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arXiv:1212.1366 · Communications in Mathematical Physics

Entropy Production for Quantum Markov Semigroups

Franco Fagnola, Rolando Rebolledo

Published Feb 19, 2015• 31 citations
arXivHTMLPDF
DOI
10.1007/s00220-015-2320-1

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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.
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.