Monotonicity of a relative Rényi entropy

We show that a recent definition of relative Rényi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of Müller–Lennert et al. Recently, Müller–Lennert et al. [12] and Wilde et al. [15] modified the traditional notion of relative Rényi entropy and showed that their new definition has several desirable properties of a relative entropy. One of the fundamental properties of a relative entropy, namely monotonicity under completely positive, trace preserving maps (quantum operations) was shown only in a limited range of parameters and conjectured for a larger range. Our goal here is to prove this conjecture. More precisely, the definition of the quantum Rényi divergence [12] or sandwiched Rényi entropy [15] is