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Advances in Mathematical Physics
Volume 2011 (2011), Article ID 625978, 16 pages
http://dx.doi.org/10.1155/2011/625978
Research Article

Quantum Dynamical Semigroups and Decoherence

1Faculty of Physics, University of Bielefeld, Universitätsstraβe 25, 33615 Bielefeld, Germany
2Bundesamt für Strahlenschutz (Federal Office for Radiation Protection), Willy-Brandt-Straße 5, 38226 Salzgitter, Germany

Received 22 June 2011; Accepted 31 August 2011

Academic Editor: Christian Maes

Copyright © 2011 Mario Hellmich. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove a version of the Jacobs-de Leeuw-Glicksberg splitting theorem for weak* continuous one-parameter semigroups on dual Banach spaces. This result is applied to give sufficient conditions for a quantum dynamical semigroup to display decoherence. The underlying notion of decoherence is that introduced by Blanchard and Olkiewicz (2003). We discuss this notion in some detail.

1. Introduction

The theory of environmental decoherence starts from the question of why macroscopic physical systems obey the laws of classical physics, despite the fact that our most fundamental physical theory—quantum theory—results in contradictions when directly applied to these objects. The infamous Schrödinger cat is a well-known illustration of this problem. This is an embarrassing situation since, from its inception in the 1920s until today, quantum theory has seen a remarkable success and an ever increasing range of applicability. Thus the question of how to reconcile quantum theory with classical physics is a fundamental one, and efforts to find answers to it persisted throughout its history. At present, the most promising and most widely discussed answer is the notion of environmental decoherence. The starting point is the contention that quantum theory is universally valid, in particular in the macroscopic domain, but that one has to take into account the fact that macroscopic objects are strongly interacting with their environment, and that precisely this interaction is the origin of classicality in the physical world. Thus classicality is a dynamically emergent phenomenon due to the essential openness of macroscopic quantum systems, that is, their interaction with other quantum systems surrounding them leads to an effective restriction of the superposition principle and results in a state space with properties different from the pure quantum case.

In order to clarify the status of decoherence and to provide a rigorous definition, Ph. Blanchard and R. Olkiewicz suggested a notion of decoherence formulated in the algebraic framework [1, 2] of quantum physics in [3], drawing on earlier work in [4]. The algebraic framework is especially useful for the discussion of decoherence, since it is able to accommodate classical systems, provides an elegant formulation of superselection rules, and can even describe systems with infinitely many degrees of freedom in a rigorous way. This is why it is becoming increasingly popular in the discussion of foundational and philosophical problems of quantum physics [5, 6].

In the present paper, we assume that the algebra of observables of the system under study is a von Neumann algebra, and due to its openness the time evolution is irreversible and hence given by a family {𝑇𝑡}𝑡0 of normal completely positive and unital linear maps on the von Neumann algebra [7, 8]. In the Markovian approximation, the family {𝑇𝑡}𝑡0 becomes a so-called quantum dynamical semigroup. It is our purpose to discuss the Blanchard-Olkiewicz notion of decoherence for quantum dynamical semigroups. To this end we study a weak* version of the so-called Jacobs-de Leeuw-Glicksberg splitting for one-parameter semigroups on dual Banach spaces. In the Markovian case, the Blanchard-Olkiewicz notion of decoherence relies on the so-called isometric-sweeping splitting, which is similar to the Jacobs-de Leeuw-Glicksberg splitting, and we will be able to prove a new criterion for the appearance of decoherence in the case of uniformly continuous quantum dynamical semigroups by examining the connection between the two asymptotic splittings.

The paper is organized as follows. In Section 2 we establish the Jacobs-de Leeuw-Glicksberg splitting for weak* continuous contractive one-parameter semigroups on dual Banach spaces. We provide a sufficient condition which ensures that the semigroup is weak* stable on the stable subspace of the splitting (Proposition 2.3). In Section 3 we turn to the study of quantum dynamical semigroups on von Neumann algebras. We begin by applying the results of Section 2 in the von Neumann algebra setting (Proposition 3.3). As a complement to Proposition 2.3, we prove Proposition 3.6, which gives another condition for weak* stability on the stable subspace of the splitting. In Section 4, we discuss a notion of decoherence which is very close to that given in [3] and establish some mathematical results related to it. In the final Section 4.2, we use the previous results to give a sufficient condition that a uniformly continuous quantum dynamical semigroup having a faithful normal invariant state displays decoherence.

2. The Jacobs-de Leeuw-Glicksberg Splitting

Suppose that 𝖷 is a Banach space and assume that it has a predual space denoted by 𝖷, that is, (𝖷)𝖷. If 𝑥𝖷 and 𝜑𝖷, we will denote the evaluation of 𝑥 at 𝜑 by 𝑥,𝜑 and consider this as a dual pairing between 𝖷 and 𝖷. The set of all bounded linear operators from 𝖷 to 𝖷, endowed with the operator norm, will be denoted by L(𝖷), and its unit ball by L(𝖷)1={𝑇L(𝖷)𝑇1}. Operators from L(𝖷)1 are called contractive. We consider the algebraic tensor product 𝖷𝖷 and endow it with the projective cross norm 𝛾; the completion of 𝖷𝖷 with respect to 𝛾 is a Banach space which will be denoted by 𝖷𝛾𝖷. Then its dual space (𝖷𝛾𝖷) is isometrically isomorphic in a canonical way with L(𝖷,(𝖷))=L(𝑋): If 𝜓(𝖷𝛾𝖷), we define Φ(𝜓)L(𝖷) by𝑥𝜑,𝜓=Φ(𝜓)(𝑥),𝜑(2.1) for all 𝑥𝖷 and 𝜑𝖷. It can now be shown that 𝜓Φ(𝜓) extends to an isometric isomorphism, and we can thus write (𝖷𝛾𝖷)L(𝖷).

We now introduce the pointwise weak* topology on L(𝖷). Let 𝑥𝖷, 𝜑𝖷, and define the seminorm L(𝖷)𝑇𝑝𝑥,𝜑(𝑇)=|𝑇(𝑥),𝜑|. The pointwise weak* topology is the locally convex topology on L(𝑋) induced by the family {𝑝𝑥,𝜑𝑥𝖷,𝜑𝖷} of seminorms. If 𝑇=Φ(𝜓)L(𝖷), we see that 𝑝𝑥,𝜑(𝑇)=|𝑇(𝑥),𝜑|=|𝑥𝜑,𝜓|; thus the pointwise weak* topology coalesces with the 𝜎(L(𝖷),𝖷𝛾𝖷) topology on L(𝖷), that is, the pointwise weak* topology is a weak* topology as well. Thus we can conclude from Alaoglu's theorem that L(𝖷)1 is compact in the pointwise weak* topology.

A linear operator 𝑇L(𝖷) will be called normal provided it is a continuous map from 𝖷 to 𝖷 when 𝖷 is endowed with the weak* topology. We denote the set of all normal operators by Ln(𝖷). We can consider the set of all normal contractive operators Ln(𝖷)1 as a semigroup under multiplication of operators, that is, if 𝑇1,𝑇2Ln(𝖷)1, then 𝑇1𝑇2 is normal and contractive; moreover, the multiplication is associative. The semigroup Ln(𝖷)1 is semitopological when endowed with the pointwise weak* topology, that is, the multiplication is separately continuous. This means that the maps 𝑇𝑇𝑆 and 𝑆𝑇𝑆 are both continuous with respect to the pointwise weak* topology. Finally we remark that it is important to note that Ln(𝖷)1 is not closed in L(𝖷)1 with respect to the pointwise weak* topology. Moreover, recall that an operator 𝑇L(𝖷) is normal if and only if there exists a (unique) predual operator 𝑇 from 𝖷 into 𝖷, defined by 𝑇(𝑥),𝜑=𝑥,𝑇(𝜑), 𝑥𝖷, 𝜑𝖷.

In this section, our goal is to study one-parameter semigroups on dual Banach spaces. A contractive one-parameter semigroup [9, 10] is a family {𝑇𝑡}𝑡0 of linear and contractive operators on 𝖷, such that 𝑇𝑠𝑇𝑡=𝑇𝑠+𝑡 for all 𝑠,𝑡0 and 𝑇0=id𝖷. The semigroup is called weak* continuous provided each 𝑇𝑡 is a normal operator and [0,[𝑡𝑇𝑡(𝑥) is weak* continuous for any 𝑥𝖷. For a weak* continuous semigroup there exists the following concept of a weak* generator 𝑍:𝑍𝑥=lim𝑡0𝑇𝑡(𝑥)𝑥𝑡intheweaktopology,(2.2)dom𝑍={𝑥𝖷thelimitin(2.2)exists}.(2.3) The predual semigroup {𝑇𝑡,}𝑡0 of a weak* continuous semigroup {𝑇𝑡}𝑡0 is strongly continuous, and the adjoint of its generator 𝑍 is equal to the weak* generator 𝑍.

Suppose now that {𝑇𝑡}𝑡0 is a weak* continuous contractive semigroup on 𝖷, and write S0={𝑇𝑡𝑡0}Ln(𝖷)1. In the following, we assume that the closure of S0 in L(𝖷)1 with respect to the pointwise weak* topology consists of normal operators, that is, we assume that S=S0Ln(𝖷)1, where the bar denotes closure in the pointwise weak* topology. Then S is a compact commutative semitopological subsemigroup of Ln(𝖷)1. We now use the fact that every compact commutative semitopological semigroup has a unique minimal ideal GS, the so-called Sushkevich kernel [9], which is given byG=𝑅S𝑅S,(2.4) and 𝑄G will denote the unit of G. We then have G=𝑄S. By compactness of G, it follows that G is, in fact, a commutative topological group. In the following, we will simplify our notation by writing 𝑇1𝑇2 instead of 𝑇1𝑇2.

We are now able to prove a weak* version of the Jacobs-de Leeuw-Glicksberg splitting theorem, originally going back to Jacobs [11] and de Leeuw and Glicksberg [12, 13], see also [14]. The present proof mimics the one given in [9] for weakly almost periodic one-parameter semigroups.

Theorem 2.1. Let S0={𝑇𝑡}𝑡0 be a weak* continuous contractive one-parameter semigroup with generator 𝑍. Assume that S=S0 consists of normal operators. Then there exist weak* closed subspaces 𝖷s,𝖷r of 𝖷 invariant under all operators 𝑇𝑡, 𝑡0, such that 𝖷=𝖷s𝖷r, and 𝖷s=𝑥𝖷0𝑇𝑡(𝑥)𝑡0w𝖷,(2.5)r=lin{𝑥dom𝑍𝛼suchthat𝑍𝑥=i𝛼𝑥}w=lin𝑥X𝛼suchthat𝑇𝑡(𝑥)=ei𝛼𝑡𝑥𝑡0w.(2.6)

Proof. Since 𝑄2=𝑄, the unit 𝑄 is a normal projection such that [𝑄,𝑇𝑡]=0 for all 𝑡0. The theorem will be established once we prove that 𝖷s=ker𝑄 and 𝖷r=ran𝑄.
Let 𝑥ker𝑄. Since 𝑄S, there is a net {𝑇𝑖}𝑖𝐼S0 such that 𝑇𝑖𝑄 relative to the pointwise weak* topology; hence 𝑇𝑖(𝑥)𝑄𝑥=0, so 0{𝑇𝑡(𝑥)𝑡0}w. Conversely, assume 0{𝑇𝑡(𝑥)𝑡0}w for some 𝑥. Then there is a net {𝑇𝑖}𝑖𝐼S0 such that 𝑇𝑖(𝑥)0 relative to the weak* topology. By compactness of S, there is a subnet {𝑇𝑗}𝑗𝐽{𝑇𝑖}𝑖𝐼 with 𝑇𝑗𝑅 relative to the pointwise weak* topology for some 𝑅S, and it follows that 𝑅𝑥=0. Hence 𝑅𝑄𝑅𝑥=0 for all 𝑅S. Choosing 𝑅 to be the inverse of 𝑄𝑅 in G, we get 𝑄𝑥=0, hence 𝑥ker𝑄. We have thus proved that 𝖷s=ker𝑄.
Let G be the character group of G. For each G𝜒 define the operator 𝖷𝑥𝑃𝜒𝑥=G𝜒(𝑆)𝑆𝑥d𝜇(𝑆),(2.7) where 𝜇 is the normalized Haar measure of G. The integral is to be understood as a weak* integral, thus 𝑃𝜒 is a well-defined bounded operator in L(𝖷) with 𝑃𝜒1. Then for all 𝑅G we get 𝑅𝑃𝜒𝑥=G𝜒(𝑆)𝑅𝑆𝑥d𝜇(𝑆)=𝜒(𝑅)G𝜒(𝑅𝑆)𝑅𝑆𝑥d𝜇(𝑆)=𝜒(𝑅)G𝜒(𝑆)𝑆𝑥d𝜇(𝑆)=𝜒(𝑅)𝑃𝜒𝑥,(2.8) in particular 𝑄𝑃𝜒=𝑃𝜒; therefore, 𝑇𝑡𝑃𝜒=𝑇𝑡𝑄𝑃𝜒=𝜒(𝑇𝑡𝑄)𝑃𝜒 for all 𝑡0. Since 𝑡𝜒(𝑇𝑡𝑄) is continuous and satisfies the functional equation 𝜒𝑇𝑡𝑄𝑇𝜒𝑠𝑄𝑇=𝜒𝑡+𝑠𝑄{𝑧|𝑧|=1}(2.9) for all 𝑡,𝑠0, we have 𝜒(𝑇𝑡𝑄)=ei𝛼𝑡 for some 𝛼. Thus 𝑇𝑡𝑃𝜒=ei𝛼𝑡𝑃𝜒, hence 𝑃𝜒𝖷dom𝑍 and 𝑍𝑃𝜒=i𝛼𝑃𝜒 for all G𝜒. We next define the subspace 𝑀=lin𝜒G𝑃𝜒𝖷w𝖷r.(2.10) We prove that ran𝑄𝑀𝖷r. Let 𝜑𝑀={𝜑𝖷𝑥,𝜑=0𝑥𝑀}. Then 𝑃𝜒𝑥,𝜑=0 for all 𝑥𝖷, G𝜒, that is, G𝜒(𝑆)𝑆𝑥,𝜑d𝜇(𝑆)=0(2.11) for all 𝑥𝖷, G𝜒. Since the character group G is total in 𝐿2(𝐺,𝜇) by the Stone-Weierstraß theorem and since 𝑆𝑆𝑥,𝜑 is continuous it follows that 𝑆𝑥,𝜑=0 for all 𝑆G and 𝑥𝖷. Take 𝑆=𝑄, then we obtain 𝜑ran𝑄 and thus 𝑀ran𝑄. By the bipolar theorem we obtain ran𝑄co𝑀=𝑀, since ran𝑄 is a weak* closed subspace. Conversely, let 𝑥dom𝑍 with 𝑍𝑥=i𝛼𝑥 for some 𝛼. It follows that 𝑇𝑡(𝑥)=ei𝛼𝑡𝑥 for all 𝑡0 and consequently 𝑅𝑥=ei𝛼𝑥 for 𝑅S. Thus there exists 𝛽 such that 𝑄𝑥=ei𝛽𝑥=𝑄2𝑥. Consequently, we must have 𝛽=0 which implies 𝑄𝑥=𝑥ran𝑄, hence 𝖷rran𝑄, and the proof is finished.

Corollary 2.2. Under the hypothesis of Theorem 2.1, there exists a weak* continuous one-parameter group {𝛼𝑡}𝑡 of isometries on 𝖷r such that 𝛼𝑡=𝑇𝑡𝖷rfor 𝑡0.

Proof. Let 𝑇S, then 𝑄𝑇G, and let 𝑅 be the inverse of 𝑄𝑇 in G, that is, 𝑅(𝑄𝑇)=𝑄. Then for all 𝑥𝖷r, we have 𝑅𝑇𝑥=𝑅𝑄𝑇𝑥=𝑄𝑥=𝑥. Now write 𝛼𝑡=𝑄𝑇𝑡 for 𝑡0 and let 𝛼𝑡 be the inverse of 𝛼𝑡 in G. The foregoing calculation shows that {𝛼𝑡}𝑡 is a one-parameter group on 𝖷r. Moreover, it is clear that it is weak* continuous and contractive. Now assume that there is 𝑥𝖷r and 𝑡0 such that 𝛼𝑡(𝑥)<𝑥. Then it follows that 𝛼𝑡>1, contradiction; thus {𝛼𝑡}𝑡 is isometric.

The subspace 𝖷r is called the reversible subspace and 𝖷s is called the stable subspace; its elements are sometimes called flight vectors.

In applications it is sometimes desirable to have a stronger characterization of the subspace 𝖷s, namely, we are interested in a stronger stability property of the elements in 𝖷s. In particular, this is relevant in the applications to decoherence we discuss in Section 4. The next result provides a sufficient condition for weak* stability to hold on 𝖷s based on the boundary spectrum spec𝑍i of the generator 𝑍.

Proposition 2.3. Assume that the hypothesis of Theorem 2.1 is satisfied and additionally that spec𝑍i is at most countable. Then the stable subspace (2.5) is given by 𝖷s=𝑥𝖷lim𝑡𝑇𝑡(𝑥)=0relativetotheweaktopology.(2.12) Moreover, the convergence in (2.12) is uniform for 𝑥 in 𝖷s𝖷1.

Proof. Consider the predual semigroup {𝑇𝑡,}𝑡0 with generator 𝑍; as already remarked, (𝑍) is 𝑍. The predual 𝑄 of 𝑄 is a projection and induces a splitting 𝖷=𝖷r,𝖷s, by way of 𝖷r,=ran𝑄 and 𝖷s,=ker𝑄. Let {𝑇𝑠𝑡,}𝑡0 be the restriction of {𝑇𝑡,}𝑡0 to 𝖷s,. Since 𝖷s, is closed; the generator 𝑍𝑠 of {𝑇s𝑡,}𝑡0 is given by the restriction 𝑍s=𝑍𝖷s,, dom𝑍s=domZ𝖷s,. A similar construction applies to the reversible subspace 𝖷r. We check that spec𝑍sspec𝑍. Let 𝜆𝜌(𝑍), that is, the map (𝜆𝟙𝑍)dom𝑍𝖷 is bijective. Then clearly the map (𝜆𝟙𝑍s)=(𝜆𝟙𝑍)𝖷𝑠,dom𝑍𝖷s,𝖷s, is injective. It is also surjective: let 𝜑𝖷s,, then there is 𝜓=𝜓s𝜓rdom𝑍 such that (𝜆𝟙𝑍)𝜓=𝜑. Now 𝜆𝟙𝑍𝜓=𝜆𝟙𝑍𝜓s𝜓r=𝜑0,(2.13) so (𝜆𝟙𝑍)𝜓r=0 and 𝜓r=0 by injectivity. Thus (𝜆𝟙𝑍s) is bijective and 𝜆𝜌(𝑍s). In particular, using spec𝑍=spec𝑍 we find that spec𝑍siiscountable.(2.14) We now see that specp𝑍si=,(2.15) for if i𝜆specp𝑍s, 𝜆, then the corresponding eigenvector 𝑥dom𝑍s𝖷s satisfying 𝑍𝑥=i𝜆𝑥 must lie in 𝖷r by (2.6), hence 𝑥=0, contradiction.
From (2.14) and (2.15), it follows by the Arendt-Batty-Lyubich-Vũ theorem [15, 16], see also [9], that the semigroup {𝑇s𝑡,}𝑡0 is strongly stable, that is, for all 𝑥𝖷s, we have lim𝑡𝑇s𝑡,(𝑥)=0. Thus if 𝑥𝖷s and 𝑥𝖷 it follows that ||𝑇𝑡(𝑥),𝑥||=||𝑄𝑇𝑡(𝑥),𝑥||=||𝑥,𝑇𝑡,𝑄𝑥||𝑥𝑇s𝑡,𝑄𝑥0(2.16) as 𝑡 uniformly for 𝑥𝖷s𝖷1, where 𝑄=id𝖷𝑄 denotes the projection onto 𝖷s.

3. Semigroups on von Neumann Algebras

The results of the previous section apply to the case of Neumann algebras. Let be a Hilbert space. A von Neumann algebra is a *-subalgebra of the Banach-*-algebra L() of all bounded linear operators acting on , which is additionally closed in the weak (or equivalently strong) operator topology. The identity operator will be denoted by 𝟙, and we will always assume that 𝟙𝔐. The ultraweak topology on 𝔐 is defined by the seminorms 𝑝𝜌(𝑥)=|tr(𝜌𝑥)|, where 𝜌 runs through the trace class operators on , it agrees with the weak operator topology on bounded portions of 𝔐. The set of all ultraweakly continuous linear functionals on 𝔐 forms a Banach space, and this Banach space is the unique (up to isomorphism) predual space of 𝔐, for this reason we denote it by 𝔐. The ultraweak topology on 𝔐 can be shown to be equivalent to the 𝜎(𝔐,𝔐) (i.e., weak*) topology. Hence the setup of the previous section applies to this case. The set of all positive operators of 𝔐 will be denoted by 𝔐+. A functional 𝜑 in 𝔐 which is positive (i.e., 𝜑(𝑥)0 provided 𝑥𝔐+) and normalized (i.e., 𝜑=1, equivalently 𝜑(𝟙)=1) will be called a normal state. A state is called faithful if 𝑥𝔐+ and 𝜔(𝑥)=0 implies 𝑥=0. For proofs of these results, we refer to [10, 17].

Let 𝑇L(𝔐). Then 𝑇 is called positive if 𝑇(𝔐+)𝔐+. A positive operator is normal (i.e., weak* continuous) if and only if for every uniformly bounded increasing net {𝑥𝑖}𝑖𝐼𝔐+ we have sup𝑖𝑇(𝑥𝑖)=𝑇(sup𝑖𝑥𝑖). Furthermore, 𝑇 is called strongly positive whenever it satisfies Kadison's inequality, that is, 𝑇(𝟙)𝑇(𝑥𝑥)𝑇(𝑥)𝑇(𝑥) for any 𝑥𝔐. Clearly strong positivity implies positivity. An even stronger notion of positivity is complete positivity: 𝑇L(𝔐) is called completely positive whenever 𝑛𝑖,𝑗=1𝑦𝑖𝑇(𝑥𝑖𝑥𝑗)𝑦𝑗0 for all 𝑛 and all 𝑥1,,𝑥𝑛 and 𝑦1,,𝑦𝑛 from 𝔐. The map 𝑇 is called unital if 𝑇(𝟙)=𝟙; a positive unital map is automatically contractive, that is, 𝑇(𝑥)𝑥 for all 𝑥𝔐.

The following result has been established in [18].

Proposition 3.1. Suppose that 𝑆Ln(𝔐)1 is a subset of normal contractive linear operators. Then the following assertions are equivalent. (1)The set {𝑇(𝜑)𝑇𝑆}𝔐 is relatively weakly compact for every 𝜑𝔐. (2)The set 𝑆 is equicontinuous when 𝔐 is endowed with the Mackey topology (i.e., the 𝜏(𝔐,𝔐) topology). (3)The pointwise weak* closure of 𝑆 consists of normal operators: 𝑆Ln(𝔐)1. Moreover, these conditions are satisfied whenever there is a faithful normal state 𝜔 on 𝔐 such that 𝜔𝑇𝑥𝑇𝑥(𝑥)𝜔𝑥forany𝑇𝑆,𝑥𝔐.(3.1) In particular, if each element in 𝑆 is strongly positive we conclude that (3.1) can be rewritten as 𝜔(𝑇(𝑥))𝜔(𝑥) for all 𝑥𝔐+, 𝑇𝑆, or briefly 𝜔𝑇𝜔, for all 𝑇𝑆.

If 𝑆Ln(𝔐)1 is a subset of normal contractive linear operators and 𝜔 a normal state, we call 𝜔 an invariant state under 𝑆 provided 𝜔(𝑇(𝑥))=𝜔(𝑥) for all 𝑥𝔐 and 𝑇𝑆. We now apply the results of Section 2 to weak* continuous semigroups on von Neumann algebras. This gives us the following result.

Corollary 3.2. Suppose that {𝑇𝑡}𝑡0 is a weak* continuous contractive strongly positive one-parameter semigroup on a von Neumann algebra 𝔐 with ultraweak generator 𝑍, and suppose that there exists a faithful normal invariant state 𝜔. Then there exist weak* closed and 𝑇𝑡-invariant subspaces 𝔐r and 𝔐s of 𝔐, given by (2.5) and (2.6), such that 𝔐=𝔐r𝔐s.

Proof. By Kadison’s inequality, (3.1) holds; thus Proposition 3.1 implies that the pointwise weak* closure S=S0, with S0={𝑇𝑡}𝑡0, consists of normal operators. Hence Theorem 2.1 applies.

It is worth pointing out that a similar result was recently established in [19] for general semigroups acting on a W*-algebra and possessing a faithful family of subinvariant states.

We now prove that 𝔐r is actually a von Neumann subalgebra. Recall that a conditional expectation 𝑄 from a C*-algebra 𝔄 onto a C*-subalgebra 𝔅𝔄 is a completely positive contraction with 𝑄(𝑥)=𝑥 for 𝑥𝔅 and 𝑄(𝑥𝑦𝑥)=𝑥𝑄(𝑦)𝑥 for 𝑥𝔅, 𝑦𝔄.

Proposition 3.3. Let {𝑇𝑡}𝑡0 be a weak* continuous semigroup of strongly positive unital operators and suppose there exists a faithful normal invariant state 𝜔. Then 𝔐r is a von Neumann subalgebra of 𝔐 and there exists a group of *-automorphisms {𝛼𝑡}𝑡 on 𝔐r such that 𝑇𝑡𝔐r=𝛼𝑡 for all 𝑡0. Moreover, there exists a normal conditional expectation 𝑄 from 𝔐 onto 𝔐r such that 𝜔𝑄=𝜔. Finally, 𝔐s is *-invariant.

Proof. Since each 𝑇𝑡 is a contraction; Corollary 3.2 applies. Let 𝑀0={𝑥𝔐𝛼suchthat𝑇𝑡(𝑥)=ei𝛼𝑡𝑥𝑡0}, that is, we have lin𝑀0w=𝔐r. As in [20] we define the sesquilinear map 𝐷𝔐×𝔐𝔐 by 𝐷(𝑥,𝑦)=𝑇𝑡(𝑥𝑦)𝑇𝑡(𝑥)𝑇𝑡(𝑦) for some fixed 𝑡0. By Kadison's inequality, the sesquilinear form 𝜑𝐷 is positive-definite for any 𝜑𝔐+, so by the Cauchy-Schwarz inequality, 𝐷(𝑥,𝑥)=0 if and only if 𝐷(𝑥,𝑦)=0 for all 𝑦𝔐. Now let 𝑥𝑀0, then 𝑇𝑡(𝑥𝑥)𝑇𝑡(𝑥)𝑇𝑡(𝑥)=ei𝛼𝑡e+i𝛼𝑡𝑥𝑥=𝑥𝑥. Thus 0𝜔(𝑇𝑡(𝑥𝑥)𝑥𝑥)𝜔(𝑥𝑥𝑥𝑥)=0, and by faithfulness 𝑇𝑡(𝑥𝑥)=𝑥𝑥, hence 𝐷(𝑥,𝑥)=0 for all 𝑥𝑀0. So 𝐷(𝑥,𝑦)=0 for all 𝑥,𝑦𝑀0, that is, 𝑇𝑡(𝑥𝑦)=𝑇𝑡(𝑥)𝑇𝑡(𝑦)=ei(𝛼2𝛼1)𝑡𝑥𝑦, and we conclude that 𝑥𝑦𝑀0 whenever 𝑥,𝑦𝑀0. It follows that lin𝑀0 is a *-subalgebra of 𝔐 (containing 𝟙) and consequently 𝔐r is a von Neumann subalgebra, and 𝑇𝑡(𝑥𝑦)=𝑇𝑡(𝑥)𝑇𝑡(𝑦) for all 𝑥,𝑦𝔐r. By Corollary 2.2, the restriction of 𝑇𝑡 to 𝔐r extends to a one-parameter group {𝛼𝑡}𝑡 of isometries and the above argument shows that 𝛼𝑡 must be a *-homomorphism. Let 𝑄 be the Sushkevich kernel of the semigroup 𝑆Ln(𝔐)1. Since 𝑄 is a projection and 𝑄=1 it follows from Tomiyama’s theorem [21] that 𝑄 is a conditional expectation; since 𝑄S; it is also clear that 𝜔𝑄=𝜔. The last assertion is clear as well.

In the following, we will be interested in the stronger characterization of 𝔐s by a stability property as in (2.12). We start by quoting the following result.

Lemma 3.4. Suppose that {𝑇𝑡}𝑡0 is a weak* continuous one-parameter semigroup of strongly positive unital operators on the von Neumann algebra 𝔐 with a faithful normal invariant state 𝜔. Introduce the subsets 𝑀=𝑥𝔐𝑇𝑡𝑥𝑥=𝑇𝑡(𝑥)𝑇𝑡,𝑀(𝑥)𝑡0=𝑥𝔐𝑇𝑡𝑥𝑥=𝑇𝑡(𝑥)𝑇𝑡(𝑥),𝔐𝑡01=𝑀𝑀.(3.2) Then 𝔐1 is a 𝑇𝑡-invariant von Neumann subalgebra of 𝔐, and there exists a group of *-automorphisms {𝛼𝑡}𝑡 on 𝔐1 such that 𝑇𝑡𝔐1=𝛼𝑡 for 𝑡0. Moreover, 𝔐1 is a maximal (in the sense of not being properly contained in a larger von Neumann subalgebra) von Neumann subalgebra on which the restriction of {𝑇𝑡}𝑡0 is given by a group of *-automorphisms.

A proof can be found in [22] (see the proof of Proposition 2). It is easy to see that we always have 𝔐r𝔐1.

Lemma 3.5. Under the assumptions of Lemma 3.4, for every 𝑥𝔐 the weak* limit points of the net {𝑇𝑡(𝑥)}𝑡+ lie in 𝔐1.

A proof of this statement is contained in the proof of Theorem 3.1 of [23].

We can now establish the following result.

Proposition 3.6. Let {𝑇𝑡}𝑡0 be a weak* continuous semigroup of strongly positive unital operators on the von Neumann algebra 𝔐 with a faithful normal invariant state 𝜔. If 𝔐r=𝔐1, it follows that 𝔐s=𝑥𝔐lim𝑡𝑇𝑡(𝑥)=0intheweaktopology.(3.3)

Proof. Let 𝑥𝔐s and assume without loss of generality that 𝑥1. By Alaoglu's theorem the net {𝑇𝑡(𝑥)}𝑡+ contained in the unit ball of 𝔐 has a limit point 𝑥0 for 𝑡. Then using Lemma 3.5, we find that 𝑥0𝔐1=𝔐𝑟. But since 𝑥𝔐s, it follows that also 𝑥0𝔐s, that is, 𝑥0𝔐s𝔐r={0}; hence 𝑥0=0. This proves that any limit point of the net {𝑇𝑡(𝑥)}𝑡+ is equal to 0; therefore we conclude that lim𝑡𝑇𝑡(𝑥)=0 in the weak* topology for all 𝑥𝔐s.

Moreover, let us remark the following: suppose that {𝑇𝑡}𝑡0 is a weak* continuous semigroup of strongly positive unital operators with generator 𝑍 having a faithful normal invariant state 𝜔, and assume that the peripheral spectrum spec𝑍i is at most countable. Then by using Proposition 2.3 the conclusion of Proposition 3.6 holds. These results will be used in the next section when we discuss the notion of decoherence for uniformly continuous quantum dynamical semigroups.

4. Applications to Decoherence

4.1. The Notion of Decoherence in the Algebraic Framework

Consider a closed quantum system whose algebra of observables is a von Neumann algebra 𝔑, and its reversible time evolution is given by a weak* continuous group of *-automorphisms {𝛽𝑡}𝑡 on 𝔑. A subsystem can be described by a von Neumann subalgebra 𝔐𝔑 containing the observables belonging to the subsystem. We will assume that there exists a normal conditional expectation 𝐸 from 𝔑 onto 𝔐. In this situation, we can define the reduced dynamics as follows:𝑇𝑡(𝑥)=𝐸𝛽𝑡(𝑥),𝑥𝔐,𝑡0.(4.1) This is the Heisenberg picture time evolution an observer whose experimental capabilities are limited to the system described by 𝔐 would witness. Since it is the time evolution of an open system it is, in general, irreversible. From (4.1) we can isolate some mathematical properties of the reduced dynamics. (1){𝑇𝑡}𝑡0 is a family of completely positive and normal linear operators on 𝔐.(2)𝑇𝑡(𝟙)=𝟙 for all 𝑡0, in particular; each 𝑇𝑡 is contractive. (3)𝑡𝑇𝑡(𝑥) is weak* continuous for any 𝑥𝔐.

In general the reduced dynamics {𝑇𝑡}𝑡0 is not Markovian, that is, memory-free, and hence the operators {𝑇𝑡}𝑡0 do not form a one-parameter semigroup. However, in many physically relevant situations it is a good approximation to describe the reduced dynamics by a semigroup satisfying the above properties (1)–(3), that is, a weak* continuous semigroup of completely positive unital maps on the von Neumann algebra 𝔐. Such a semigroup is called a quantum dynamical semigroup. We remark that in many physically relevant models we have the following structure: 𝔑=𝔐𝔐0, acting on a tensor product 0 of two Hilbert spaces, where 𝔐0 describes the environment of the system (e.g., a heat bath). The time evolution of the system and environment is Hamiltonian, that is, 𝛽𝑡(𝑥)=ei𝑡𝐻𝑥ei𝑡𝐻 with 𝐻=𝐻1𝟙+𝟙𝐻2+𝐻int, where 𝐻1 and 𝐻2 are the Hamiltonians belonging to the system and its environment, and 𝐻int is an interaction term. Moreover, the conditional expectation 𝐸𝜔 is given with respect to a reference state 𝜔 of the environment, that is, 𝜑𝜔(𝑥)=𝜑(𝐸𝜔(𝑥)) for all 𝑥𝔑 and 𝜑𝔐. In this situation, the predual time evolution is given by the familiar formula𝑇𝑡,(𝜑)=tr2ei𝑡𝐻(𝜑𝜔)ei𝑡𝐻,(4.2) where 𝜑 is a normal state on 𝔐 and tr2 denotes the partial trace with respect to the degrees of freedom of the environment.

Since the reduced dynamics is in general not reversible, new phenomena like the approach to equilibrium can appear. In this paper, we are particularly interested in an effect called decoherence. The following general and mathematically rigorous characterization of decoherence in the algebraic framework was introduced by Blanchard and Olkiewicz [3, 4]. Its present form is taken from [24]; the relation of this form and that given in [3] is discussed in [25].

Definition 4.1. We say that the reduced dynamics {𝑇𝑡}𝑡0 displays decoherence if the following assertions are satisfied: there exists a 𝑇𝑡-invariant von Neumann subalgebra 𝔐1 of 𝔐 and an weak* continuous group {𝛼𝑡}𝑡 of *-automorphisms on 𝔐1 such that 𝑇𝑡𝔐1=𝛼𝑡 for 𝑡0, and a 𝑇𝑡-invariant and *-invariant weak* closed subspace 𝔐2 of 𝔐 such that 𝔐=𝔐1𝔐2,(4.3)lim𝑡𝑇𝑡(𝑥)=0intheweaktopologyforany𝑥𝔐2.(4.4) Moreover, we require that 𝔐1 is a maximal von Neumann subalgebra of 𝔐 (in the sense of not being properly contained in any larger von Neumann subalgebra) on which {𝑇𝑡}𝑡0 extends to a group of *-automorphisms. We call 𝔐1 the algebra of effective observables.

The physical interpretation of this definition is rather clear: if decoherence takes place there is a (maximal) von Neumann subalgebra on which the reduced dynamics is reversible, that is, given by an automorphism group, and a complementary subspace on which the expectation values with respect to any normal state of all its elements tend to zero in time. Thus any observable 𝑥𝔐 has a decomposition 𝑥=𝑥1+𝑥2, where 𝑥𝑖𝔐𝑖, such that 𝜑(𝑇𝑡(𝑥2))0 as 𝑡 for any normal state 𝜑 on 𝔐. Hence after a sufficiently long time the system behaves effectively like a closed system described by 𝔐1 and {𝛼𝑡}𝑡. By analyzing the structure of the algebra of effective observables 𝔐1 and the reversible dynamics {𝛼𝑡}𝑡 various physically relevant and well-known phenomena of decoherence can be identified, like the appearance of pointer states, environment-induced superselection rules, and classical dynamical systems. In this way it is possible to obtain an exhaustive classification of possible decoherence scenarios, see [3] for a thorough discussion of this point. Particularly interesting is the case when 𝔐1 is a factor, that is, after decoherence we still have a system of pure quantum character. This is of interest in the context of quantum computation since in this way one may obtain a system which retains its quantum character despite decoherence.

According to Definition 4.1, if {𝑇𝑡}𝑡0 is a group of automorphisms, decoherence takes place and the splitting (4.3) is trivial with 𝔐2={0}. However, we will keep this slightly unfortunate terminology since it simplifies the statements of theorems, keeping in mind that physically decoherence corresponds to the case when 𝔐2{0}. This can only happen if {𝑇𝑡}𝑡0 is irreversible.

We remark that the algebra 𝔐1 has been studied in [26] and explicit representations of 𝔐1 are obtained for quantum dynamical semigroups with unbounded generators. Moreover, in [27] a different notion of decoherence for quantum dynamical semigroups is introduced in a mathematically rigorous way, and its connection to the Blanchard-Olkiewicz notion is briefly discussed in [28]. In [29] an asymptotic property similar to (4.3) and (4.4) is discussed under the designation “limited relaxation.”

In connection with Definition 4.1, the following question arises: if there exists a maximal von Neumann subalgebra of 𝔐 on which {𝑇𝑡}𝑡0 extends to a group of automorphisms, is this subalgebra necessarily unique? The following theorem answers this question.

Theorem 4.2. Let {𝑇𝑡}t0 be a quantum dynamical semigroup and suppose it has a faithful normal invariant state 𝜔. Then there exists a unique maximal von Neumann subalgebra on which {𝑇𝑡}𝑡0 extends to a group of automorphisms.

Proof. Let 𝔐𝑖, where 𝑖𝐼 is some index set, be a collection of von Neumann subalgebras on each of which {𝑇𝑡}𝑡0 extends to a group of automorphisms. Then we put 𝔅=𝑖𝐼𝔐𝑖=𝑥lin𝑖1𝑥𝑖𝑘𝑖1,,𝑖𝑘𝐼,𝑘,𝑥𝑖𝔐𝑖,(4.5) where the closure is taken in the ultraweak topology. We proceed as in the proof of Proposition 3.3 and introduce 𝐷(𝑥,𝑦)=𝑇𝑡(𝑥𝑦)𝑇𝑡(𝑥)𝑇𝑡(𝑦) for a fixed 𝑡0 and 𝑥,𝑦𝔐. Then 𝐷(𝑥,𝑥)=0 if and only if 𝐷(𝑥,𝑦)=0 for all 𝑦𝔐. Now if 𝑥𝔐𝑖, we get 𝐷(𝑥,𝑥)=0, thus 𝐷(𝑥,𝑦)=0 for 𝑦𝔐, where 𝑖𝐼, that is, 𝑇𝑡(𝑥𝑦)=𝑇𝑡(𝑥)𝑇𝑡(𝑦). Proceeding inductively we have 𝑇𝑡𝑥𝑖1𝑥𝑖𝑘=𝑇𝑡𝑥𝑖1𝑇𝑡𝑥𝑖𝑘,(4.6) for an arbitrary monomial 𝑥𝑖1𝑥𝑖𝑘, where 𝑥𝑖𝔐𝑖, 𝑖1,,𝑖𝑘𝐼. Therefore, if 𝐵0=lin{𝑥𝑖1𝑥𝑖𝑘𝑖1,,𝑖𝑘𝐼,𝑘}, then 𝑇𝑡𝐵0𝐵0 is a *-homomorphism of the *-subalgebra 𝐵0, and 𝑇𝑡𝔅𝔅 is a *-homomorphism as well.
Next we note that 𝑇𝑡𝔅 is injective. Namely, if 𝑥ker(𝑇𝑡𝔅), we get 𝜔(𝑥𝑥)=𝜔(𝑇𝑡(𝑥𝑥))=𝜔(𝑇𝑡(𝑥)𝑇𝑡(𝑥))=0, thus 𝑥=0 by faithfulness of 𝜔 and so ker(𝑇𝑡𝔅)={0}.
We now establish that 𝑇𝑡𝔅𝔅 is surjective. Since 𝑇𝑡 is an injective *-homomorphism on the C*-algebra 𝔅, it follows that 𝑇𝑡(𝑥)=𝑥 for all 𝑥𝔅. Moreover, notice that 𝑇𝑡𝐵0𝐵0 is invertible since each restriction 𝑇𝑡𝔐𝑖 is invertible. This implies 𝑇𝑡1(𝑥)=𝑇𝑡(𝑇𝑡1(𝑥))=𝑥 for any 𝑥𝐵0. Now for the proof of surjectivity choose 𝑦𝔅. By the Kaplansky density theorem, there exists a net {𝑦𝑗}𝑗𝐽𝐵0 such that 𝑦𝑗𝐶 for all 𝑗𝐽, and such that lim𝑦𝑗=𝑦. Put 𝑥𝑗=𝑇𝑡1(𝑦𝑗). Then 𝑥𝑗=𝑦𝑗𝐶, and by Alaoglu's theorem the net {𝑥𝑗}𝑗𝐽 has an ultraweak limit point 𝑥0. Thus there is a subnet {𝑥𝑗}{𝑥𝑗}𝑗𝐽 such that lim𝑥𝑗=𝑥0, hence 𝑦𝑗=𝑇𝑡𝑥𝑗𝑦=𝑇𝑡𝑥0.(4.7) We conclude by injectivity that any ultraweak limit point of the net {𝑥𝑗}𝑗𝐽 is equal to 𝑥0, hence this net is convergent to 𝑥0, and 𝑇𝑡(𝑥0)=𝑦, establishing surjectivity. We have thus proved that 𝑇𝑡𝔅 is a *-automorphism.
To finish the proof, we have to choose the collection {𝔐𝑖𝑖𝐼} in (4.5) to consist of all von Neumann subalgebras on which {𝑇𝑡}𝑡0 extends to a group of automorphisms.

We next prove that in certain cases the splitting (4.3) is always given by a conditional expectation.

Proposition 4.3. Suppose that the (not necessarily Markovian) reduced dynamics {𝑇𝑡}𝑡0 on the von Neumann algebra 𝔐 displays decoherence. If 𝑇𝑡𝔐1=id𝔐1, then there exists a normal conditional expectation 𝐸 from 𝔐 onto 𝔐1.

Proof. Let 𝑥𝔐 and write 𝑥=𝑥1+𝑥2 with 𝑥𝑖𝔐𝑖, 𝑖=1,2, and define 𝐸𝔐𝔐1 by 𝐸(𝑥)=𝑥1. Then 𝐸(𝔐)=𝔐1 and 𝐸2=𝐸. Since 𝔐2 is *-invariant, 𝐸(𝑥)=𝐸(𝑥1+𝑥2)=𝑥1=𝐸(𝑥), hence 𝐸(𝔐sa)𝔐sa. Now let 𝑥𝔐+, consider the decomposition 𝑥=𝑥1+𝑥2, 𝑥𝑖𝔐𝑖, 𝑖=1,2, and suppose that 𝑥1𝔐sa is not positive. Then there exists 𝜑𝔐+ such that 𝜑(𝑥1)<0. This implies 𝑇0𝜑𝑡𝑇(𝑥)=𝜑𝑡𝑥1𝑇+𝜑𝑡𝑥2𝑥=𝜑1𝑇+𝜑𝑡𝑥2,(4.8) and letting 𝑡 yields 𝜑(𝑇𝑡(𝑥2))0, a contradiction. Thus 𝐸(𝔐+)𝔐+, and since 𝐸(𝟙)=𝟙 it follows that 𝐸1. From 𝐸2=𝐸, we get 𝐸=1, so 𝐸 is a projection of norm 1 and hence by Tomiyama's theorem a conditional expectation. Since ker𝐸={𝑥𝔐𝐸(𝑥)=𝑥1=0}=𝔐2 is ultraweakly closed, we obtain by a theorem of Tomiyama [30] that 𝐸 is normal.

Given a reduced dynamics, the question arises of under what conditions decoherence will occur. In the Markovian case, sufficient conditions for the appearance of decoherence in the sense of Definition 4.1 have been formulated in [31]. In fact, we have the following theorem.

Theorem 4.4. Let {𝑇𝑡}𝑡0 be a weak* continuous one-parameter semigroup with a faithful normal state 𝜔. Assume that the following conditions are satisfied. (1)Each 𝑇𝑡, 𝑡0, is strongly positive and unital. (2)Let {𝜎𝜔𝑡}𝑡 denote the modular group (see, e.g., [10]) corresponding to the state 𝜔. Assume that [𝑇𝑡,𝜎𝜔𝑠]=0 for all 𝑠 and 𝑡0. Then {𝑇𝑡}𝑡0 displays decoherence and there exists a normal conditional expectation 𝐸 from 𝔐 onto 𝔐1 such that [𝑇𝑡,𝐸]=0 for all 𝑡0 and 𝜔𝐸=𝜔.

The splitting 𝔐=𝔐1𝔐2 of {𝑇𝑡}𝑡0 provided by this theorem is called the isometric-sweeping splitting. In [31] the theorem was proved under the more general hypothesis that 𝜔 is only a faithful semifinite normal weight. Then some additional technical assumptions about {𝑇𝑡}𝑡0 are necessary. A simpler proof for the theorem as stated above has been given in [24].

The Jacobs-de Leeuw-Glicksberg splitting for weak* continuous semigroups on von Neumann algebras established in Corollary 3.2 can now be applied to establish decoherence.

Corollary 4.5. Let {𝑇𝑡}𝑡0 be a weak* continuous one-parameter semigroup of strongly positive unital operators with a faithful normal state 𝜔. Assume that 𝔐r=𝔐1. Then {𝑇𝑡}𝑡0 displays decoherence.

Proof. According to Corollary 3.2, the Jacobs-de Leeuw-Glicksberg splitting exists, and by Proposition 3.6 we conclude that the requirements of Definition 4.1 are satisfied.

Remark 4.6. Whenever a conditional expectation 𝐸 from a von Neumann algebra 𝔐 onto a von Neumann subalgebra 𝔐1 satisfies 𝜔𝐸=𝜔 for a faithful normal state 𝜔, it is uniquely determined by these conditions [17, Corollary II.6.10]. Since in case of the isometric-sweeping splitting as given by Theorem 4.4 we have 𝔐1=𝐸𝔐 and 𝔐2=(𝟙𝐸)𝔐, and in the Jacobs-de Leeuw-Glicksberg splitting as given by Corollary 3.2 we have 𝔐r=𝑄𝔐 and 𝔐s=(𝟙𝑄)𝔐, it follows that the isometric-sweeping and Jacobs-de Leeuw-Glicksberg splittings agree whenever 𝔐r=𝔐1.

4.2. Uniformly Continuous Semigroups

The purpose of this section is to show how the Jacobs-de Leeuw-Glicksberg splitting can be applied to establish decoherence of quantum dynamical semigroups in the sense of Definition 4.1. To avoid complications arising from unbounded generators, we will concentrate on the case of uniformly continuous quantum dynamical semigroups. We will arrive at a result which avoids assumption (2) in Theorem 4.4.

Let 𝔐 be a von Neumann algebra acting on a separable Hilbert space , and let {𝑇𝑡}𝑡0 be a quantum dynamical semigroup such that 𝑡𝑇𝑡 is continuous in the uniform topology. Then by [32] the generator 𝑍 of {𝑇𝑡}𝑡0, which is a bounded operator on 𝔐, is given by𝑍𝑥=𝐺𝑥+𝑥𝐺+Φ(𝑥),(4.9) where 𝐺L() and Φ𝔐L() is a normal completely positive map. Since we have 𝑇𝑡(𝟙)=𝟙 for all 𝑡0, it follows that 𝑍𝟙=0 which forces 𝐺=𝐺Φ(𝟙). Upon introducing the operator 𝐻=i𝐺+(1/2)iΦ(𝟙), it is seen that 𝐻 is a bounded selfadjoint operator on and 𝑍 may be written as[]1𝑍𝑥=i𝐻,𝑥2{Φ(𝟙),𝑥}+Φ(𝑥),(4.10) where {,} denotes the anticommutator. Let us suppose now that Φ has a Kraus decompositionΦ(𝑥)=𝑛=1𝐴𝑛𝑥𝐴𝑛,(4.11) where {𝐴𝑛}𝑛 is a sequence of bounded linear operators on and the series converges in the weak* topology. This is always the case if 𝔐 is injective or equivalently, by the Connes theorem [33] and separability of , that 𝔐 is hyperfinite (this includes the case 𝔐=L()). The preadjoint operator of 𝑍 on 𝔐 then has the familiar Lindblad form [34]𝑍[]1𝜌=i𝐻,𝜌2𝑛=1𝜌𝐴𝑛𝐴𝑛+𝐴𝑛𝐴𝑛𝜌+𝑛=1𝐴𝑛𝜌𝐴𝑛,(4.12) where 𝜌𝔐. We are now able to prove the following theorem.

Theorem 4.7. Let {𝑇𝑡}𝑡0 be as above. Assume that there is a faithful normal state 𝜔 such that 𝜔𝑇𝑡=𝜔 for all 𝑡0 and that 𝐻 in (4.10) has pure point spectrum. Then {𝑇𝑡}𝑡0 displays decoherence, and for the effective subalgebra 𝔐1 we have 𝔐1𝐴𝑛,𝐴𝑛𝑛𝔐,(4.13) where the prime denotes the commutant. Moreover, there exists a normal conditional expectation 𝑄 from 𝔐 onto 𝔐1 such that 𝜔𝑄=𝜔. If the derivation 𝑥i[𝐻,𝑥] leaves the subalgebra {𝐴𝑛,𝐴𝑛𝑛}𝔐 invariant, equality holds in (4.13).

Proof. First note that the assumptions of Proposition 3.3 are satisfied, that is, 𝔐𝑟 is a von Neumann subalgebra. Consider the subalgebra 𝔐1 defined in Lemma 3.4, then {𝑇𝑡}𝑡0 restricted to 𝔐1 extends to a group of automorphisms. We start by proving (4.13). By a simple calculation as in [20], one obtains 𝑍𝑥𝑥𝑥𝑍𝑥𝑥𝑍(𝑥)=𝑥Φ𝑥(𝟙)𝑥+Φ𝑥𝑥Φ𝑥𝑥Φ=(𝑥)𝑛=1𝐴𝑛,𝑥𝐴𝑛.,𝑥(4.14) The generator 𝑍, when restricted to 𝔐1, is a *-derivation; thus if 𝑥𝔐1, then 𝑥0=𝑍𝑥𝑥𝑥𝑍(𝑥)𝑍𝑥=𝑛=1𝐴𝑛,𝑥𝐴𝑛,𝑥,(4.15) that is, [𝐴𝑛,𝑥]=0 for all 𝑛, and, moreover, [𝐴𝑛,𝑥]=0 for all 𝑛 since 𝔐1 is a *-subalgebra. This proves that 𝔐1{𝐴𝑛,𝐴𝑛𝑛}𝔐. Conversely, under the assumption that i[𝐻,] leaves the right-hand side of (4.13) invariant, we have 𝑍𝑥=i[𝐻,𝑥] or 𝑇𝑡(𝑥)=ei𝑡𝐻𝑥ei𝑡𝐻 on {𝐴𝑛,𝐴𝑛𝑛}𝔐, which implies equality in (4.13).
Now let 𝑥𝔐s𝔐1, 𝑥0. Then there exist eigenvectors 𝜉,𝜂 of 𝐻 with corresponding eigenvalues 𝐸𝜉 and 𝐸𝜂 such that 𝜉,𝑥𝜂0, thus 𝜉,𝑇𝑡e(𝑥)𝜂=i𝑡𝐻𝜉,𝑥ei𝑡𝐻𝜂=ei𝑡(𝐸𝜉𝐸𝜂)𝜉,𝑥𝜂(4.16) is bounded away from 0, so 0 is not a weak* limit point of {𝑇𝑡(𝑥)𝑡0}. Since𝑥𝔐𝑠, this is a contradiction in view of (2.5), hence 𝔐s𝔐1={0}. Now let 𝑥𝔐1 and write 𝑥=𝑥s+𝑥r𝔐s𝔐r. Since 𝔐r𝔐1, we have 𝑥r𝔐1, and 𝑥s=𝑥𝑥r𝔐s𝔐1={0}, thus 𝑥𝔐r. This proves 𝔐1=𝔐r and it follows by Proposition 3.6 that 𝔐s has the property (3.3). So we conclude that {𝑇𝑡}𝑡0 displays decoherence. The last assertion is clear from Proposition 3.3.

We remark that in [29, equation (34)], a class of generators has been given for which equality in (4.13) always holds. As a corollary, we obtain the following result which is similar to the one proved in [35] and is also contained in [36].

Corollary 4.8. Let {𝑇𝑡}𝑡0 be a uniformly continuous semigroup on 𝔐 consisting of normal completely positive and contractive operators, and suppose it has a faithful normal invariant state 𝜔. If {𝐴𝑛,𝐴𝑛𝑛}=𝟙, then lim𝑡𝑇𝑡(𝑥)=𝜔(𝑥)𝟙intheweaktopology(4.17) for any 𝑥𝔐.

Thus if 𝔐1 is trivial the semigroup describes the approach to equilibrium.

We remark that the last theorem can be generalized to certain cases when the semigroup {𝑇𝑡}𝑡0 is not uniformly continuous but has an unbounded generator of the form (4.10).

The existence of a faithful normal invariant state of a quantum dynamical semigroup as required by Theorems 4.7 and 4.4 has been discussed in the literature. It is particularly simple in the case of a finite-dimensional von Neumann algebra, that is, a matrix algebra. Suppose 𝔐=𝑀(𝑑) is the 𝑑×𝑑-matrix algebra and consider a quantum dynamical semigroup {𝑇𝑡}𝑡0 on 𝔐, then its generator is given by (4.10) and its preadjoint by (4.12), thus if the 𝐴𝑛 are normal, 𝜌0=(1/𝑑)𝟙 is a faithful normal invariant state for {𝑇𝑡}𝑡0. Such generators arise, for example, in the singular coupling limit of 𝑁-level systems, see [8].

Acknowledgments

Thanks are due to Professor Ph. Blanchard (Bielefeld) for various discussions and to the anonymous referees for a number of remarks which helped to improve the paper and for pointing out an error in it.

References

  1. R. Haag, Local Quantum Physics, Springer, Berlin, Germany, 2nd edition, 1996.
  2. F. Strocchi, An Introduction to the Mathematical Structure of Quantum Mechanics, World Scientific, Hackensack, NJ, USA, 2005.
  3. P. Blanchard and R. Olkiewicz, “Decoherence induced transition from quantum to classical dynamics,” Reviews in Mathematical Physics, vol. 15, no. 3, pp. 217–243, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. Olkiewicz, “Environment-induced superselection rules in Markovian regime,” Communications in Mathematical Physics, vol. 208, no. 1, pp. 245–265, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. N. P. Landsman, “When champions meet: rethinking the Bohr-Einstein debate,” Studies in History and Philosophy of Science B, vol. 37, no. 1, pp. 212–242, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  6. N. P. Landsman, “Between classical and quantum,” in Handbook of the Philosophy of Science: Philosophy of Physics, J. Butterfield and J. Earman, Eds., North Holland, Amsterdam, The Netherlands, 2007.
  7. E. B. Davies, Quantum Theory of Open Systems, Academic Press, London, UK, 1976.
  8. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, New York, NY, USA, 2002.
  9. K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, NY, USA, 2000.
  10. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum-Statistical Mechanics I, Springer, New York, NY, USA, 2nd edition, 1987.
  11. K. Jacobs, “Fastperiodizitätseigenschaften allgemeiner Halbgruppen in Banach-Räumen,” Mathematische Zeitschrift, vol. 67, pp. 83–92, 1957. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. K. De Leeuw and I. Glicksberg, “Almost periodic compactifications,” Bulletin of the American Mathematical Society, vol. 65, pp. 134–139, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. K. De Leeuw and I. Glicksberg, “Applications of almost periodic compactifications,” Acta Mathematica, vol. 105, pp. 63–97, 1961. View at Publisher · View at Google Scholar
  14. J. van Neerven, The Asymptotic Behaviour of Semigroups of Linear Operators, Birkhäuser, Basel, Switzerland, 1996.
  15. W. Arendt and C. J. K. Batty, “Tauberian theorems and stability of one-parameter semigroups,” Transactions of the American Mathematical Society, vol. 306, no. 2, pp. 837–852, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Y. I. Lyubich and Q. P. Vû, “Asymptotic stability of linear differential equations in Banach spaces,” Studia Mathematica, vol. 88, no. 1, pp. 37–42, 1988. View at Zentralblatt MATH
  17. B. Blackadar, Operator Algebras, Springer, Berlin, Germany, 2006.
  18. B. Kümmerer and R. Nagel, “Mean ergodic semigroups on W-algebras,” Acta Scientiarum Mathematicarum, vol. 41, no. 1-2, pp. 151–159, 1979.
  19. A. Bátkai, U. Groh, C. Kunszenti-Kovács, and M. Schreiber, “Decomposition of operator semigroups on W-algebras,” http://arxiv.org/abs/1106.0287v1. View at Publisher · View at Google Scholar
  20. D. E. Evans, “Irreducible quantum dynamical semigroups,” Communications in Mathematical Physics, vol. 54, no. 3, pp. 293–297, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. J. Tomiyama, “On the projection of norm one in W-algebras,” Proceedings of the Japan Academy, vol. 33, pp. 125–129, 1957.
  22. D. W. Robinson, “Strongly positive semigroups and faithful invariant states,” Communications in Mathematical Physics, vol. 85, no. 1, pp. 129–142, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. A. Frigerio, “Stationary states of quantum dynamical semigroups,” Communications in Mathematical Physics, vol. 63, no. 3, pp. 269–276, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. M. Hellmich, Decoherence in infinite quantum systems, Ph.D. thesis, University of Bielefeld, Bielefeld, Germany, 2009.
  25. R. Carbone, E. Sasso, and V. Umanità, “Decoherence for positive semigroups on M2(),” Journal of Mathematical Physics, vol. 52, no. 3, Article ID 032202, 2011.
  26. A. Dhahri, F. Fagnola, and R. Rebolledo, “The decoherence-free subalgebra of a quantum Markov semigroup with unbounded generator,” Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol. 13, no. 3, pp. 413–433, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. R. Rebolledo, “Decoherence of quantum Markov semigroups,” Annales de l'Institut Henri Poincar é, vol. 41, no. 3, pp. 349–373, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. R. Rebolledo, “A view on decoherence via master equations,” Open Systems and Information Dynamics, vol. 12, no. 1, pp. 37–54, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. R. Alicki, “Controlled quantum open systems,” Lecture Notes in Physics, vol. 622, pp. 121–139, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. J. Tomiyama, “On the projection of norm one in W-algebras II,” The Tôhoku Mathematical Journal, vol. 10, pp. 125–129, 1958.
  31. P. Ługiewicz and R. Olkiewicz, “Classical properties of infinite quantum open systems,” Communications in Mathematical Physics, vol. 239, no. 1-2, pp. 241–259, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. E. Christensen, “Generators of semigroups of completely positive maps,” Communications in Mathematical Physics, vol. 62, no. 2, pp. 167–171, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. A. Connes, “Classification of injective factors,” Annals of Mathematics, vol. 104, no. 1, pp. 73–115, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  34. G. Lindblad, “On the generators of quantum dynamical semigroups,” Communications in Mathematical Physics, vol. 48, no. 2, pp. 119–130, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  35. H. Spohn, “An algebraic condition for the approach to equilibrium of an open N-level system,” Letters in Mathematical Physics, vol. 2, no. 1, pp. 33–38, 1977.
  36. A. Frigerio and M. Verri, “Long-time asymptotic properties of dynamical semigroups on W-algebras,” Mathematische Zeitschrift, vol. 180, no. 2, pp. 275–286, 1982. View at Publisher · View at Google Scholar