# Geometric relative entropies and barycentric Rényi divergences

We give a systematic way of defining quantum relative entropies and quantum Rényi α-divergences for α∈(0,1) with good mathematical properties. In the case of quantum relative entropies, we start from a given quantum relative entropy D^q, and define a family of quantum relative entropies by D^q,#_γ(ρσ):=1/1-γD^q(ρσ#_γρ), γ∈(0,1), where σ#_γρ is the Kubo-Ando γ-weighted geometric mean. In the case of Rényi divergences, we start from two quantum relative entropies D^q_0 and D^q_1, and define a quantum Rényi α-divergence by the variational formula D^b,(q_0,q_1) :=inf_ω∈𝒮(ℋ){α/1-αD^q_0(ωρ)+D^q_1(ωσ) }. We analyze the properties of these quantities in detail, and illustrate the general constructions by various concrete examples.

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