Some continuity properties of quantum Rényi divergences

09/01/2022

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by   Milán Mosonyi, et al.

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In the problem of binary quantum channel discrimination with product inputs, the supremum of all type II error exponents for which the optimal type I errors go to zero is equal to the Umegaki channel relative entropy, while the infimum of all type II error exponents for which the optimal type I errors go to one is equal to the infimum of the sandwiched channel Rényi α-divergences over all α>1. Here we prove that these two threshold values coincide, thus proving, in particular, the strong converse property for this problem. Our proof method uses a minimax argument, which crucially relies on a newly established continuity property of the sandwiched Rényi α-divergences for α∈(1,2]. Motivated by this, we also prove various other continuity properties for quantum (channel) Rény divergences, which may be of independent interest.