Max-Relative Entropy of Entanglement, Alias Log Robustness

Properties of the max-relative entropy of entanglement, defined in Ref. 10, are investigated, and its significance as an upper bound to the one-shot rate for perfect entanglement dilution, under a particular class of quantum operations, is discussed. It is shown that it is in fact equal to another known entanglement monotone, namely the log robustness, defined in Ref. 7. It is known that the latter is not asymptotically continuous and it is not known whether it is weakly additive. However, by suitably modifying the max-relative entropy of entanglement we obtain a quantity which is seen to satisfy both these properties. In fact, the modified quantity is shown to be equal to the regularized relative entropy of entanglement.