This article presents in a self-contained way A. Uhlmann's celebrated Theorem of monotonicity of the relative entropy under completely positive and trace preserving maps. The Theorem is presented in its more general form and meaningful examples are given.
Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this paper we prove several new monotonicity results, the first of which is a monotonicity theorem that has as a simple corollary a celebrated concavity theorem of Epstein. Our starting points are the monotonicity versions of the Lieb Concavity and the Lieb Convexity Theorems. We also give two new proofs of these in their general forms using interpolation. We then prove our new monotonicity theorems by several duality arguments.
We revisit the monotonicity of relative entropy under the action of quantum channels, a foundational result in quantum information theory. Among the several available proofs, we focus on those by Petz and Uhlmann, which we reformulate within a unified, finite-dimensional operator-theoretic framework. In the first part, we examine Petz's strategy, identify a subtle flaw in his original use of Jensen's contractive operator inequality, and point out how it was corrected to restore the validity of his line of reasoning. In the second part, we develop Uhlmann's approach, which is based on interpolations of positive sesquilinear forms and applies automatically to non-invertible density operators. By comparing these two approaches, we highlight their complementary strengths: Petz's method is more direct and clear; Uhlmann's method is more abstract and general. Our treatment aims to clarify the mathematical structure underlying the monotonicity of relative entropy and to make these proofs more accessible to a broader audience interested in both the foundations and applications of quantum information theory.