This paper explores various perspectives on Quantum Detailed Balance and the Entropy Production Rate within the framework of Quantum Markov Semigroups. Using the generators of these semigroups, formulated according to the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) theorem, and their respective adjoints, we identify two contrasting families of Quantum Markov Semigroups. The first family demonstrates a situation where the condition for Quantum Detailed Balance is violated, yet the entropy production rate is zero. In contrast, the second family reveals cases where the quantum entropy production rate aligns with an interpretation of Quantum Detailed Balance. These findings provide insights into the relationship between quantum detailed balance and entropy production rate in open systems.This paper explores various perspectives on Quantum Detailed Balance and the Entropy Production Rate within the framework of Quantum Markov Semigroups. Using the generators of these semigroups, formulated according to the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) theorem, and their respective adjoints, we identify two contrasting families of Quantum Markov Semigroups. The first family demonstrates a situation where the condition for Quantum Detailed Balance is violated, yet the entropy production rate is zero. In contrast, the second family reveals cases where the quantum entropy production rate aligns with an interpretation of Quantum Detailed Balance. These findings provide insights into the relationship between quantum detailed balance and entropy production rate in open systems.

Suplemento de la Revista Mexicana de FísicaJun 12, 20250 citations

Generalized Poincaré inequality for quantum Markov semigroups

Marius Junge, Jia Wang

We prove a noncommutative $p$ -Poincaré inequality for GNS-detailed-balance quantum Markov semigroups (QMSs) on non-tracial $σ$ -finite von Neumann algebras, assuming only the existence of a spectral gap. Extending the semi-commutative results of Huang and Tropp, we first establish the inequality for tracial von Neumann algebras. We use Markov dilations to obtain chain-rule estimates for Dirichlet forms and amalgamated free products to define an appropriate noncommutative derivation. We then extend the result to QMSs satisfying GNS detailed balance on non-tracial $σ$ -finite von Neumann algebras, using Haagerup's reduction and Kosaki's interpolation theorem. As applications, we recover sub-exponential concentration inequalities and estimate the Lipschitz and completely bounded Lipschitz diameters of the QMSs.We prove a noncommutative $p$-Poincaré inequality for GNS-detailed-balance quantum Markov semigroups (QMSs) on non-tracial $σ$-finite von Neumann algebras, assuming only the existence of a spectral gap. Extending the semi-commutative results of Huang and Tropp, we first establish the inequality for tracial von Neumann algebras. We use Markov dilations to obtain chain-rule estimates for Dirichlet forms and amalgamated free products to define an appropriate noncommutative derivation. We then extend the result to QMSs satisfying GNS detailed balance on non-tracial $σ$-finite von Neumann algebras, using Haagerup's reduction and Kosaki's interpolation theorem. As applications, we recover sub-exponential concentration inequalities and estimate the Lipschitz and completely bounded Lipschitz diameters of the QMSs.

math.OACross-listedarXiv:2601.06005Jan 12, 2026

Bimodule Quantum Markov Semigroups

Jinsong Wu, Zishuo Zhao

We present a systematic investigation of bimodule quantum Markov semigroups within the framework of quantum Fourier analysis. We introduce the concepts of bimodule detailed balance conditions and bimodule KMS symmetry, which not only generalize the classical notions of detailed balance but also expose interesting structures of quantum channels. We demonstrate that the evolution of densities governed by the bimodule quantum Markov semigroup with bimodule detailed balance is the gradient flow for the relative entropy with respect to the hidden density. Consequently, we obtain a modified logarithmic Sobolev inequality and a Talagrand inequality with respect to a hidden density from higher dimensional structure. Furthermore, we establish a Poincaré inequality for irreducible inclusions and relative ergodic bimodule quantum semigroups.We present a systematic investigation of bimodule quantum Markov semigroups within the framework of quantum Fourier analysis. We introduce the concepts of bimodule detailed balance conditions and bimodule KMS symmetry, which not only generalize the classical notions of detailed balance but also expose interesting structures of quantum channels. We demonstrate that the evolution of densities governed by the bimodule quantum Markov semigroup with bimodule detailed balance is the gradient flow for the relative entropy with respect to the hidden density. Consequently, we obtain a modified logarithmic Sobolev inequality and a Talagrand inequality with respect to a hidden density from higher dimensional structure. Furthermore, we establish a Poincaré inequality for irreducible inclusions and relative ergodic bimodule quantum semigroups.

arXiv:2504.09576Apr 13, 20250 citations

We prove a noncommutative

p

-Poincaré inequality for GNS-detailed-balance quantum Markov semigroups (QMSs) on non-tracial

σ

-finite von Neumann algebras, assuming only the existence of a spectral gap. Extending the semi-commutative results of Huang and Tropp, we first establish the inequality for tracial von Neumann algebras. We use Markov dilations to obtain chain-rule estimates for Dirichlet forms and amalgamated free products to define an appropriate noncommutative derivation. We then extend the result to QMSs satisfying GNS detailed balance on non-tracial

σ

-finite von Neumann algebras, using Haagerup's reduction and Kosaki's interpolation theorem. As applications, we recover sub-exponential concentration inequalities and estimate the Lipschitz and completely bounded Lipschitz diameters of the QMSs.