  Properties of the max- relative entropy of entanglement 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. 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 regularised relative entropy of
entanglement.
