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International Journal of Quantum Information

W-Entropy of quantum Markov semigroups in terms of quantum Bernoulli noises

Jinshu Chen, Jie Hao, Nana Yang

Published Jan 18, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0219749925500029

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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.
Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal-time. In this paper, we aim to investigate quantum Feller semigroups in terms of QBN. We first investigate local structure of the algebra generated by identity operator and QBN. We then use our new results obtained here to construct a class of quantum Markov semigroups from QBN which enjoy Feller property. As an application of our results, we examine a special quantum Feller semigroup associated with QBN, when it reduced to a certain Abelian subalgebra, the semigroup gives rise to the semigroup generated by Ornstein–Uhlenbeck operator. Finally, we find a sufficient condition for the existence of faithful invariant states that are diagonal for the semigroup.
Quantum Bernoulli noises (QBNs) refer to the annihilation and creation operators acting on the space [Formula: see text] of square integrable Bernoulli functionals, which satisfy the canonical anti-commutation relation (CAR) in equal time. In this paper, we consider the Markov evolution of an open quantum system interacting with QBNs. Let [Formula: see text] be the system space of an open quantum system interacting with QBNs. Then [Formula: see text] just describes the coupled quantum system. In the framework of [Formula: see text], we first construct a quantum Markov semigroup that respects the interactions between the system and QBNs, and then we prove that under some mild conditions the semigroup has faithful invariant states. To support our main results mentioned above, we prove several technical propositions and theorems about operators defined in [Formula: see text]. Some other results are also obtained.
Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy the canonical anti-commutation relation (CAR) in equal time. This paper aimed to discuss the classical reduction and ergodicity of quantum exclusion semigroups constructed by QBN. We first study the classical reduction of the quantum semigroups to an Abelian algebra of diagonal elements and the space of off-diagonal elements. We then provide an explicit representation formula by separating the action on off-diagonal and diagonal operators, on which they are reduced to the semigroups of classical Markov chains. Finally, we prove that the asymptotic behavior of the quantum semigroups is equivalent to one of its associated Markov chains, and that the semigroups restricted to the off diagonal space of operators have a zero limit.
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.
Quantum Bernoulli noises (QBN for short) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anticommutation relation in equal-time. In this paper, by using QBN, we first introduce a class of self-adjoint operators acting on Bernoulli functionals, which we call the weighted number operators. We then make clear spectral decompositions of these operators and establish their commutation relations with the annihilation as well as the creation operators. We also obtain a necessary and sufficient condition for a weighted number operator to be bounded. Finally, as an application of the above results, we construct a class of quantum Markov semigroups associated with the weighted number operators, which belong to the category of quantum exclusion semigroups. Some basic properties of these quantum Markov semigroups are shown and examples are given.