Telescopic relative entropy

Abstract

We introduce the telescopic relative entropy (TRE), which is a new regularisation of the relative entropy related to smoothing, to overcome the problem that the relative entropy between pure states is either zero or infinity and therefore useless as a distance measure in this case. We study basic properties of this quantity, and find interesting relationships between the TRE and the trace norm distance. We then exploit the same techniques to obtain a new and shorter proof of an upper bound on the relative Tsallis entropies in terms of the trace norm distance, 1 - Tr ρ1-p σp ≤ ρ - σ 1 / 2. © 2014 Springer-Verlag Berlin Heidelberg.