Generators of quantum Markov semigroups

Abstract

Quantum Markov Semigroups (QMSs) originally arose in the study of the evolutions of irreversible open quantum systems. Mathematically, they are a generalization of classical Markov semigroups where the underlying function space is replaced by anon-commutative operator algebra. In the case when the QMS is uniformly continuous, theorems due to Lindblad [14], Stinespring [19], and Kraus [13] imply that the generator of the semigroup has the formwhere Vn and G are elements of the underlying operator algebra. In the present paper we investigate the form of the generators of QMSs which are not necessarily uniformly continuous and act on the bounded operators of a Hilbert space. We prove that the generators of such semigroups have forms that reflect the results of Lindblad and Stinespring.We also make some progress towards forms reflecting Kraus' result. Lastly we look at several examples to clarify our findings and verify that some of the unbounded operators we are using have dense domains. Motivation and Overview of our ResultsIn this section we motivate and overview our results while precise definitions appear in section 2. In the early seventies, R.S. Ingarden and A. Kossakowski (see [11] and [12]) postulated that the time evolution of a statistically open system, in the Schrodinger picture, be given by a one-parameter semigroup of linear operators acting on the trace-class operators of a separable Hilbert space H satisfying certain conditions. In the Heisenberg picture the situation translates to a one-parameter semigroup (T t ) t≥0 acting on B(H) (the bounded operators on a Hilbert space H) where each T t is positive and σ-weakly continuous, satisfying T t (1) = 1 for all t ≥ 0, and where the map t → T t A is σ-weakly continuous for eachThe article is part of the second author's Ph.D. thesis which is prepared at the University of South Carolina under the supervision of the first author. 1 2 GEORGE ANDROULAKIS AND MATTHEW ZIEMKE A ∈ B(H). In 1976, G. Lindblad [14] added to the formulation the condition that each T t be completely positive rather than simply positive, a condition which he justified physically. Results of Stinespring [19, Theorem 4] and Arveson [1, Proposition 1.2.2] further justify this condition by proving that if an operator has a commutative domain or target space then positivity and complete positivity are equivalent. Further, under the assumption that the map t → T t is uniformly continuous, the semigroup is called a uniformly continuous QMS, the generator L of the semigroup is bounded, and Lindblad was able to write L in the form L(A) = φ(A) + G * A + AG where φ is completely positive and G ∈ B(H). Using an earlier theorem of Stinespring [19] we can then write φ in the form φ(A) = V * φ(A)V where V : H → K for some Hilbert space K and π : B(H) → B(K) is a normal representation. Further, a theorem due to Kraus [13] lets us write π in the form π(A) = ∞ n=1 W * n AW n where W n : K → H is a bounded linear operator. When we combine Stinespring's and Kraus' results we are then able to write φ in th...

The closedness of the generator of a semigroup

“…Extensions of the result of Lindblad were made by Davies [5,6], Holevo [12], and recently by the authors [1]. The article [1] studies QMSs on the von Neuman algebra B(H) (where H is a Hilbert space) without assuming that the semigroup is uniformly continuous. An important tool in the work of [1] is the domain algebra A of the generator L of a semigroup, which was introduced by Arveson [2].…”

“…V t (g)(x) = g(x − t) for g ∈ H, and e is a smooth function with compact support). The main result of [1] states that there exists a Hilbert space K, a unital * -representation π : A → B(K) and linear maps G : U → H and V : U → K such that Hence, x ∈ Dom(G * ). Next, if we fix A ∈ A and x ∈ U , then for any y ∈ U ,…”

The Induced Semigroup of Schwarz Maps to the Space of Hilbert-Schmidt Operators

“…Remark 4.10. The generator of a semigroup with respect to an orthonormal basis that we defined above is related to the form generator which was defined by Davies [17] and was further studied in [18,19,20,21,22,23,24,25]. If (T t ) t≥0 is a weak * continuous semigroup on B(H) for some Hilbert space H, then a form generator is the map φ : K × B(H) × K → C where K is a dense linear subspace of H, defined by…”

“…QMSs have been extensively studied since the 1970s with the exact form for the generators being one of the topics which has garnered a fair amount of attention. See for example [6], [5], [35], [36], [21], [37], [24], and [25]. The generator of a QMS is a generally unbounded operator defined on a weak * dense linear subspace of B(H).…”

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“…We note that the quantum dynamical semigroup has been studied in the contexts of quantum Markov processes (see Section 1.1). It is shown [22,49,50] that any unital quantum dynamical semigroup is generated by a Liouvillian L :…”