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Suplemento de la Revista Mexicana de Física

Quantum entropy production rate of quantum markov semigroups

Jorge Bolaños

Published Jun 12, 2025• 0 citations

Abstract

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.

DOI: 10.31349/suplrevmexfis.6.011309

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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.

For a quantum Markov semigroup [Formula: see text] on the algebra [Formula: see text] with a faithful invariant state ρ, we can define an adjoint [Formula: see text] with respect to the scalar product determined by ρ. In this paper, we solve the open problems of characterizing adjoints [Formula: see text] that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators H, Lk in the Gorini–Kossakowski–Sudarshan–Lindblad representation [Formula: see text] of the generator of [Formula: see text]. We study the adjoint semigroup with respect to both scalar products 〈a, b〉 = tr (ρa*b) and 〈a, b〉 = tr (ρ1/2a*ρ1/2 b).

We give an explicit entropy production formula for a class of quantum Markov semigroups, arising in the weak coupling limit of a system coupled with reservoirs, whose generators [Formula: see text] are sums of other generators [Formula: see text] associated with positive Bohr frequencies [Formula: see text] of the system. As a consequence, we show that any such semigroup satisfies the quantum detailed balance condition with respect to an invariant state if and only if all semigroups generated by each [Formula: see text] do so with respect to the same invariant state.

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.

A well established approach to describe open quantum system dynamics is the celebrated Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation. In this article we aim to shed light on semigroup dynamics that are not of this form. Such dynamics may appear in real physical systems describing asymptotic long-time dynamics. Thus, they correspond to semigroup dynamics applicable to a restricted set of initial states only. As a way of motivating our studies, first we review known results about the single-mode Gaussian Quantum Brownian motion channel and corresponding master equations. Next we discuss a conceptually simpler family of qubit semigroup generators and identify parameter regions of completely positive, merely positive, and non-positive dynamics. We then focus on the latter case to which we will refer to as restricted-positive, as the semigroup generator may well be physically meaningful when applied to a restricted set of admissible initial states.