalizing previous results limited to specific parameter ranges.

Findings

All quantum ^2-divergences are convex functions.

Convexity holds regardless of the specific quantum ^2-divergence chosen.

Supports the mathematical robustness of quantum divergence measures.

Abstract

The quantum \chi^2-divergence has recently been introduced and applied to quantum channels (quantum Markov processes). In contrast to the classical setting the quantum \chi^2-divergence is not unique but depends on the choice of quantum statistics. In the reference [11] a special one-parameter family of quantum \chi^2_\alpha(\rho,\sigma)-divergences for density matrices were studied, and it was established that they are convex functions in (\rho,\sigma) for parameter values \alpha\in [0,1], thus mirroring the classical theorem for the \chi^2(p,q)-divergence for probability distributions (p,q). We prove that any quantum \chi^2-divergence is a convex function in its two arguments.