Convexity of quantum χ2\chi^2-divergence

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.Comment: Proof clarified, typos correcte