Geometric relative entropies and barycentric Rényi divergences

We give a systematic way of defining quantum relative entropies and quantum R\'enyi $\alpha$-divergences for $\alpha\in(0,1)$ 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 \begin{align*} D^{q,\#_{\gamma}}(\rho\|\sigma):=\frac{1}{1-\gamma}D^q(\rho\|\sigma\#_{\gamma}\rho),\quad\quad \gamma\in(0,1), \end{align*} where $\sigma\#_{\gamma}\rho$ is the Kubo-Ando $\gamma$-weighted geometric mean. In the case of R\'enyi divergences, we start from two quantum relative entropies $D^{q_0}$ and $D^{q_1}$, and define a quantum R\'enyi $\alpha$-divergence by the variational formula $$ D^{\mathrm{b},(q_0,q_1)} :=\inf_{\omega\in\mathcal{S}(\mathcal{H})}\left\{ \frac{\alpha}{1-\alpha}D^{q_0}(\omega\|\rho)+D^{q_1}(\omega\|\sigma) \right\}.$$ We analyze the properties of these quantities in detail, and illustrate the general constructions by various concrete examples.