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arXiv:2512.06707 · Journal of Statistical Physics

Interplay between Standard Quantum Detailed Balance and Thermodynamically Consistent Entropy Production

Xin-Hai Tong, Kohei Yoshimura, Tan Van Vu, Naruo Ohga

quant-phcond-mat.stat-mechmath-phSubmitted Dec 7, 2025• Announced Dec 9, 2025• 0 citations• Revised May 3, 2026 (v3)

v1 · Dec 7, 2025 v2 · Jan 18, 2026 v3 · May 3, 2026

Abstract

We demonstrate that, for a quantum Markovian semigroup on a finite-dimensional Hilbert space, if it satisfies the standard quantum detailed balance condition, its generator admits a special representation that yields a vanishing entropy production rate. Conversely, if the generator admits a special representation adhering to the condition of thermodynamic consistency and leading to a vanishing entropy production rate, then the corresponding quantum Markovian semigroup must satisfy the standard quantum detailed balance condition. In this context, we adopt the definition of entropy production rate that is motivated by the physics literature and standard for thermodynamically consistent Lindbladians.

Comments: 15 pages

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An invariant state of a quantum Markov semigroup is an equilibrium state if it satisfies a quantum detailed balance condition. In this paper, we introduce a notion of entropy production for faithful normal invariant states of a quantum Markov semigroup on B(h) as a numerical index measuring "how much far" they are from equilibrium. The entropy production is defined as the derivative of the relative entropy of the one-step forward and backward evolution in analogy with the classical probabilistic concept. We prove an explicit trace formula expressing the entropy production in terms of the completely positive part of the generator of a norm continuous quantum Markov semigroup showing that it turns out to be zero if and only if a standard quantum detailed balance condition holds.

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