Telescopic Relative Entropy--II Triangle inequalities

TL;DR

This paper establishes upper bounds resembling triangle inequalities for the telescopic relative entropy, a regularized quantum relative entropy, addressing its stability under argument variations.

Contribution

It introduces and proves bounds on the variation of the telescopic relative entropy when its arguments change, extending its mathematical understanding.

Findings

Derived upper bounds on TRE variation with argument changes

Established bounds are similar to triangle inequalities

Addressed limitations of traditional quantum relative entropy

Abstract

In previous work (see arxiv:1102.3040), we have defined the telescopic relative entropy (TRE), which is a regularisation of the quantum relative entropy $S(\rho||\sigma)=\trace\rho(\log\rho-\log\sigma)$, by replacing the second argument $\sigma$ by a convex combination of the first and the second argument, $\tau=a\rho+(1-a)\sigma$ and dividing the result by $-\log a$. We also explored some basic properties of the TRE. In this follow-up paper we state and prove two upper bounds on the variation of the TRE when either the first or the second argument changes. These bounds are close in spirit to a triangle inequality. For the ordinary relative entropy no such bounds are possible due to the fact that the variation could be infinite.