  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.
