A limit of the quantum Renyi divergence

Abstract: Recently, an interesting quantity called the quantum Renyi divergence (or "sandwiched" Renyi relative entropy) was defined for pairs of positive semi-definite operators $\rho$ and $\sigma$. It depends on a parameter $\alpha$ and acts as a parent quantity for other relative entropies which have important operational significances in quantum information theory: the quantum relative entropy and the min- and max-relative entropies. There is, however, another relative entropy, called the 0-relative Renyi entropy, which plays a key role in the analysis of various quantum information-processing tasks in the one-shot setting. We prove that the 0-relative Renyi entropy is obtainable from the quantum Renyi divergence only if $\rho$ and $\sigma$ have equal supports. This, along with existing results in the literature, suggests that it suffices to consider two essential parent quantities from which operationally relevant entropic quantities can be derived - the quantum Renyi divergence with parameter $\alpha \ge 1/2$, and the $\alpha$-relative R\'enyi entropy with $\alpha\in [0,1)$.