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Entropy Production for Quantum Markov Semigroups

F. Fagnola, Rolando Rebolledo

Published Aug 16, 2016• 37 citations
OpenAlex

Abstract

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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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.
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.
In this study, we first deal with the canonical anti-commutation relation algebra in which the Bernoulli creation and annihilation operators generate the structure of ultra hyperfinite C[Formula: see text]-algebra. This observation simplifies the construction of the minimal quantum Markov semigroup construct from quantum Bernoulli noises by taking a filter through the family gamma of finite subsets of the natural numbers. As a result, we obtain a Feller quantum Markov semigroup defined on the C[Formula: see text]-algebra, which in turn is extended to the entire algebra of bounded operators. Then, one condition for us to obtain a faithful normal invariant state of this semigroup is an equilibrium state (it satisfies the detailed balance conditions). Finally, we use the quantum Wasserstein distance to discuss the W-entropy rate of this invariant state.
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.
We prove that hypercontractivity implies entropy contraction for a single quantum channel, without a detailed balance condition. For primitive quantum Markov semigroups that are KMS-symmetric with respect to a faithful invariant state σσσ, we obtain the modified Log-Sobolev bound α1≥λ(2+log⁡∥σ−1∥∞)α_1\geq \fracλ{(2+\log\|σ^{-1}\|_\infty)}α1​≥(2+log∥σ−1∥∞​)λ​ where λλλ is the spectral gap. This removes the assumption of LpL_pLp​-regularity for the comparison through the log-Sobolev constant. As an application, we show that the hypercontractivity of an average of two conditional expectations implies the approximate tensorization of relative entropy.