Convexity of quantum ${\chi }^2$-divergence

Abstract

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

Proof clarified, typos corrected

References in corpus (5)

• Metric adjusted skew information
• A Matrix Convexity Approach to Some Celebrated Quantum Inequalities
• The ${\chi }^2$-divergence and Mixing times of quantum Markov processes
• Operator log-convex functions and operator means
• Inequalities for quantum skew information

Cited by in corpus (3)

• WYD-like skew information measures
• Convexity of quasi-entropy type functions: Lieb's and Ando's convexity theorems revisited
• Maps on positive definite operators preserving the quantum ${\chi }_{\alpha }^2$-divergence