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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:2210.13093Oct 24, 20220 citations

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

Juan Manuel Pérez-Pardo

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.

WORLD SCIENTIFIC eBooksApr 1, 20231 citation

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, 2013177 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:1306.5358Journal of Mathematical PhysicsDec 1, 2013218 citations

Monotonicity of the Quantum Relative Entropy Under Positive Maps

Alexander Müller‐Hermes, David Reeb

arXiv:1512.06117Annales Henri PoincaréJan 27, 2017110 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.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:2205.02342Linear Algebra and its ApplicationsSep 7, 20227 citations

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