An optimal problem for relative entropy

TL;DR

This paper investigates the extremal values of quantum relative entropy under unitary transformations, establishing the maximum and minimum values and showing the set of all such entropies forms a continuous interval.

Contribution

It provides explicit formulas for the maximum and minimum relative entropy values under unitary conjugation of quantum states.

Findings

Derived formulas for extremal relative entropy values.

Showed the set of all relative entropies under unitary transformations forms a full interval.

Enhanced understanding of quantum relative entropy optimization.

Abstract

Relative entropy is an essential tool in quantum information theory. There are so many problems which are related to relative entropy. In this article, the optimal values which are defined by $\displaystyle\max_{U\in{U(\cX_{d})}} S(U\rho{U^{\ast}}\parallel\sigma)$ and $\displaystyle\min_{U\in{U(\cX_{d})}} S(U\rho{U^{\ast}}\parallel\sigma)$ for two positive definite operators $\rho,\sigma\in{\textmd{Pd}(\cX)}$ are obtained. And the set of $S(U\rho{U^{\ast}}\parallel\sigma)$ for every unitary operator $U$ is full of the interval $[\displaystyle\min_{U\in{U(\cX_{d})}} S(U\rho{U^{\ast}}\parallel\sigma),\displaystyle\max_{U\in{U(\cX_{d})}} S(U\rho{U^{\ast}}\parallel\sigma)]$