Geometric relative entropies and barycentric R\'enyi divergences

We give systematic ways of defining monotone quantum relative entropies and (multi-variate) quantum R\'enyi divergences starting from a set of monotone quantum relative entropies. Despite its central importance in information theory, only two additive and monotone quantum extensions of the classical relative entropy have been known so far, the Umegaki and the Belavkin-Staszewski relative entropies. Here we give a general procedure to construct monotone and additive quantum relative entropies from a given one with the same properties; in particular, when starting from the Umegaki relative entropy, this gives a new one-parameter family of monotone and additive quantum relative entropies interpolating between the Umegaki and the Belavkin-Staszewski ones on full-rank states. In a different direction, we use a generalization of a classical variational formula to define multi-variate quantum R\'enyi quantities corresponding to any finite set of quantum relative entropies $(D^{q_x})_{x\in X}$ and signed probability measure $P$, as $$ Q_P^{\mathrm{b},\mathbf{q}}((\rho_x)_{x\in X}):=\sup_{\tau\ge 0}\left\{\text{Tr}\,\tau-\sum_xP(x)D^{q_x}(\tau\|\rho_x)\right\}. $$ We show that monotone quantum relative entropies define monotone R\'enyi quantities whenever $P$ is a probability measure. With the proper normalization, the negative logarithm of the above quantity gives a quantum extension of the classical R\'enyi $\alpha$-divergence in the 2-variable case ($X=\{0,1\}$, $P(0)=\alpha$). We show that if both $D^{q_0}$ and $D^{q_1}$ are monotone and additive quantum relative entropies, and at least one of them is strictly larger than the Umegaki relative entropy then the resulting barycentric R\'enyi divergences are strictly between the log-Euclidean and the maximal R\'enyi divergences, and hence they are different from any previously studied quantum R\'enyi divergence.