Belavkin–Staszewski Relative Entropy, Conditional Entropy, and Mutual Information

Belavkin-Staszewski relative entropy can naturally characterize the effects of the possible noncommutativity of quantum states. In this paper, two new conditional entropy terms and four new mutual information terms are first defined by replacing quantum relative entropy with Belavkin-Staszewski relative entropy. Next, their basic properties are investigated, especially in classical-quantum settings. In particular, we show the weak concavity of the Belavkin-Staszewski conditional entropy and obtain the chain rule for the Belavkin-Staszewski mutual information. Finally, the subadditivity of the Belavkin-Staszewski relative entropy is established, i.e., the Belavkin-Staszewski relative entropy of a joint system is less than the sum of that of its corresponding subsystems with the help of some multiplicative and additive factors. Meanwhile, we also provide a certain subadditivity of the geometric Rényi relative entropy.