A limit of the quantum Renyi divergence

Recently, an interesting quantity called the quantum Renyi divergence (or 'sandwiched' Renyi relative entropy) was defined for pairs of positive semidefinite operators rho and sigma. It depends on a parameter alpha and acts as a parent quantity for other relative entropies which have important operational significance 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 >= 1/2, and the alpha-relative Renyi entropy with alpha is an element of [0, 1).