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Convexity of quantum $χ^2$-divergence

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.