Telescopic Relative Entropy

Understand

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 a lower bound on the relative Renyi entropies in terms of the trace norm distance, $\trace\rho^{1-p}\sigma^p \ge 1-(||\rho-\sigma||_1)/2$.