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 $$ Q_P^{b,q}((\rho_x)_{x\in X}):=\sup_{\tau\ge 0}\left\{\Tr\tau-\sum_xP(x)D^{q_x}(\tau\|\rho_x)\right\}. $$ 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.

翻译：我们给出了从一组单调量子相对熵出发，系统定义单调量子相对熵及（多变量）量子Rényi散度的方法。尽管相对熵在信息理论中具有核心重要性，但迄今为止仅知道两种可加且单调的经典相对熵量子扩展，即Umegaki相对熵和Belavkin-Staszewski相对熵。本文提出了一种通用方法，可从已知的具有相同性质的量子相对熵出发，构造新的单调且可加的量子相对熵；特别地，当以Umegaki相对熵为起点时，该方法可生成一个单参数族的单调且可加量子相对熵，该族在满秩态上插值于Umegaki与Belavkin-Staszewski相对熵之间。另一方面，我们利用经典变分公式的推广，定义了与任意有限量子相对熵集$(D^{q_x})_{x\in X}$及带号概率测度$P$相对应的多变量量子Rényi量，其形式为$$ Q_P^{b,q}((\rho_x)_{x\in X}):=\sup_{\tau\ge 0}\left\{\Tr\tau-\sum_xP(x)D^{q_x}(\tau\|\rho_x)\right\}. $$ 我们证明，当$P$为概率测度时，单调量子相对熵自然定义了单调Rényi量。在适当归一化下，上述量的负对数给出了经典Rényi $\alpha$-散度在2变量情形（$X=\{0,1\}$，$P(0)=\alpha$）的量子扩展。我们进一步证明，若$D^{q_0}$与$D^{q_1$均为单调且可加的量子相对熵，且其中至少一个严格大于Umegaki相对熵，则所得重心Rényi散度严格介于对数Euclidean散度与最大Rényi散度之间，因此它们与任何先前研究的量子Rényi散度均不相同。