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Reviews in Mathematical Physics

Quantum Markov semigroup for open quantum system interacting with quantum Bernoulli noises

L Zhang, Caishi Wang

Published May 13, 2023• 2 citations
Publisher

Abstract

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.

DOI
10.1142/s0129055x23500150

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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.
Being the annihilation and creation operators on the space h of square integrable Bernoulli functionals, quantum Bernoulli noises (QBN) satisfy the canonical anti-commutation relation (CAR) in equal time. Let K be the Hilbert space of an open quantum system interacting with QBN (the environment). Then K⊗h just describes the coupled quantum system. In this paper, we introduce and investigate an interacting stochastic Schrödinger equation (SSE) in the framework K⊗h, which might play a role in describing the evolution of the open quantum system interacting with QBN (the environment). We first prove some technical propositions about operators in K⊗h. In particular, we obtain the spectral decomposition of the tensor operator IK⊗N, where IK means the identity operator on K and N is the number operator in h, and give a representation of IK⊗N in terms of operators IK⊗∂k*∂k, k≥0, where ∂k and ∂k* are the annihilation and creation operators on h, respectively. Based on these technical propositions as well as Mora and Rebolledo’s results on a general SSE, we show that under some mild conditions, our interacting SSE has a unique solution admitting some regularity properties. Some other results are also proven.
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 (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.
Quantum Bernoulli noises (QBNs) are the family of annihilation and creation operators acting on Bernoulli functionals, which can describe a two-level quantum system with infinitely many sites. In this paper, we consider the problem to construct quantum Markov semigroups (QMSs) directly from QBNs. We first establish several new theorems concerning QBNs. In particular, we define the number operator acting on Bernoulli functionals by using the canonical orthonormal basis, prove its self-adjoint property, and describe precisely its connections with QBN in a mathematically rigorous way. We then show the possibility to construct QMS directly from QBN. This is done by combining the general results on QMS with our new results on QBN obtained here. Finally, we examine some properties of QMS constructed from QBN.
Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation. In this paper, we study linear stochastic Schrödinger equations (LSSEs) associated with QBN in the space of square integrable complex-valued Bernoulli functionals. We first rigorously prove a formula concerning the number operator N on Bernoulli functionals. And then, by using this formula as well as Mora and Rebolledo’s results on a general LSSE [C. M. Mora and R. Rebolledo, Infinite. Dimens. Anal. Quantum Probab. Relat. Top. 10, 237–259 (2007)], we obtain an easily checking condition for a LSSE associated with QBN to have a unique Nr-strong solution of mean square norm conservation for given r≥0. Finally, as an application of this condition, we examine a special class of LSSEs associated with QBN and some further results are proven.