Chain rules for a mutual information based on Rényi zero-relative entropy

Quantum R\'enyi relative entropies play a significant part in characterizing operational tasks in quantum information theory. In this paper, we first give some fundamental properties of the zero-relative entropy, and we find that the geometric R\'enyi relative entropy reduces to the zero-relative entropy in the limit case. Then, we define a new mutual information via the zero-relative entropy, and it is the so-called zero-mutual information. In particular, the zero-mutual information is a lower bound of the Petz-R\'enyi and geometric generalized mutual information. We establish the chain rules for the unsmoothed and smoothed versions of the zero-mutual information. As an application, we discuss generalized mutual information with the Petz-R\'enyi, sandwiched R\'enyi, and geometric R\'enyi types, our result gives a uniform chain rule inequality for quantum generalized mutual information.