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Infinite Dimensional Analysis Quantum Probability and Related Topics

The Θ-KMS adjoint and time reversed quantum Markov semigroups

Jorge R. Bolaños-Servín, Roberto B. Quezada

Published Aug 17, 2015• 9 citations
Publisher

Abstract

We introduce the notion of Θ-KMS adjoint of a quantum Markov semigroup, which is identified with the time reversed semigroup. The break of Θ-KMS symmetry, or Θ-standard quantum detailed balance in the sense of Fagnola–Umanità,11 is measured by means of the von Neumann relative entropy of states associated with the semigroup and its Θ-KMS adjoint.

DOI
10.1142/s0219025715500162

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We introduce the notion of Θ-KMS adjoint of a quantum Markov semigroup, which is identified with the time reversed semigroup. The break of Θ-KMS symmetry, or Θ-standard quantum detailed balance in the sense of Fagnola–Umanità,11 is measured by means of the von Neumann relative entropy of states associated with the semigroup and its Θ-KMS adjoint.
We propose a definition of infinite dimensional Choi-Jamiolkowski state associated with a completely positive trace preserving map. We introduce the notion of Theta-KMS adjoint of a quantum Markov semigroup, which is identified with the time reversed semigroup. The break down of Theta-KMS symmetry (or Theta-standard quantum detailed balance in the sense of Fagnola-Umanita) is measured by means of the von Neumann relative entropy of the Choi-Jamiolkowski states associated with the semigroup and its Theta-KMS adjoint.
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.
Generators of KMS Symmetric Markov Semigroups on B(h){\mathcal{B}({\rm h})}B(h) Symmetry and Quantum Detailed Balance
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 classify generators of quantum Markov semigroups T\mathcal TT on B(h)\mathcal{B}(\mathsf h)B(h), with h{\mathsf{h}}h finite-dimensional and with a faithful normal invariant state ρ\rhoρ satisfying the standard quantum detailed balance condition with an anti