Pré-Publication, Document De Travail Année : 2025

On the Monotonicity of relative entropy: A Comparative Study of Petz's and Uhlmann's Approaches

Santiago Matheus
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Francesco Bottacin
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Résumé

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 provide a correction that not only restores the validity of his line of reasoning, but also allows it to be extended to non-invertible density operators. In the second part, we develop Uhlmann's approach, which is based on interpolations of positive sesquilinear forms and applies automatically also 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 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 the applications of quantum information theory.

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Dates et versions

hal-05215210 , version 1 (19-08-2025)
hal-05215210 , version 2 (14-09-2025)

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  • HAL Id : hal-05215210 , version 1

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Santiago Matheus, Francesco Bottacin, Edoardo Provenzi. On the Monotonicity of relative entropy: A Comparative Study of Petz's and Uhlmann's Approaches. 2025. ⟨hal-05215210v1⟩
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