Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

WORLD SCIENTIFIC eBooks

On Uhlmann’s Proof of the Monotonicity of the Relative Entropy

Juan Manuel Pérez-Pardo

Published Apr 1, 2023• 1 citation
Publisher

Abstract

This chapter 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.
DOI
10.1142/9789811270437_0014

Related papers

On Uhlmann's proof of the Monotonicity of the Relative Entropy

Juan Manuel Pérez-Pardo

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.

arXiv:2210.13093•Oct 24, 2022•0 citations

Particles, Fields and Topology

T. R. Govindarajan, Giuseppe Marmo, V. P. Nair +3 more

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.

WORLD SCIENTIFIC eBooks•Nov 3, 2022•1 citation

On the Monotonicity of Relative Entropy: A Comparative Study of Petz’s and Uhlmann’s Approaches

Santiago Matheus, Francesco Bottacin, Edoardo Provenzi

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.

arXiv:2509.11221•Entropy•Sep 14, 2025•0 citations

Monotonicity of a relative Renyi entropy

Rupert L. Frank, Elliott, H. Lieb

We show that a recent definition of relative R\'enyi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of M\"uller-Lennert et al.

•Jan 1, 2013•177 citations

Monotonicity versions of Epstein's Concavity Theorem and related inequalities

Eric A. Carlen, Haonan Zhang

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.

arXiv:2205.02342•Linear Algebra and its Applications•Sep 7, 2022•7 citations

Monotonicity of a relative Rényi entropy

Rupert L. Frank, Élliott H. Lieb

We show that a recent definition of relative Rényi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of Müller-Lennert et al. [“On quantum Rényi entropies: A new definition, some properties,” J. Math. Phys. 54, 122203 (2013); e-print arXiv:1306.3142v1; see also e-print arXiv:1306.3142].

arXiv:1306.5358•Journal of Mathematical Physics•Dec 1, 2013•218 citations
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.
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.