A limit of the quantum Renyi divergence

Understand

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.