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Advances in Mathematical Physics
Volume 2011 (2011), Article ID 625978, 16 pages
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.
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.
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 , drawing on earlier work in . 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 of normal completely positive and unital linear maps on the von Neumann algebra [7, 8]. In the Markovian approximation, the family 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  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 is a Banach space and assume that it has a predual space denoted by , that is, . If and , we will denote the evaluation of at by and consider this as a dual pairing between and . The set of all bounded linear operators from to , endowed with the operator norm, will be denoted by , and its unit ball by . Operators from are called contractive. We consider the algebraic tensor product and endow it with the projective cross norm ; the completion of with respect to is a Banach space which will be denoted by . Then its dual space is isometrically isomorphic in a canonical way with : If , we define by for all and . It can now be shown that extends to an isometric isomorphism, and we can thus write .
We now introduce the pointwise weak* topology on . Let , , and define the seminorm . The pointwise weak* topology is the locally convex topology on induced by the family of seminorms. If , we see that ; thus the pointwise weak* topology coalesces with the topology on , that is, the pointwise weak* topology is a weak* topology as well. Thus we can conclude from Alaoglu's theorem that is compact in the pointwise weak* topology.
A linear operator will be called normal provided it is a continuous map from to when is endowed with the weak* topology. We denote the set of all normal operators by . We can consider the set of all normal contractive operators as a semigroup under multiplication of operators, that is, if , then is normal and contractive; moreover, the multiplication is associative. The semigroup is semitopological when endowed with the pointwise weak* topology, that is, the multiplication is separately continuous. This means that the maps and are both continuous with respect to the pointwise weak* topology. Finally we remark that it is important to note that is not closed in with respect to the pointwise weak* topology. Moreover, recall that an operator is normal if and only if there exists a (unique) predual operator from into , defined by , , .
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 of linear and contractive operators on , such that for all and . The semigroup is called weak* continuous provided each is a normal operator and is weak* continuous for any . For a weak* continuous semigroup there exists the following concept of a weak* generator : The predual semigroup of a weak* continuous semigroup is strongly continuous, and the adjoint of its generator is equal to the weak* generator .
Suppose now that is a weak* continuous contractive semigroup on , and write . In the following, we assume that the closure of S0 in with respect to the pointwise weak* topology consists of normal operators, that is, we assume that , where the bar denotes closure in the pointwise weak* topology. Then is a compact commutative semitopological subsemigroup of . We now use the fact that every compact commutative semitopological semigroup has a unique minimal ideal , the so-called Sushkevich kernel , which is given by and will denote the unit of . We then have . By compactness of , it follows that is, in fact, a commutative topological group. In the following, we will simplify our notation by writing instead of .
We are now able to prove a weak* version of the Jacobs-de Leeuw-Glicksberg splitting theorem, originally going back to Jacobs  and de Leeuw and Glicksberg [12, 13], see also . The present proof mimics the one given in  for weakly almost periodic one-parameter semigroups.
Theorem 2.1. Let be a weak* continuous contractive one-parameter semigroup with generator . Assume that consists of normal operators. Then there exist weak* closed subspaces of invariant under all operators , , such that , and
Proof. Since , the unit is a normal projection such that for all . The theorem will be established once we prove that and .
Let . Since , there is a net such that relative to the pointwise weak* topology; hence , so . Conversely, assume for some . Then there is a net such that relative to the weak* topology. By compactness of , there is a subnet with relative to the pointwise weak* topology for some , and it follows that . Hence for all . Choosing to be the inverse of in , we get , hence . We have thus proved that .
Let be the character group of . For each define the operator where is the normalized Haar measure of . The integral is to be understood as a weak* integral, thus is a well-defined bounded operator in with . Then for all we get in particular ; therefore, for all . Since is continuous and satisfies the functional equation for all , we have for some . Thus , hence and for all . We next define the subspace We prove that . Let . Then for all , , that is, for all , . Since the character group is total in by the Stone-Weierstraß theorem and since is continuous it follows that for all and . Take , then we obtain and thus . By the bipolar theorem we obtain , since is a weak* closed subspace. Conversely, let with for some . It follows that for all and consequently for . Thus there exists such that . Consequently, we must have which implies , hence , and the proof is finished.
Corollary 2.2. Under the hypothesis of Theorem 2.1, there exists a weak* continuous one-parameter group of isometries on such that for .
Proof. Let , then , and let be the inverse of in , that is, . Then for all , we have . Now write for and let be the inverse of in . The foregoing calculation shows that is a one-parameter group on . Moreover, it is clear that it is weak* continuous and contractive. Now assume that there is and such that . Then it follows that , contradiction; thus is isometric.
The subspace is called the reversible subspace and 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 , namely, we are interested in a stronger stability property of the elements in . 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 based on the boundary spectrum of the generator .
Proposition 2.3. Assume that the hypothesis of Theorem 2.1 is satisfied and additionally that is at most countable. Then the stable subspace (2.5) is given by Moreover, the convergence in (2.12) is uniform for in .
Proof. Consider the predual semigroup with generator ; as already remarked, is . The predual of is a projection and induces a splitting by way of and . Let be the restriction of to . Since is closed; the generator of is given by the restriction , . A similar construction applies to the reversible subspace . We check that . Let , that is, the map is bijective. Then clearly the map is injective. It is also surjective: let , then there is such that . Now
so and by injectivity. Thus is bijective and . In particular, using we find that
We now see that
for if , , then the corresponding eigenvector satisfying must lie in by (2.6), hence , contradiction.
From (2.14) and (2.15), it follows by the Arendt-Batty-Lyubich-Vũ theorem [15, 16], see also , that the semigroup is strongly stable, that is, for all we have . Thus if and it follows that as uniformly for , where denotes the projection onto .
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 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 , and we will always assume that . The ultraweak topology on is defined by the seminorms , where runs through the trace class operators on , it agrees with the weak operator topology on bounded portions of . The set of all ultraweakly continuous linear functionals on forms a Banach space, and this Banach space is the unique (up to isomorphism) predual space of , for this reason we denote it by . The ultraweak topology on can be shown to be equivalent to the (i.e., weak*) topology. Hence the setup of the previous section applies to this case. The set of all positive operators of will be denoted by . A functional in which is positive (i.e., provided ) and normalized (i.e., , equivalently ) will be called a normal state. A state is called faithful if and implies . For proofs of these results, we refer to [10, 17].
Let . Then is called positive if . A positive operator is normal (i.e., weak* continuous) if and only if for every uniformly bounded increasing net we have . Furthermore, is called strongly positive whenever it satisfies Kadison's inequality, that is, for any . Clearly strong positivity implies positivity. An even stronger notion of positivity is complete positivity: is called completely positive whenever for all and all and from . The map is called unital if ; a positive unital map is automatically contractive, that is, for all .
The following result has been established in .
Proposition 3.1. Suppose that is a subset of normal contractive linear operators. Then the following assertions are equivalent. (1)The set is relatively weakly compact for every . (2)The set is equicontinuous when is endowed with the Mackey topology (i.e., the topology). (3)The pointwise weak* closure of consists of normal operators: . Moreover, these conditions are satisfied whenever there is a faithful normal state on such that In particular, if each element in is strongly positive we conclude that (3.1) can be rewritten as for all , , or briefly , for all .
If is a subset of normal contractive linear operators and a normal state, we call an invariant state under provided for all and . 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 is a weak* continuous contractive strongly positive one-parameter semigroup on a von Neumann algebra with ultraweak generator , and suppose that there exists a faithful normal invariant state . Then there exist weak* closed and -invariant subspaces and of , given by (2.5) and (2.6), such that .
It is worth pointing out that a similar result was recently established in  for general semigroups acting on a W*-algebra and possessing a faithful family of subinvariant states.
We now prove that is actually a von Neumann subalgebra. Recall that a conditional expectation from a C*-algebra onto a C*-subalgebra is a completely positive contraction with for and for , .
Proposition 3.3. Let be a weak* continuous semigroup of strongly positive unital operators and suppose there exists a faithful normal invariant state . Then is a von Neumann subalgebra of and there exists a group of *-automorphisms on such that for all . Moreover, there exists a normal conditional expectation from onto such that . Finally, is *-invariant.
Proof. Since each is a contraction; Corollary 3.2 applies. Let , that is, we have . As in  we define the sesquilinear map by for some fixed . By Kadison's inequality, the sesquilinear form is positive-definite for any , so by the Cauchy-Schwarz inequality, if and only if for all . Now let , then . Thus , and by faithfulness , hence for all . So for all , that is, , and we conclude that whenever . It follows that is a *-subalgebra of (containing ) and consequently is a von Neumann subalgebra, and for all . By Corollary 2.2, the restriction of to extends to a one-parameter group of isometries and the above argument shows that must be a *-homomorphism. Let be the Sushkevich kernel of the semigroup . Since is a projection and it follows from Tomiyama’s theorem  that is a conditional expectation; since ; it is also clear that . The last assertion is clear as well.
In the following, we will be interested in the stronger characterization of by a stability property as in (2.12). We start by quoting the following result.
Lemma 3.4. Suppose that is a weak* continuous one-parameter semigroup of strongly positive unital operators on the von Neumann algebra with a faithful normal invariant state . Introduce the subsets Then is a -invariant von Neumann subalgebra of , and there exists a group of *-automorphisms on such that for . Moreover, 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 is given by a group of *-automorphisms.
A proof can be found in  (see the proof of Proposition 2). It is easy to see that we always have .
Lemma 3.5. Under the assumptions of Lemma 3.4, for every the weak* limit points of the net lie in .
A proof of this statement is contained in the proof of Theorem 3.1 of .
We can now establish the following result.
Proposition 3.6. Let be a weak* continuous semigroup of strongly positive unital operators on the von Neumann algebra with a faithful normal invariant state . If , it follows that
Proof. Let and assume without loss of generality that . By Alaoglu's theorem the net contained in the unit ball of has a limit point for . Then using Lemma 3.5, we find that . But since , it follows that also , that is, ; hence . This proves that any limit point of the net is equal to 0; therefore we conclude that in the weak* topology for all .
Moreover, let us remark the following: suppose that is a weak* continuous semigroup of strongly positive unital operators with generator having a faithful normal invariant state , and assume that the peripheral spectrum 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 , and its reversible time evolution is given by a weak* continuous group of *-automorphisms on . A subsystem can be described by a von Neumann subalgebra containing the observables belonging to the subsystem. We will assume that there exists a normal conditional expectation from onto . In this situation, we can define the reduced dynamics as follows: This is the Heisenberg picture time evolution an observer whose experimental capabilities are limited to the system described by 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) is a family of completely positive and normal linear operators on .(2) for all , in particular; each is contractive. (3) is weak* continuous for any .
In general the reduced dynamics is not Markovian, that is, memory-free, and hence the operators 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 . Such a semigroup is called a quantum dynamical semigroup. We remark that in many physically relevant models we have the following structure: , acting on a tensor product of two Hilbert spaces, where describes the environment of the system (e.g., a heat bath). The time evolution of the system and environment is Hamiltonian, that is, with , where and are the Hamiltonians belonging to the system and its environment, and is an interaction term. Moreover, the conditional expectation is given with respect to a reference state of the environment, that is, for all and . In this situation, the predual time evolution is given by the familiar formula where is a normal state on and 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 ; the relation of this form and that given in  is discussed in .
Definition 4.1. We say that the reduced dynamics displays decoherence if the following assertions are satisfied: there exists a -invariant von Neumann subalgebra of and an weak* continuous group of *-automorphisms on such that for , and a -invariant and *-invariant weak* closed subspace of such that Moreover, we require that is a maximal von Neumann subalgebra of (in the sense of not being properly contained in any larger von Neumann subalgebra) on which extends to a group of *-automorphisms. We call 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 has a decomposition , where , such that as for any normal state on . Hence after a sufficiently long time the system behaves effectively like a closed system described by and . By analyzing the structure of the algebra of effective observables and the reversible dynamics 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  for a thorough discussion of this point. Particularly interesting is the case when 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 is a group of automorphisms, decoherence takes place and the splitting (4.3) is trivial with . 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 . This can only happen if is irreversible.
We remark that the algebra has been studied in  and explicit representations of are obtained for quantum dynamical semigroups with unbounded generators. Moreover, in  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 . In  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 on which extends to a group of automorphisms, is this subalgebra necessarily unique? The following theorem answers this question.
Theorem 4.2. Let be a quantum dynamical semigroup and suppose it has a faithful normal invariant state . Then there exists a unique maximal von Neumann subalgebra on which extends to a group of automorphisms.
Proof. Let , where is some index set, be a collection of von Neumann subalgebras on each of which extends to a group of automorphisms. Then we put
where the closure is taken in the ultraweak topology. We proceed as in the proof of Proposition 3.3 and introduce for a fixed and . Then if and only if for all . Now if , we get , thus for , where , that is, . Proceeding inductively we have
for an arbitrary monomial , where , . Therefore, if , then is a *-homomorphism of the *-subalgebra , and is a *-homomorphism as well.
Next we note that is injective. Namely, if , we get , thus by faithfulness of and so .
We now establish that is surjective. Since is an injective *-homomorphism on the C*-algebra , it follows that for all . Moreover, notice that is invertible since each restriction is invertible. This implies for any . Now for the proof of surjectivity choose . By the Kaplansky density theorem, there exists a net such that for all , and such that . Put . Then , and by Alaoglu's theorem the net has an ultraweak limit point . Thus there is a subnet such that , hence We conclude by injectivity that any ultraweak limit point of the net is equal to , hence this net is convergent to , and , establishing surjectivity. We have thus proved that is a *-automorphism.
To finish the proof, we have to choose the collection in (4.5) to consist of all von Neumann subalgebras on which 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 on the von Neumann algebra displays decoherence. If , then there exists a normal conditional expectation from onto .
Proof. Let and write with , , and define by . Then and . Since is *-invariant, , hence . Now let , consider the decomposition , , , and suppose that is not positive. Then there exists such that . This implies and letting yields , a contradiction. Thus , and since it follows that . From , we get , so is a projection of norm 1 and hence by Tomiyama's theorem a conditional expectation. Since is ultraweakly closed, we obtain by a theorem of Tomiyama  that 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 . In fact, we have the following theorem.
Theorem 4.4. Let be a weak* continuous one-parameter semigroup with a faithful normal state . Assume that the following conditions are satisfied. (1)Each , , is strongly positive and unital. (2)Let denote the modular group (see, e.g., ) corresponding to the state . Assume that for all and . Then displays decoherence and there exists a normal conditional expectation from onto such that for all and .
The splitting of provided by this theorem is called the isometric-sweeping splitting. In  the theorem was proved under the more general hypothesis that is only a faithful semifinite normal weight. Then some additional technical assumptions about are necessary. A simpler proof for the theorem as stated above has been given in .
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 be a weak* continuous one-parameter semigroup of strongly positive unital operators with a faithful normal state . Assume that . Then displays decoherence.
Remark 4.6. Whenever a conditional expectation from a von Neumann algebra onto a von Neumann subalgebra satisfies for a faithful normal state , it is uniquely determined by these conditions [17, Corollary ]. Since in case of the isometric-sweeping splitting as given by Theorem 4.4 we have and , and in the Jacobs-de Leeuw-Glicksberg splitting as given by Corollary 3.2 we have and , it follows that the isometric-sweeping and Jacobs-de Leeuw-Glicksberg splittings agree whenever .
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 be a von Neumann algebra acting on a separable Hilbert space , and let be a quantum dynamical semigroup such that is continuous in the uniform topology. Then by  the generator of , which is a bounded operator on , is given by where and is a normal completely positive map. Since we have for all , it follows that which forces . Upon introducing the operator , it is seen that is a bounded selfadjoint operator on and may be written as where denotes the anticommutator. Let us suppose now that has a Kraus decomposition where is a sequence of bounded linear operators on and the series converges in the weak* topology. This is always the case if is injective or equivalently, by the Connes theorem  and separability of , that is hyperfinite (this includes the case ). The preadjoint operator of on then has the familiar Lindblad form  where . We are now able to prove the following theorem.
Theorem 4.7. Let be as above. Assume that there is a faithful normal state such that for all and that in (4.10) has pure point spectrum. Then displays decoherence, and for the effective subalgebra we have where the prime denotes the commutant. Moreover, there exists a normal conditional expectation from onto such that . If the derivation leaves the subalgebra invariant, equality holds in (4.13).
Proof. First note that the assumptions of Proposition 3.3 are satisfied, that is, is a von Neumann subalgebra. Consider the subalgebra defined in Lemma 3.4, then restricted to extends to a group of automorphisms. We start by proving (4.13). By a simple calculation as in , one obtains
The generator , when restricted to , is a *-derivation; thus if , then
that is, for all , and, moreover, for all since is a *-subalgebra. This proves that . Conversely, under the assumption that leaves the right-hand side of (4.13) invariant, we have or on , which implies equality in (4.13).
Now let , . Then there exist eigenvectors of with corresponding eigenvalues and such that , thus is bounded away from 0, so 0 is not a weak* limit point of . Since, this is a contradiction in view of (2.5), hence . Now let and write . Since , we have , and , thus . This proves and it follows by Proposition 3.6 that has the property (3.3). So we conclude that 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  and is also contained in .
Corollary 4.8. Let be a uniformly continuous semigroup on consisting of normal completely positive and contractive operators, and suppose it has a faithful normal invariant state . If , then for any .
Thus if is trivial the semigroup describes the approach to equilibrium.
We remark that the last theorem can be generalized to certain cases when the semigroup 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 is the -matrix algebra and consider a quantum dynamical semigroup on , then its generator is given by (4.10) and its preadjoint by (4.12), thus if the are normal, is a faithful normal invariant state for . Such generators arise, for example, in the singular coupling limit of -level systems, see .
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.
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