# Quantum Dynamical Semigroups and Decoherence

Authors: Mario Hellmich
Journal: Advances in Mathematical Physics
 (2011)

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## 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.


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## Body

## 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}t≥0 of normal completely positive and unital linear maps on the von Neumann algebra [7, 8]. In the Markovian approximation, the family {Tt}t≥0 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 Section2 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 thatX is a Banach space and assume that it has a predual space denoted by X*, that is, (X*)*≅X. If x∈X and φ∈X*, 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={T∈L(X):∥T∥≤1}. Operators from L(X)1 are called contractive. We consider the algebraic tensor product X⊙X* and endow it with the projective cross norm γ; the completion of X⊙X* with respect to γ is a Banach space which will be denoted by X⊗γX*. Then its dual space (X⊗γX*)* is isometrically isomorphic in a canonical way with L(X,(X*)*)=L(X): If ψ∈(X⊗γX*)*, we define Φ(ψ)∈L(X) by(2.1)〈x⊗φ,ψ〉=〈Φ(ψ)(x),φ〉
for all x∈X and φ∈X*. It can now be shown that ψ↦Φ(ψ) extends to an isometric isomorphism, and we can thus write (X⊗γX*)*≅L(X).We now introduce the pointwise weak* topology onL(X). Let x∈X, φ∈X*, and define the seminorm L(X)∋T↦px,φ(T)=|〈T(x),φ〉|. The pointwise weak* topology is the locally convex topology on L(X) induced by the family {px,φ:x∈X,φ∈X*} of seminorms. If T=Φ(ψ)∈L(X), we see that px,φ(T)=|〈T(x),φ〉|=|〈x⊗φ,ψ〉|; thus the pointwise weak* topology coalesces with the σ(L(X),X⊗γX*) 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 operatorT∈L(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,T2∈Ln(X)1, then T1∘T2 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 T↦T∘S and S↦T∘S 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 T∈L(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)〉, x∈X, φ∈X*.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}t≥0 of linear and contractive operators on X, such that Ts∘Tt=Ts+t for all s,t≥0 and T0=idX. The semigroup is called weak* continuous provided each Tt is a normal operator and [0,∞[∋t↦Tt(x) is weak* continuous for any x∈X. For a weak* continuous semigroup there exists the following concept of a weak* generatorZ:(2.2)Zx=lim⁡t↓0Tt(x)-xtintheweak*topology,(2.3)dom⁡Z={x∈X:thelimitin(2.2)exists}.
The predual semigroup {Tt,*}t≥0 of a weak* continuous semigroup {Tt}t≥0 is strongly continuous, and the adjoint of its generator Z* is equal to the weak* generator Z.Suppose now that{Tt}t≥0 is a weak* continuous contractive semigroup on X, and write S0={Tt:t≥0}⊆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=S0¯⊆Ln(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 G⊆S, the so-called Sushkevich kernel [9], which is given by(2.4)G=⋂R∈SR∘S,
and Q∈G will denote the unit of G. We then have G=Q∘S. 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 T1∘T2.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.
LetS0={Tt}t≥0 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, t≥0, such that X=Xs⊕Xr, and
(2.5)Xs={x∈X:0∈{Tt(x):t≥0}¯w*},(2.6)Xr=lin⁡⁡{x∈dom⁡Z:∃α∈RsuchthatZx=iαx}¯w*=lin⁡⁡{x∈X:∃α∈RsuchthatTt(x)=eiαtx∀t≥0}¯w*.Proof.
SinceQ2=Q, the unit Q is a normal projection such that [Q,Tt]=0 for all t≥0. The theorem will be established once we prove that Xs=kerQ and Xr=ranQ.
Letx∈kerQ. Since Q∈S, there is a net {Ti}i∈I⊆S0 such that Ti→Q relative to the pointwise weak* topology; hence Ti(x)→Qx=0, so 0∈{Tt(x):t≥0}¯w*. Conversely, assume 0∈{Tt(x):t≥0}¯w* for some x. Then there is a net {Ti}i∈I⊆S0 such that Ti(x)→0 relative to the weak* topology. By compactness of S, there is a subnet {Tj}j∈J⊆{Ti}i∈I with Tj→R relative to the pointwise weak* topology for some R∈S, and it follows that Rx=0. Hence R′QRx=0 for all R′∈S. Choosing R′ to be the inverse of QR in G, we get Qx=0, hence x∈kerQ. We have thus proved that Xs=kerQ.
LetĜ be the character group of G. For each χ∈Ĝ define the operator
(2.7)X∋x⟼Pχx=∫Gχ(S)¯Sxdμ(S),
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 ∥Pχ∥≤1. Then for all R∈G we get
(2.8)RPχx=∫Gχ(S)¯RSxdμ(S)=χ(R)∫Gχ(RS)¯RSxdμ(S)=χ(R)∫Gχ(S)¯Sxdμ(S)=χ(R)Pχx,
in particular QPχ=Pχ; therefore, TtPχ=TtQPχ=χ(TtQ)Pχ for all t≥0. Since t↦χ(TtQ) is continuous and satisfies the functional equation
(2.9)χ(TtQ)⋅χ(TsQ)=χ(Tt+sQ)∈{z∈C:|z|=1}
for all t,s≥0, we have χ(TtQ)=eiαt for some α∈ℝ. Thus TtPχ=eiαtPχ, hence PχX⊆domZ and ZPχ=iαPχ for all χ∈Ĝ. We next define the subspace
(2.10)M=lin⁡{⋃χ∈ĜPχX}¯w*⊆Xr.
We prove that ranQ⊆M⊆Xr. Let φ∈M⊥={φ∈X*:〈x,φ〉=0∀x∈M}. Then 〈Pχx,φ〉=0 for all x∈X, χ∈Ĝ, that is,
(2.11)∫Ĝχ(S)¯〈Sx,φ〉dμ(S)=0
for all x∈X, χ∈Ĝ. Since the character group Ĝ is total in L2(G,μ) by the Stone-Weierstraß theorem and since S↦〈Sx,φ〉 is continuous it follows that 〈Sx,φ〉=0 for all S∈G and x∈X. Take S=Q, then we obtain φ∈ranQ⊥ and thus M⊥⊆ranQ⊥. By the bipolar theorem we obtain ranQ⊆co¯M=M, since ranQ is a weak* closed subspace. Conversely, let x∈domZ with Zx=iαx for some α∈ℝ. It follows that Tt(x)=eiαtx for all t≥0 and consequently Rx=eiαx for R∈S. Thus there exists β∈ℝ such that Qx=eiβx=Q2x. Consequently, we must have β=0 which implies Qx=x∈ranQ, hence Xr⊆ranQ, and the proof is finished.Corollary 2.2.
Under the hypothesis of Theorem2.1, there exists a weak* continuous one-parameter group {αt}t∈ℝ of isometries on Xr such that αt=Tt↾Xrfor t≥0.Proof.
LetT∈S, then QT∈G, and let R be the inverse of QT in G, that is, R(QT)=Q. Then for all x∈Xr, we have RTx=RQTx=Qx=x. Now write αt=QTt for t≥0 and let α-t be the inverse of αt in G. The foregoing calculation shows that {αt}t∈ℝ is a one-parameter group on Xr. Moreover, it is clear that it is weak* continuous and contractive. Now assume that there is x∈Xr and t≥0 such that ∥αt(x)∥<∥x∥. Then it follows that ∥α-t∥>1, contradiction; thus {αt}t∈ℝ is isometric.The subspaceXr 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 subspaceXs, 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 specZ∩iℝ of the generator Z.Proposition 2.3.
Assume that the hypothesis of Theorem2.1 is satisfied and additionally that specZ∩iℝ is at most countable. Then the stable subspace (2.5) is given by
(2.12)Xs={x∈X:lim⁡t→∞Tt(x)=0relativetotheweak*topology}.
Moreover, the convergence in (2.12) is uniform for x in Xs∩X1.Proof.
Consider the predual semigroup{Tt,*}t≥0 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 {Tt,*s}t≥0 be the restriction of {Tt,*}t≥0 to Xs,*. Since Xs,* is closed; the generator Z*s of {Tt,*s}t≥0 is given by the restriction Z*s=Z*↾Xs,*, domZ*s=domZ*∩Xs,*. A similar construction applies to the reversible subspace Xr. We check that specZ*s⊆specZ*. Let λ∈ρ(Z*), that is, the map (λ1-Z*):domZ*→X* is bijective. Then clearly the map (λ1-Z*s)=(λ1-Z*)↾Xs,*:domZ*∩Xs,*→Xs,* is injective. It is also surjective: let φ∈Xs,*, then there is ψ=ψs⊕ψr∈domZ* such that (λ1-Z*)ψ=φ. Now
(2.13)(λ1-Z*)ψ=(λ1-Z*)(ψs⊕ψr)=φ⊕0,
so (λ1-Z*)ψr=0 and ψr=0 by injectivity. Thus (λ1-Z*s) is bijective and λ∈ρ(Z*s). In particular, using specZ=specZ* we find that
(2.14)specZ*s∩iRiscountable.
We now see that
(2.15)specpZs∩iR=∅,
for if iλ∈specpZs, λ∈ℝ, then the corresponding eigenvector x∈domZs⊆Xs 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 {Tt,*s}t≥0 is strongly stable, that is, for all x*∈Xs,* we have limt→∞∥Tt,*s(x*)∥=0. Thus if x∈Xs and x*∈X* it follows that
(2.16)|〈Tt(x),x*〉|=|〈Q⊥(Tt(x)),x*〉|=|〈x,Tt,*(Q*⊥(x*))〉|≤∥x∥⋅∥Tt,*s(Q*⊥(x*))∥⟶0
as t→∞ uniformly for x∈Xs∩X1, where Q⊥=idX-Q denotes the projection onto Xs.


## 3. Semigroups on von Neumann Algebras

The results of the previous section apply to the case of Neumann algebras. Letℋ be a Hilbert space. A von Neumann algebra is a *-subalgebra of the Banach-*-algebra L(ℋ) of all bounded linear operators acting on ℋ, 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 1∈𝔐. The ultraweak topology on 𝔐 is defined by the seminorms pρ(x)=|tr(ρx)|, where ρ runs through the trace class operators on ℋ, 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 x∈𝔐+) and normalized (i.e., ∥φ∥=1, equivalently φ(1)=1) will be called a normal state. A state is called faithful if x∈𝔐+ and ω(x)=0 implies x=0. For proofs of these results, we refer to [10, 17].LetT∈L(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}i∈I⊆𝔐+ we have supiT(xi)=T(supixi). Furthermore, T is called strongly positive whenever it satisfies Kadison's inequality, that is, ∥T(1)∥T(x*x)≥T(x)*T(x) for any x∈𝔐. Clearly strong positivity implies positivity. An even stronger notion of positivity is complete positivity: T∈L(𝔐) is called completely positive whenever ∑i,j=1nyi*T(xi*xj)yj≥0 for all n∈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 x∈𝔐.The following result has been established in [18].Proposition 3.1.
Suppose thatS⊆Ln(𝔐)1 is a subset of normal contractive linear operators. Then the following assertions are equivalent.  (1)
The set{T*(φ):T∈S}⊆𝔐* is relatively weakly compact for every φ∈𝔐*.(2)
The setS is equicontinuous when 𝔐 is endowed with the Mackey topology (i.e., the τ(𝔐,𝔐*) topology).(3)
The pointwise weak* closure ofS consists of normal operators: S¯⊆Ln(𝔐)1.Moreover, these conditions are satisfied whenever there is a faithful normal state ω on 𝔐 such that
(3.1)ω(T(x*)T(x))≤ω(x*x)foranyT∈S,x∈M.
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 x∈𝔐+, T∈S, or briefly ω∘T≤ω, for all T∈S.IfS⊆Ln(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 x∈𝔐 and T∈S. 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}t≥0 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=𝔐r⊕𝔐s.Proof.
By Kadison’s inequality, (3.1) holds; thus Proposition 3.1 implies that the pointwise weak* closure S=S0¯, with S0={Tt}t≥0, 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 that𝔐r is actually a von Neumann subalgebra. Recall that a conditional expectation Q from a C*-algebra 𝔄 onto a C*-subalgebra 𝔅⊆𝔄 is a completely positive contraction with Q(x)=x for x∈𝔅 and Q(xyx)=xQ(y)x for x∈𝔅, y∈𝔄.Proposition 3.3.
Let{Tt}t≥0 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∈ℝ on 𝔐r such that Tt↾𝔐r=αt for all t≥0. Moreover, there exists a normal conditional expectation Q from 𝔐 onto 𝔐r such that ω∘Q=ω. Finally, 𝔐s is *-invariant.Proof.
Since eachTt is a contraction; Corollary 3.2 applies. Let M0={x∈𝔐:∃α∈ℝsuchthatTt(x)=eiαtx∀t≥0}, that is, we have linM0¯w*=𝔐r. As in [20] we define the sesquilinear map D:𝔐×𝔐→𝔐 by D(x,y)=Tt(x*y)-Tt(x)*Tt(y) for some fixed t≥0. By Kadison's inequality, the sesquilinear form φ∘D is positive-definite for any φ∈𝔐*+, so by the Cauchy-Schwarz inequality, D(x,x)=0 if and only if D(x,y)=0 for all y∈𝔐. Now let x∈M0, then Tt(x*x)≥Tt(x)*Tt(x)=e-iαte+iαtx*x=x*x. Thus 0≤ω(Tt(x*x)-x*x)≤ω(x*x-x*x)=0, and by faithfulness Tt(x*x)=x*x, hence D(x,x)=0 for all x∈M0. So D(x,y)=0 for all x,y∈M0, that is, Tt(x*y)=Tt(x)*Tt(y)=ei(α2-α1)tx*y, and we conclude that xy∈M0 whenever x,y∈M0. It follows that linM0 is a *-subalgebra of 𝔐 (containing 1) and consequently 𝔐r is a von Neumann subalgebra, and Tt(x*y)=Tt(x*)Tt(y) for all x,y∈𝔐r. By Corollary 2.2, the restriction of Tt to 𝔐r extends to a one-parameter group {αt}t∈ℝ of isometries and the above argument shows that αt must be a *-homomorphism. Let Q be the Sushkevich kernel of the semigroup S⊆Ln(𝔐)1. Since Q is a projection and ∥Q∥=1 it follows from Tomiyama’s theorem [21] that Q is a conditional expectation; since Q∈S; it is also clear that ω∘Q=ω. The last assertion is clear as well.In the following, we will be interested in the stronger characterization of𝔐s by a stability property as in (2.12). We start by quoting the following result.Lemma 3.4.
Suppose that{Tt}t≥0 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
(3.2)M={x∈M:Tt(x*x)=Tt(x)*Tt(x)∀t≥0},M*={x∈M:Tt(xx*)=Tt(x)Tt(x)*∀t≥0},M1=M∩M*.
Then 𝔐1 is a Tt-invariant von Neumann subalgebra of 𝔐, and there exists a group of *-automorphisms {αt}t∈ℝ on 𝔐1 such that Tt↾𝔐1=αt for t≥0. 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}t≥0 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 𝔐r⊆𝔐1.Lemma 3.5.
Under the assumptions of Lemma3.4, for every x∈𝔐 the weak* limit points of the net {Tt(x)}t∈ℝ+ 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}t≥0 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
(3.3)Ms={x∈M:lim⁡t→∞Tt(x)=0intheweak*topology}.Proof.
Letx∈𝔐s and assume without loss of generality that ∥x∥≤1. By Alaoglu's theorem the net {Tt(x)}t∈ℝ+ contained in the unit ball of 𝔐 has a limit point x0 for t→∞. Then using Lemma 3.5, we find that x0∈𝔐1=𝔐r. But since x∈𝔐s, it follows that also x0∈𝔐s, that is, x0∈𝔐s∩𝔐r={0}; hence x0=0. This proves that any limit point of the net {Tt(x)}t∈ℝ+ is equal to 0; therefore we conclude that limt→∞Tt(x)=0 in the weak* topology for all x∈𝔐s.Moreover, let us remark the following: suppose that{Tt}t≥0 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 specZ∩iℝ 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 algebra𝔑, and its reversible time evolution is given by a weak* continuous group of *-automorphisms {βt}t∈ℝ on 𝔑. A subsystem can be described by a von Neumann subalgebra 𝔐⊆𝔑 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:(4.1)Tt(x)=E∘βt(x),x∈M,t≥0.
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}t≥0 is a family of completely positive and normal linear operators on 𝔐.(2)
Tt(1)=1 for all t≥0, in particular; each Tt is contractive.(3)
t↦Tt(x) is weak* continuous for any x∈𝔐.In general the reduced dynamics{Tt}t≥0 is not Markovian, that is, memory-free, and hence the operators {Tt}t≥0 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=𝔐⊗¯𝔐0, acting on a tensor product ℋ⊗ℋ0 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)=eitHxe-itH with H=H1⊗1+1⊗H2+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, φ⊗ω(x)=φ(Eω(x)) for all x∈𝔑 and φ∈𝔐*. In this situation, the predual time evolution is given by the familiar formula(4.2)Tt,*(φ)=tr⁡2[e-itH(φ⊗ω)eitH],
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}t≥0 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∈ℝ of *-automorphisms on 𝔐1 such that Tt↾𝔐1=αt for t≥0, and a Tt-invariant and *-invariant weak* closed subspace 𝔐2 of 𝔐 such that
(4.3)M=M1⊕M2,(4.4)lim⁡t→∞Tt(x)=0intheweak*topologyforanyx∈M2.
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}t≥0 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 observablex∈𝔐 has a decomposition x=x1+x2, where xi∈𝔐i, 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∈ℝ. By analyzing the structure of the algebra of effective observables 𝔐1 and the reversible dynamics {αt}t∈ℝ 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 Definition4.1, if {Tt}t≥0 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}t≥0 is irreversible.We remark that the algebra𝔐1 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 Definition4.1, the following question arises: if there exists a maximal von Neumann subalgebra of 𝔐 on which {Tt}t≥0 extends to a group of automorphisms, is this subalgebra necessarily unique? The following theorem answers this question.Theorem 4.2.
Let{Tt}t≥0 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}t≥0 extends to a group of automorphisms.Proof.
Let𝔐i, where i∈I is some index set, be a collection of von Neumann subalgebras on each of which {Tt}t≥0 extends to a group of automorphisms. Then we put
(4.5)B=⋁i∈IMi=lin⁡¯{xi1⋯xik:i1,…,ik∈I,k∈N,xil∈Mil},
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(x*y)-Tt(x)*Tt(y) for a fixed t≥0 and x,y∈𝔐. Then D(x,x)=0 if and only if D(x,y)=0 for all y∈𝔐. Now if x∈𝔐i, we get D(x,x)=0, thus D(x,y)=0 for y∈𝔐, where i∈I, that is, Tt(xy)=Tt(x)Tt(y). Proceeding inductively we have
(4.6)Tt(xi1⋯xik)=Tt(xi1)⋯Tt(xik),
for an arbitrary monomial xi1⋯xik, where xiℓ∈𝔐iℓ, i1,…,ik∈I. Therefore, if B0=lin{xi1⋯xik:i1,…,ik∈I,k∈ℕ}, then Tt:B0→B0 is a *-homomorphism of the *-subalgebra B0, and Tt:𝔅→𝔅 is a *-homomorphism as well.
Next we note thatTt↾𝔅 is injective. Namely, if x∈ker(Tt↾𝔅), we get ω(x*x)=ω(Tt(x*x))=ω(Tt(x)*Tt(x))=0, thus x=0 by faithfulness of ω and so ker(Tt↾𝔅)={0}.
We now establish thatTt:𝔅→𝔅 is surjective. Since Tt is an injective *-homomorphism on the C*-algebra 𝔅, it follows that ∥Tt(x)∥=∥x∥ for all x∈𝔅. Moreover, notice that Tt:B0→B0 is invertible since each restriction Tt↾𝔐i is invertible. This implies ∥Tt-1(x)∥=∥Tt(Tt-1(x))∥=∥x∥ for any x∈B0. Now for the proof of surjectivity choose y∈𝔅. By the Kaplansky density theorem, there exists a net {yj}j∈J⊆B0 such that ∥yj∥≤C for all j∈J, and such that limyj=y. Put xj=Tt-1(yj). Then ∥xj∥=∥yj∥≤C, and by Alaoglu's theorem the net {xj}j∈J has an ultraweak limit point x0. Thus there is a subnet {xjℓ}⊆{xj}j∈J such that limxjℓ=x0, hence
(4.7)yjl=Tt(xjl)⟶y=Tt(x0).
We conclude by injectivity that any ultraweak limit point of the net {xj}j∈J is equal to x0, hence this net is convergent to x0, and Tt(x0)=y, establishing surjectivity. We have thus proved that Tt↾𝔅 is a *-automorphism.
To finish the proof, we have to choose the collection{𝔐i:i∈I} in (4.5) to consist of all von Neumann subalgebras on which {Tt}t≥0 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}t≥0 on the von Neumann algebra 𝔐 displays decoherence. If Tt↾𝔐1=id𝔐1, then there exists a normal conditional expectation E from 𝔐 onto 𝔐1.Proof.
Letx∈𝔐 and write x=x1+x2 with xi∈𝔐i, i=1,2, and define E:𝔐→𝔐1 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 x∈𝔐+, consider the decomposition x=x1+x2, xi∈𝔐i, i=1,2, and suppose that x1∈𝔐sa is not positive. Then there exists φ∈𝔐*+ such that φ(x1)<0. This implies
(4.8)0≤φ(Tt(x))=φ(Tt(x1))+φ(Tt(x2))=φ(x1)+φ(Tt(x2)),
and letting t→∞ yields φ(Tt(x2))→0, a contradiction. Thus E(𝔐+)⊆𝔐+, and since E(1)=1 it follows that ∥E∥≤1. 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={x∈𝔐:E(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 Definition4.1 have been formulated in [31]. In fact, we have the following theorem.Theorem 4.4.
Let{Tt}t≥0 be a weak* continuous one-parameter semigroup with a faithful normal state ω. Assume that the following conditions are satisfied.  (1)
EachTt, t≥0, is strongly positive and unital.(2)
Let{σtω}t∈ℝ denote the modular group (see, e.g., [10]) corresponding to the state ω. Assume that [Tt,σsω]=0 for all s∈ℝ and t≥0.Then {Tt}t≥0 displays decoherence and there exists a normal conditional expectation E from 𝔐 onto 𝔐1 such that [Tt,E]=0 for all t≥0 and ω∘E=ω.The splitting𝔐=𝔐1⊕𝔐2 of {Tt}t≥0 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}t≥0 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 Corollary3.2 can now be applied to establish decoherence.Corollary 4.5.
Let{Tt}t≥0 be a weak* continuous one-parameter semigroup of strongly positive unital operators with a faithful normal state ω. Assume that 𝔐r=𝔐1. Then {Tt}t≥0 displays decoherence.Proof.
According to Corollary3.2, 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 expectationE from a von Neumann algebra 𝔐 onto a von Neumann subalgebra 𝔐1 satisfies ω∘E=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=(1-E)𝔐, and in the Jacobs-de Leeuw-Glicksberg splitting as given by Corollary 3.2 we have 𝔐r=Q𝔐 and 𝔐s=(1-Q)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 Definition4.1. 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.Let𝔐 be a von Neumann algebra acting on a separable Hilbert space ℋ, and let {Tt}t≥0 be a quantum dynamical semigroup such that t↦Tt is continuous in the uniform topology. Then by [32] the generator Z of {Tt}t≥0, which is a bounded operator on 𝔐, is given by(4.9)Zx=G*x+xG+Φ(x),
where G∈L(ℋ) and Φ:𝔐→L(ℋ) is a normal completely positive map. Since we have Tt(1)=1 for all t≥0, it follows that Z1=0 which forces G*=-G-Φ(1). Upon introducing the operator H=iG+(1/2)iΦ(1), it is seen that H is a bounded selfadjoint operator on ℋ and Z may be written as(4.10)Zx=i[H,x]-12{Φ(1),x}+Φ(x),
where {·,·} denotes the anticommutator. Let us suppose now that Φ has a Kraus decomposition(4.11)Φ(x)=∑n=1∞An*xAn,
where {An}n∈ℕ is a sequence of bounded linear operators on ℋ 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 ℋ, that 𝔐 is hyperfinite (this includes the case 𝔐=L(ℋ)). The preadjoint operator of Z on 𝔐* then has the familiar Lindblad form [34](4.12)Z*ρ=-i[H,ρ]-12∑n=1∞(ρAn*An+An*Anρ)+∑n=1∞AnρAn*,
where ρ∈𝔐*. We are now able to prove the following theorem.Theorem 4.7.
Let{Tt}t≥0 be as above. Assume that there is a faithful normal state ω such that ω∘Tt=ω for all t≥0 and that H in (4.10) has pure point spectrum. Then {Tt}t≥0 displays decoherence, and for the effective subalgebra 𝔐1 we have
(4.13)M1⊆{An,An*:n∈N}′∩M,
where the prime denotes the commutant. Moreover, there exists a normal conditional expectation Q from 𝔐 onto 𝔐1 such that ω∘Q=ω. If the derivation x↦i[H,x] leaves the subalgebra {An,An*:n∈ℕ}′∩𝔐 invariant, equality holds in (4.13).Proof.
First note that the assumptions of Proposition3.3 are satisfied, that is, 𝔐r is a von Neumann subalgebra. Consider the subalgebra 𝔐1 defined in Lemma 3.4, then {Tt}t≥0 restricted to 𝔐1 extends to a group of automorphisms. We start by proving (4.13). By a simple calculation as in [20], one obtains
(4.14)Z(x*x)-Z(x*)x-x*Z(x)=x*Φ(1)x+Φ(x*x)-Φ(x*)x-x*Φ(x)=∑n=1∞[An,x]*[An,x].
The generator Z, when restricted to 𝔐1, is a *-derivation; thus if x∈𝔐1, then
(4.15)0=Z(x*x)-x*Z(x)-Z(x*)x=∑n=1∞[An,x]*[An,x],
that is, [An,x]=0 for all n∈ℕ, and, moreover, [An*,x]=0 for all n∈ℕ since 𝔐1 is a *-subalgebra. This proves that 𝔐1⊆{An,An*:n∈ℕ}′∩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)=eitHxe-itH on {An,An*:n∈ℕ}′∩𝔐, which implies equality in (4.13).
Now letx∈𝔐s∩𝔐1, x≠0. Then there exist eigenvectors ξ,η∈ℋ of H with corresponding eigenvalues Eξ and Eη such that 〈ξ,xη〉≠0, thus
(4.16)〈ξ,Tt(x)η〉=〈e-itHξ,xe-itHη〉=eit(Eξ-Eη)〈ξ,xη〉
is bounded away from 0, so 0 is not a weak* limit point of {Tt(x):t≥0}. Sincex∈𝔐s, this is a contradiction in view of (2.5), hence 𝔐s∩𝔐1={0}. Now let x∈𝔐1 and write x=xs+xr∈𝔐s⊕𝔐r. Since 𝔐r⊆𝔐1, we have xr∈𝔐1, and xs=x-xr∈𝔐s∩𝔐1={0}, thus x∈𝔐r. This proves 𝔐1=𝔐r and it follows by Proposition 3.6 that 𝔐s has the property (3.3). So we conclude that {Tt}t≥0 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}t≥0 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,An*:n∈ℕ}′=ℂ1, then
(4.17)lim⁡t→∞Tt(x)=ω(x)1intheweak*topology
for any x∈𝔐.Thus if𝔐1 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}t≥0 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 Theorems4.7 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=Mℂ(d) is the d×d-matrix algebra and consider a quantum dynamical semigroup {Tt}t≥0 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}t≥0. Such generators arise, for example, in the singular coupling limit of N-level systems, see [8].


## 4.1. The Notion of Decoherence in the Algebraic Framework

Consider a closed quantum system whose algebra of observables is a von Neumann algebra𝔑, and its reversible time evolution is given by a weak* continuous group of *-automorphisms {βt}t∈ℝ on 𝔑. A subsystem can be described by a von Neumann subalgebra 𝔐⊆𝔑 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:(4.1)Tt(x)=E∘βt(x),x∈M,t≥0.
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}t≥0 is a family of completely positive and normal linear operators on 𝔐.(2)
Tt(1)=1 for all t≥0, in particular; each Tt is contractive.(3)
t↦Tt(x) is weak* continuous for any x∈𝔐.In general the reduced dynamics{Tt}t≥0 is not Markovian, that is, memory-free, and hence the operators {Tt}t≥0 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=𝔐⊗¯𝔐0, acting on a tensor product ℋ⊗ℋ0 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)=eitHxe-itH with H=H1⊗1+1⊗H2+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, φ⊗ω(x)=φ(Eω(x)) for all x∈𝔑 and φ∈𝔐*. In this situation, the predual time evolution is given by the familiar formula(4.2)Tt,*(φ)=tr⁡2[e-itH(φ⊗ω)eitH],
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}t≥0 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∈ℝ of *-automorphisms on 𝔐1 such that Tt↾𝔐1=αt for t≥0, and a Tt-invariant and *-invariant weak* closed subspace 𝔐2 of 𝔐 such that
(4.3)M=M1⊕M2,(4.4)lim⁡t→∞Tt(x)=0intheweak*topologyforanyx∈M2.
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}t≥0 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 observablex∈𝔐 has a decomposition x=x1+x2, where xi∈𝔐i, 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∈ℝ. By analyzing the structure of the algebra of effective observables 𝔐1 and the reversible dynamics {αt}t∈ℝ 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 Definition4.1, if {Tt}t≥0 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}t≥0 is irreversible.We remark that the algebra𝔐1 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 Definition4.1, the following question arises: if there exists a maximal von Neumann subalgebra of 𝔐 on which {Tt}t≥0 extends to a group of automorphisms, is this subalgebra necessarily unique? The following theorem answers this question.Theorem 4.2.
Let{Tt}t≥0 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}t≥0 extends to a group of automorphisms.Proof.
Let𝔐i, where i∈I is some index set, be a collection of von Neumann subalgebras on each of which {Tt}t≥0 extends to a group of automorphisms. Then we put
(4.5)B=⋁i∈IMi=lin⁡¯{xi1⋯xik:i1,…,ik∈I,k∈N,xil∈Mil},
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(x*y)-Tt(x)*Tt(y) for a fixed t≥0 and x,y∈𝔐. Then D(x,x)=0 if and only if D(x,y)=0 for all y∈𝔐. Now if x∈𝔐i, we get D(x,x)=0, thus D(x,y)=0 for y∈𝔐, where i∈I, that is, Tt(xy)=Tt(x)Tt(y). Proceeding inductively we have
(4.6)Tt(xi1⋯xik)=Tt(xi1)⋯Tt(xik),
for an arbitrary monomial xi1⋯xik, where xiℓ∈𝔐iℓ, i1,…,ik∈I. Therefore, if B0=lin{xi1⋯xik:i1,…,ik∈I,k∈ℕ}, then Tt:B0→B0 is a *-homomorphism of the *-subalgebra B0, and Tt:𝔅→𝔅 is a *-homomorphism as well.
Next we note thatTt↾𝔅 is injective. Namely, if x∈ker(Tt↾𝔅), we get ω(x*x)=ω(Tt(x*x))=ω(Tt(x)*Tt(x))=0, thus x=0 by faithfulness of ω and so ker(Tt↾𝔅)={0}.
We now establish thatTt:𝔅→𝔅 is surjective. Since Tt is an injective *-homomorphism on the C*-algebra 𝔅, it follows that ∥Tt(x)∥=∥x∥ for all x∈𝔅. Moreover, notice that Tt:B0→B0 is invertible since each restriction Tt↾𝔐i is invertible. This implies ∥Tt-1(x)∥=∥Tt(Tt-1(x))∥=∥x∥ for any x∈B0. Now for the proof of surjectivity choose y∈𝔅. By the Kaplansky density theorem, there exists a net {yj}j∈J⊆B0 such that ∥yj∥≤C for all j∈J, and such that limyj=y. Put xj=Tt-1(yj). Then ∥xj∥=∥yj∥≤C, and by Alaoglu's theorem the net {xj}j∈J has an ultraweak limit point x0. Thus there is a subnet {xjℓ}⊆{xj}j∈J such that limxjℓ=x0, hence
(4.7)yjl=Tt(xjl)⟶y=Tt(x0).
We conclude by injectivity that any ultraweak limit point of the net {xj}j∈J is equal to x0, hence this net is convergent to x0, and Tt(x0)=y, establishing surjectivity. We have thus proved that Tt↾𝔅 is a *-automorphism.
To finish the proof, we have to choose the collection{𝔐i:i∈I} in (4.5) to consist of all von Neumann subalgebras on which {Tt}t≥0 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}t≥0 on the von Neumann algebra 𝔐 displays decoherence. If Tt↾𝔐1=id𝔐1, then there exists a normal conditional expectation E from 𝔐 onto 𝔐1.Proof.
Letx∈𝔐 and write x=x1+x2 with xi∈𝔐i, i=1,2, and define E:𝔐→𝔐1 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 x∈𝔐+, consider the decomposition x=x1+x2, xi∈𝔐i, i=1,2, and suppose that x1∈𝔐sa is not positive. Then there exists φ∈𝔐*+ such that φ(x1)<0. This implies
(4.8)0≤φ(Tt(x))=φ(Tt(x1))+φ(Tt(x2))=φ(x1)+φ(Tt(x2)),
and letting t→∞ yields φ(Tt(x2))→0, a contradiction. Thus E(𝔐+)⊆𝔐+, and since E(1)=1 it follows that ∥E∥≤1. 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={x∈𝔐:E(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 Definition4.1 have been formulated in [31]. In fact, we have the following theorem.Theorem 4.4.
Let{Tt}t≥0 be a weak* continuous one-parameter semigroup with a faithful normal state ω. Assume that the following conditions are satisfied.  (1)
EachTt, t≥0, is strongly positive and unital.(2)
Let{σtω}t∈ℝ denote the modular group (see, e.g., [10]) corresponding to the state ω. Assume that [Tt,σsω]=0 for all s∈ℝ and t≥0.Then {Tt}t≥0 displays decoherence and there exists a normal conditional expectation E from 𝔐 onto 𝔐1 such that [Tt,E]=0 for all t≥0 and ω∘E=ω.The splitting𝔐=𝔐1⊕𝔐2 of {Tt}t≥0 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}t≥0 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 Corollary3.2 can now be applied to establish decoherence.Corollary 4.5.
Let{Tt}t≥0 be a weak* continuous one-parameter semigroup of strongly positive unital operators with a faithful normal state ω. Assume that 𝔐r=𝔐1. Then {Tt}t≥0 displays decoherence.Proof.
According to Corollary3.2, 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 expectationE from a von Neumann algebra 𝔐 onto a von Neumann subalgebra 𝔐1 satisfies ω∘E=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=(1-E)𝔐, and in the Jacobs-de Leeuw-Glicksberg splitting as given by Corollary 3.2 we have 𝔐r=Q𝔐 and 𝔐s=(1-Q)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 Definition4.1. 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.Let𝔐 be a von Neumann algebra acting on a separable Hilbert space ℋ, and let {Tt}t≥0 be a quantum dynamical semigroup such that t↦Tt is continuous in the uniform topology. Then by [32] the generator Z of {Tt}t≥0, which is a bounded operator on 𝔐, is given by(4.9)Zx=G*x+xG+Φ(x),
where G∈L(ℋ) and Φ:𝔐→L(ℋ) is a normal completely positive map. Since we have Tt(1)=1 for all t≥0, it follows that Z1=0 which forces G*=-G-Φ(1). Upon introducing the operator H=iG+(1/2)iΦ(1), it is seen that H is a bounded selfadjoint operator on ℋ and Z may be written as(4.10)Zx=i[H,x]-12{Φ(1),x}+Φ(x),
where {·,·} denotes the anticommutator. Let us suppose now that Φ has a Kraus decomposition(4.11)Φ(x)=∑n=1∞An*xAn,
where {An}n∈ℕ is a sequence of bounded linear operators on ℋ 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 ℋ, that 𝔐 is hyperfinite (this includes the case 𝔐=L(ℋ)). The preadjoint operator of Z on 𝔐* then has the familiar Lindblad form [34](4.12)Z*ρ=-i[H,ρ]-12∑n=1∞(ρAn*An+An*Anρ)+∑n=1∞AnρAn*,
where ρ∈𝔐*. We are now able to prove the following theorem.Theorem 4.7.
Let{Tt}t≥0 be as above. Assume that there is a faithful normal state ω such that ω∘Tt=ω for all t≥0 and that H in (4.10) has pure point spectrum. Then {Tt}t≥0 displays decoherence, and for the effective subalgebra 𝔐1 we have
(4.13)M1⊆{An,An*:n∈N}′∩M,
where the prime denotes the commutant. Moreover, there exists a normal conditional expectation Q from 𝔐 onto 𝔐1 such that ω∘Q=ω. If the derivation x↦i[H,x] leaves the subalgebra {An,An*:n∈ℕ}′∩𝔐 invariant, equality holds in (4.13).Proof.
First note that the assumptions of Proposition3.3 are satisfied, that is, 𝔐r is a von Neumann subalgebra. Consider the subalgebra 𝔐1 defined in Lemma 3.4, then {Tt}t≥0 restricted to 𝔐1 extends to a group of automorphisms. We start by proving (4.13). By a simple calculation as in [20], one obtains
(4.14)Z(x*x)-Z(x*)x-x*Z(x)=x*Φ(1)x+Φ(x*x)-Φ(x*)x-x*Φ(x)=∑n=1∞[An,x]*[An,x].
The generator Z, when restricted to 𝔐1, is a *-derivation; thus if x∈𝔐1, then
(4.15)0=Z(x*x)-x*Z(x)-Z(x*)x=∑n=1∞[An,x]*[An,x],
that is, [An,x]=0 for all n∈ℕ, and, moreover, [An*,x]=0 for all n∈ℕ since 𝔐1 is a *-subalgebra. This proves that 𝔐1⊆{An,An*:n∈ℕ}′∩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)=eitHxe-itH on {An,An*:n∈ℕ}′∩𝔐, which implies equality in (4.13).
Now letx∈𝔐s∩𝔐1, x≠0. Then there exist eigenvectors ξ,η∈ℋ of H with corresponding eigenvalues Eξ and Eη such that 〈ξ,xη〉≠0, thus
(4.16)〈ξ,Tt(x)η〉=〈e-itHξ,xe-itHη〉=eit(Eξ-Eη)〈ξ,xη〉
is bounded away from 0, so 0 is not a weak* limit point of {Tt(x):t≥0}. Sincex∈𝔐s, this is a contradiction in view of (2.5), hence 𝔐s∩𝔐1={0}. Now let x∈𝔐1 and write x=xs+xr∈𝔐s⊕𝔐r. Since 𝔐r⊆𝔐1, we have xr∈𝔐1, and xs=x-xr∈𝔐s∩𝔐1={0}, thus x∈𝔐r. This proves 𝔐1=𝔐r and it follows by Proposition 3.6 that 𝔐s has the property (3.3). So we conclude that {Tt}t≥0 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}t≥0 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,An*:n∈ℕ}′=ℂ1, then
(4.17)lim⁡t→∞Tt(x)=ω(x)1intheweak*topology
for any x∈𝔐.Thus if𝔐1 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}t≥0 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 Theorems4.7 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=Mℂ(d) is the d×d-matrix algebra and consider a quantum dynamical semigroup {Tt}t≥0 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}t≥0. Such generators arise, for example, in the singular coupling limit of N-level systems, see [8].



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