Recently, an interesting quantity called the quantum Renyi divergence (or
"sandwiched" Renyi relative entropy) was defined for pairs of positive
semi-definite operators ρ and σ. It depends on a parameter α
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 ρ and σ 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 α≥1/2, and the α-relative R\'enyi entropy with
α∈[0,1).Comment: 8 pages; v2 slight change in the Abstract and Conclusion