Convexity of quantum χ²-divergence

The general quantum χ²-divergence has recently been introduced by Temme et al. [Temme K, Kastoryano M, Ruskai M, Wolf M, Verstrate F (2010) J Math Phys 51:122201] and applied to quantum channels (quantum Markov processes). The quantum χ²-divergence is not unique, as opposed to the classical χ²-divergence, but depends on the choice of quantum statistics. It was noticed that the elements in a particular one-parameter family χ_α²(ρ,σ) of quantum χ²-divergences are convex functions in the density matrices (ρ,σ), thus mirroring the convexity of the classical χ²(p,q)-divergence in probability distributions (p,q). We prove that any quantum χ²-divergence is a convex function in its two arguments.