A Survey of Recent Inequalities for Relative Operator Entropy

Abstract

The concepts of relative operator entropy and operator entropy play an important role in different subjects, such as statistical mechanics, information theory, dynamical systems and ergodic theory, biology, economics, human and social sciences. They are closely related to the problem of the quantification of entanglement, the distinguishability of quantum states and to thermodynamical ideas. In this paper we survey some recent inequalities obtained by the author for the relative operator entropy \(S\left (\cdot \vert \cdot \right ),\) for positive invertible operators A and B in general, and, in particular when they satisfy the boundedness condition mA ≤ B ≤ MA for some m, M with 0 < m < M. Natural applications for the operator entropy \(\eta \left (\cdot \right )\) are provided. In the end, some trace inequalities for trace class operators A and B that satisfy the normality condition \(\mathop{\mathrm{tr}}\left (A\right ) =\mathop{ \mathrm{tr}} \left (B\right ) = 1\) are also given.