ON THE RECURRENCE OF QUANTUM DYNAMICAL SEMIGROUPS ...

ON THE RECURRENCE OF QUANTUM DYNAMICAL SEMIGROUPS ROLANDO REBOLLEDO

Abstract. The mathematical description of the evolution of Quantum Open Systems seems to reach a suitable formalism within the Theory of Quantum Dynamical Semigroups extensively developed during the last two decades. Moreover, from a probabilistic point of view, this theory provides a natural non commutative extension of Markov Processes. Following that line, we discuss the notion of recurrence and summarize a number of results on the large time behavior of Quantum Dynamical Semigroups.

1. Ergodic theorems on von Neumann algebras During the seventies several authors proved ergodic theorems for semigroups acting on a von Neumann algebra. Namely, Frigerio[16] started in 1977 a systematic study of the large time asymptotic behavior of semigroups of completely positive linear maps of a von Neumann algebra. He pursued this investigation in [17] while Watanabe[40], working independently, published in 1979 various ergodic theorems for semigroups of linear positive maps of a von Neumann algebra into itself. In all the references before, a crucial hypothesis was the assumption on the existence of a faithful normal invariant state. Later, in 1982, Frigerio and Verri[18], succeeded in obtaining an ergodic-type theorem with no assumption on the existence of a faithful normal invariant state. Indeed, their result establishes the convergence of the Cesàro mean for a suitable reduced semigroup of completely positive maps. On the other hand, as it has been noticed by several authors (see e.g. [27],[25]), completely positive maps are a natural extension of transition probabilities in the theory of Markov semigroups. So that, when dealing with ergodic theorems we follow the ideas of Frigerio and introduce the following definition. Definition 1.1. A Quantum Dynamical Semigroup (QDS) of a von Neumann algebra M is a weakly*–continuous one–parameter semigroup (Tt )t≥0 of completely positive linear normal maps of M into itself which preserve the identity. In addition, it is assumed that T0 coincides with the identity map I. 1

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ROLANDO REBOLLEDO

Later on we will specialize the von Neumann algebra M to associate a Quantum Flow to our QDS. Now we provide an alternative proof of the ergodic behavior for general QDS. We denote F(T ) the set of fixed points of T in M. If the existence of a faithful, normal, stationary state ω is assumed, then F(T ) becomes a von Neumann subalgebra of M. Moreover, this subalgebra is globally invariant under the modular automorphism σtω introduced in the theory of Tomita and Takesaki (see [3], [19]), since σtω and Tt commute. Therefore, there exists a faithful normal conditional expectation E F (T ) which satisfies CE1: E F (T ) : M → F(T ) is linear, w∗ –continuous, completely positive, CE2: E F (T ) (I) = I, where I is the unit of M, CE3: ω ◦ E F (T ) = ω, and E F (T ) (aE F (T ) (b)) = E F (T ) (a)E F (T ) (b), for all a, b ∈ M. The above characterization contains the Ergodic Theorem for QDS. Indeed E F (T ) is unique since, given any other map E which satisfies CE1, CE2, CE3, it follows that E F (T ) = E ◦ E F (T ) = E F (T ) ◦ E = E. More precisely, Theorem 1.1. If ω is a faithful, normal state which is invariant under T , then there exists a unique normal conditional expectation E F (T ) onto F(T ). In addition, E F (T ) ◦ Tt = E F (T ) for all t ≥ 0; for any element a ∈ M, E F (T ) (a) belongs to the w∗ –closure co(T (a)) of the convex hull of the orbit T (a) = (Tt (a))t≥0 . Moreover, invariant states under the action of the predual semigroup (T∗t )t≥0 , are elements of the form ϕ ◦ E F (T ) of the predual M∗ , where ϕ run over all states defined on the algebra F(T ). Proof. To prove the equality E F (T ) ◦ Tt = E F (T ) , it suffices to notice that E F (T ) ◦ Tt satisfies CE1, CE2 and CE3 as well. This means that both , E F (T ) ◦ Tt and E F (T ) , are conditional expectations onto F(T ), so that they coincide. The following easy lemma (whose proof is omitted) allows to better handle the second part of the theorem. Lemma 1.1. Given a bounded continuous function f defined on [0, ∞[, for any element α of the closure of the convex hull of the range of f , there exists a sequence (µn )n of borelian probability measures on [0, ∞[ R such that limn f dµn exists and Z ∞ α = lim f dµn . n

0

ON THE RECURRENCE OF QUANTUM DYNAMICAL SEMIGROUPS

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Given any probability measure µ on the positive real line (endowed with the Borel σ–field) one obtains (from CE3 and the first part of the theorem): (1) Z Z ∞

∞

Tt (a)dµ(t)

ω

= ω ◦ E F (T )

Tt (a)dµ(t)

0

= ω(E F (T ) (a)),

0

where the integrals of operators are understood in the weak* sense. Due to the faithfulness of ω and the lemma before, the above equation shows that E F (T ) (a) cannot be on the complement of co(T (a)). So that E F (T ) (a) ∈ co(T (a)) and it is the unique element which belongs to F(T ). Finally, if ϕ is an invariant state defined on F(T ), it follows straightforward that ϕ ◦ E F (T ) is an invariant state on M. Let us prove the reciprocal property. Indeed, given any invariant state ψ on M and an element a ∈ M, write E F (T ) (a) like in the lemma 1.1: Z F (T ) ∗ E (a) = w − lim Tt (a)dµn (t). n

Therefore, ψ(E

F (T )

Z

∗

(a)) = w − lim n

ψ(Tt (a))dµn (t) = ψ(a).

So that it suffices to take ϕ as the state–restriction of ψ to F(T ) to conclude the proof. It is worth noticing that co(T (a)) ∩ F(T ) = {E F (T ) (a)} and E F (T ) (a) can be computed, for instance through the equivalent formulae: Z 1 T F (T ) ∗ (2) Tt (a)dt E (a) = w − lim T →∞ T 0 Z ∞ (3) = w∗ − lim λ e−λt Tt (a)dt. λ→0

0

Definition 1.2. According to the classification of non commutative dynamical systems developed by Dang Ngoc [6], [19], the QDS T is finite if there is enough normal invariant states, that is, if for all nonzero positive element X of M, there exists a normal state ω in the space of fixed points F(T∗ ) of the predual semigroup, such that ω(X) > 0. The QDS is infinite if it is not finite. Moreover, a given a projection π ∈ F(T ) is said to be finite if the induced semigroup Ttπ (X) = πTt (X)π, for all X ∈ πMπ is finite, where πMπ is the algebra of elements of the form X = πX0 π, X0 ∈ M.

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ROLANDO REBOLLEDO

We recall here for easy reference the following result of Dang Ngoc [6]. For a given state ϕ, we denote S(ϕ) its support projection. Theorem 1.2. Let denote e the least upper bound of S(ω), when ω run over the set of all normal invariant states. Then e belongs to the center of F(T ) and coincides with the least upper bound of finite projections in F(T ). The semigroup has no invariant state if e = 0. On the other hand, the following theorem summarizes the characterization of finite QDS. Theorem 1.3. The following conditions are equivalent: (1) The Quantum Dynamical Semigroup T is finite; (2) e = I; (3) E F (T ) (M) = F(T ); (4) E F (T ) is faithful; (5) There exists an invariant conditional expectation from M to F(T ). Moreover there exists at most one conditional expectation from M to F(T ) and if it exists, it coincides with E F (T ) . (6) The orbits of T∗ in M∗ are relatively weakly compact. The equivalence of condition 1 with 2, 3 and 4 is a straightforward consequence of Dang Ngoc’s theorem 1.2. Kovács and Sz¨ ucs [23] proved that condition 5 is necessary and sufficient for a semigroup to be finite. Finally, the characterization 6 is due to Störmer [36]. Since the paper of S.Ch.Moy [26], several authors have studied the construction of conditional expectations in a non-commutative framework, in particular Umegaki [38], Takesaki [37], Accardi and Cechini [1] (see eg. the survey included in the work of Petz [29]). As we have pointed out along this section, the concept of a conditional expectation is crucially related to the existence of invariant states and Cesàro (or Abel) limits. 2. Markovian flow Now we will focus on a QDS defined over the von Neumann algebra M = B(h) of all bounded linear operators on a complex separable Hilbert space h. We denote L the infinitesimal generator of the semigroup T , whose domain is given by the set D(L) of all X ∈ B(h) for which the w∗ –limit of t−1 (Tt (X) − X) exists when t → 0, and we define L(X) such a limit. The space h contains the “initial data” of the Quantum Markov process associated to T . To define the corresponding flow, we need a bigger space which is the tensor product H = h ⊗ Γ(L2 (R+ ; K))


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where the first factor is the initial space and the second, the Fock space associated to L2 (R+ ; K), where K is a complex separable Hilbert space with an orthonormal basis denoted by (zk ; k ≥ 1). The exponential vector on Γ(L2 (R+ ; K)), associated to a function f ∈ L2 (R+ ; K), is denoted e(f ). Furthermore, we introduce a canonical projection E defined by Eu ⊗ e(f ) = u ⊗ e(0). For simplicity we write down the vector u⊗e(f ) in the form ue(f ), and we identify any operator X on h with its canonical extension X ⊗ I to the whole space H. We call B the von Neumann algebra B(H) of all bounded linear operators on H. Moreover, a family of intermediate von Neumann algebras and Hilbert spaces are introduced as follows. Call B0 = M and Bt the von Neumann algebra generated by {X ⊗ B : X ∈ M, B ∈ B(Γ(L2 ([0, t]; K))} and by Bt] = {X ⊗ I[t : X ∈ Mt , }, I[t denoting the identity operator in Γ(L2 ([t, ∞[; K)), for all t ≥ 0. Finally, we need a family of conditional expectations E t] : B → Bt] , for all t ≥ 0. One could argue again on the existence of these conditional expectations by means of Tomita-Takesaki Theory, assuming the existence of an invariant faithful normal state on B. Then, by Stinespring’s Theorem, each E t] should have a representation E t] (Z) = Et ZEt , (Z ∈ B(H)), where Et is a projection from H to Ht = h ⊗ Γ(L2 ([0, t]; K). However, the particular von Neumann algebras we fixed within this section allow to built up the projections Et explicitly like Et u ⊗ e(f ) = u ⊗ e(f I[0,t] ), without any additional hypothesis on the existence of invariant states. Notice that E0 = E, the canonical projection introduced before. Now we are in position of introducing Markovian flows. Definition 2.1. A Quantum Markovian Flow associated to the semigroup T = (Tt )t≥0 , is a family J = (jt )t≥0 such that each jt : M → Bt] is a ∗ –homomorphism, completely positive, normal, identity preserving and satisfying the Markov property: (4)

js (Tt−s (X)) = E s] (jt (X)), j0 (X) = X ⊗ I, (X ∈ M).

for all 0 ≤ s ≤ t. The flow may be represented by means of a Quantum Markovian Cocycle V = (Vt ; t ≥ 0) in B(H), such that each Vt is a unitary contraction and jt (X) = Vt∗ X ⊗ IVt , (X ∈ M, t ≥ 0). We say that Z ∈ B(H) is an invariant element of the flow, if jt (E 0] (Z)) coincides with Z on the whole space H, for all t ≥ 0. So that two invariant elements of the flow, say Z, Z 0 , coincide if E 0] (Z) = E 0] (Z 0 ). Call I the space of invariant elements of the flow.

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Proposition 2.1. The map Z 7→ E 0] (Z) sets up a one-to-one and onto correspondence between I and F(T ). Moreover, given X ∈ F(T ), Z ∈ I may be chosen as Z = w∗ − lim jt (X). t

Proof. Indeed, E 0] is a normal, faithful, completely positive conditional expectation and if Z is invariant, denoting X = E 0] (Z), then for all t ≥ 0, Tt (X) = E 0] (jt (X)) = E 0] (X ⊗ I) = X. Therefore, X ∈ F(T ). On the other hand, if X ∈ F(T ), the Markov property implies that js (X) = E s] (jt (X)), so that (jt (X))t is a bounded martingale and the limit w∗ − limt jt (X) exists (see for instance Exercise 28.10, p. 246 in the book of Parthasarathy [27]). Call Z this limit and notice that E s] (Z) = js (X), for all s ≥ 0. So that X = E 0] (Z). The existence of Quantum Markovian Cocycles has been analyzed by several authors. We refer to [27], [25] and [11] for a more detailed study on the connection with quantum stochastic differential equations (Evans-Hudson flows). Given Ran event or projection π of M, we denote j∞ (π) = w∗ − t limt→∞ 1t 0 js (π)ds, whenever this limit exists. It is worth noticing that the above limit exists as a bounded operator on H as soon as the conditional expectation E F (T ) (π) exists. Indeed, the family of all Cesàro mean of the flow is uniformly bounded in norm, so that it is weakly* relatively compact. Moreover, since E 0] is faithful, the set of limit points of the above Cesàro mean family is reduced to a single element and E F (T ) (π) = E 0] (j∞ (π)). As a result, the operator j∞ (π) exists at least when the QDS is finite. We denote R(π) the recurrence subspace of π defined by the elements x ∈ H for which j∞ (π)x = x, that is, R(π) = Ker(j∞ (π) − I). Intuitively, if R(π) coincides with H, then π is “visited” by the flow infinitely often. Notice that j∞ (π) is an invariant element since E 0] (j∞ (π)) = E F (T ) (π) ∈ F(T ).


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Definition 2.2. We say that the event π ∈ M is recurrent if j∞ (π) = I; it is transient if j∞ (π) = 0. In general, j∞ (π) of the above definition does not need to be a projection. However, each jt (π) does and if the w∗ limit of jt (π) exists, it coincides with j∞ (π) being a projection. After a result of Frigerio and Verri [18], the above limit exists in particular when F(T ) is trivial. Theorem 2.1. If ω is a faithful, normal state which is invariant under T , the following three statements are equivalent: (1) F(T ) = C I; (2) I = C I; (3) All event π ∈ M is either recurrent or transient. Proof. The equivalence between 1 and 2 follows straightforward from Proposition 2.1. Now, assume that proposition 1 holds. Given any event π, since E F (T ) (π) = E 0] (j∞ (π)) ∈ F(T ), it yields to E F (T ) (π) = ω(E F (T ) (π))I = ω(π) I. As a result, one has two possibilities: either ω(π) = 1 and π is recurrent; either ω(π) = 0 and the event is transient. Finally, we prove that condition 3 implies 1. For this, we recall that F(T ) is a von Neumann algebra so that, any element of it being a linear combination of selfadjoint operators, the projections in F(T ) span a norm-dense subspace of F(T ). Therefore, it suffices to prove that projections in F(T ) are trivial if the statement 3 holds. Take an arbitrary projection π ∈ F(T ). Assume first that j∞ (π) = I. Since E 0] (jt (π)) = π = E F (T ) (π), one obtains that I = E 0] (j∞ (π)) = π. On the other hand, if j∞ (π) = 0, the above argument implies π = 0. Therefore, in both cases π is trivial.

3. Recurrence and Sojourn time We keep denoting a faithful normal, stationary state by ω. Even though, its existence will not been assumed unless explicitly stated. Furthermore, we introduce the following definitions. Definition 3.1. Given a normal state ϕ ∈ M∗ and a projection π ∈ M, the Mean Sojourn Time of the QDS on π, measured on the state ϕ, is given by Z 1 τ (5) T (ϕ, π) = lim sup ϕ ◦ Tt (π)dt. τ 0 τ

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ROLANDO REBOLLEDO

We write π < I to indicate that π is a finite rank projection (notice that our algebra M is semi-finite since there exists a net of increasing finite rank projections which tends to I), and introduce the following classification of states: • The normal state ϕ is said to be scattering if T (ϕ, π) = 0 for all π < I. • The normal state ϕ is bound if the orbit T∗ (ϕ) = (T∗t (ϕ))t≥0 is relatively weakly compact in M∗ . The convex cone of scattering (respectively bound) states is denoted by S sc (respectively S bd . For the specific von Neumann algebras we have introduced, any state ϕ may be identified with a unique positive trace-class operator ρ of unit trace through the formula ϕ(X) = tr (ρX). In [2] a classification of states has been studied, which is expressed in terms of the functional Z 1 τ tr (ρTt (ρ))dt, Q(ρ) = lim sup τ 0 τ defined on trace–class operators. If the QDS is finite then, the limit of the Cesàro mean involved in the definition of T (ϕ, π) exists for all state ϕ and T (ϕ, π) = ϕ ◦ E F (T ) (π) = ϕ ◦ E 0] (j∞ (π)). Theorem 3.1. The cones S bd and S sc are disjoint. Proof. Given ω ∈ S bd , the conditional expectation E F (T∗ ) (ω) exists. Indeed, since the orbit T∗ (ω) is weakly compact, there is a unique element in co(T∗ (ω)) ∩ F(T∗ ), which is the conditional expectation E F (T∗ ) (ω) (rephrasing the proof of Theorem 1.3). Therefore T (ω, π) = ω ◦ E F (T ) (π), for all event π. As a result, for any π ∈ F(T ), we have T (ω, π) = ω(π) and there exists at least one π < I such that T (ω, π) > 0. Therefore, ω 6∈ S sc . It is worth noticing that for a bound state ω, its conditional expectation E F (T∗ ) (ω) provides an invariant state for the QDS. In particular, invariant states belong to S bd . Proposition 3.1. A state ϕ is scattering if and only if its support projection S(ϕ) belongs to F(T )⊥ . If the QDS has a faithful scattering state, then there is no invariant state and all finite rank events are transient. On the other hand, if the QDS is finite there is no scattering state.


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Proof. If ϕ is scattering, then for all π < I, ϕ ◦ Tt (π) = 0 for almost all t ≥ 0 and hence, for all t ≥ 0 since t 7→ ϕ ◦ Tt (π) is a continuous map. So that, given any π ∈ F(T ), ϕ(π) = 0, which implies that S(ϕ) is orthogonal to F(T ). Reciprocally, if S(ϕ) is orthogonal to F(T ), then given any finite rank projection π, S(ϕ) is orthogonal to all elements of the convex closure co(T (ϕ)). It follows that ϕ ◦ Tt (π) = 0 for almost all t and all finite rank projection π. Therefore, T (ϕ, π) = 0 for all π < I. If there is a faithful scattering state ϕ, then for all π ∈ F(T ), π < I, we obtain π = 0, since ϕ(π) = 0, (π < I). Therefore e = 0 and there is no invarant state. In addition, for all π < I, j∞ (π) = 0 and all events are transient. If the QDS is finite, then e = I and one cannot have a state with projection support orthogonal to e. So that there is no scattering state.

Proposition 3.2. If ω is a normal faithful bound state, then T (ω, π) = ω ◦ E F (T ) (π). Therefore, π < I is transient if and only T (ω, π) = 0; and π < I is recurrent if and only if T (ω, π) = 1. Proof. The equality T (ω, π) = ω ◦ E F (T ) (π) has been derived in the proof of Theorem 3.1. Furthermore, notice that j∞ (π) is 0 if and only if ω(E F (T ) (π)) = 0, which is equivalent to T (ω, π) = 0. Similarly, j∞ (π) = I if and only if ω(E F (T ) (π)) = 1 = T (ω, π). Definition 3.2. The quantum flow is Harris recurrent if there exists a normal, faithful, invariant state ω on M such that all events π for which ω(π) > 0 are recurrent. Proposition 3.3. If there exists a normal faithful invariant state and every event π > 0 is recurrent, then the flow is Harris recurrent. Proof. It suffices to remark that faithfulness implies that all π > 0 are exactly the projections for which ω(π) > 0. Theorem 3.2. If the flow is Harris recurrent, the algebra of fixed points F(T ) is trivial, the QDS is finite and there is no scattering state. Proof. It suffices to remark that under the hypothesis any event π is either recurrent (ω(π) > 0) or transient (ω(π) = 0). Theorem 2.1 yields then to the conclusion.

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4. Examples 4.1. Jaynes–Cummings model in Quantum Optics. In [12] and [13] the quantum dynamical semigroup associated to master equations in Quantum Optics (see [21]) has been obtained. The initial space is h = `2 (N) endowed with the creation (resp. annihilation) operator a† (resp. a), and the number operator denoted N . In addition, the canonical form of the Lindblad generator L of the semigroup is defined by four dissipative coefficients Lk (k = 1, . . . , 4) and a self–adjoint operator G given by the expressions √ L1 = µa, L2 = λa† , L3 = R cos(φ aa† ), √ 4 1X ∗ sin(φ aa† ) † √ L Lk , L4 = Ra , G=− 2 k=1 k aa† where the parameters φ, R ≥ 0, λ < µ, specify the physical model. Indeed, the generator of the semigroup is determined as a bilinear form (6)

ˆ hv, L(X)ui = hGv, Xui +

4 X

hLk v, XLk ui + hv, XGui,

k=1

where u, v ∈ D, the domain of the number operator. Moreover, the existence of a stationary state for T has been proved in [12]: if λ < µ, then T has a faithful stationary state given by ρ∞ =

∞ X

πn |en ihen |

n=0

where (πn )n≥0 is the sequence defined by π0 = c,

πn = c

√ n Y λ2 k + R2 sin2 (φ k) k=1

µ2 k

(n ≥ 1).

where c is a suitable normalization constant. In [14], [15] the convergence to the equilibrium has been proved for this model as a corollary of the main result appearing in those papers. Indeed, for this model one has F(T ) = C I. Therefore, Theorem 2.1 implies that any event is either recurrent, either transient. Moreover, for the invariant state ω(·) = tr (ρ∞ ·), we have T (ω, π) = ω(π) from which we obtain that all π > 0 are recurrent. As a result, the quantum flow is Harris recurrent.


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4.2. The closed quantum system. If the semigroup is given by (7)

Tt (X) = eitH Xe−itH ,

where H is a self-adjoint operator of B(h) and X ∈ B(h), then our classification of states is an extension of the one used in Scattering Theory (see eg. [31] p.15 and following). It turns out that pure bound states ψ in the sense of Perry leads to pure bound states |ψihψ| in our (weak) sense. Moreover, pure states that leave any compact subset in the time mean, in Perry’s sense, correspond to our pure scattering states. Some other authors use the wording outgoing state for the same concept. Within this framework it is well-known the crucial role played by the Wiener Tauberian Theorem (see [32] Theorem XI.114). Indeed, to simplify notations, given a unit vector ψ in h, we write |ψihψ| the projection in the direction of ψ and the (tracial) pure state ωψ (·) = tr (|ψihψ| · ) induced by ψ ∈ h. Denote ξ(dx) the spectral measure of H and µψ (dx) = tr (|ψihψ|ξ(dx) = hψ, ξ(dx)ψi. Like in Scattering Theory, we define hpp (H) (respectively, hcont (H), hac (H), hsc (H)) as the space of ψ ∈ h for which µψ is a pure point (respectively, continuous, absolutely continuous, singular continuous) measure and refer to as the pure point (respectively continuous, absolutely continuous, singular continuous) spectral subspaces. Therefore, the following characterization holds (see [31], Theorem 1.2): Theorem 4.1. Under the above hypothesis on ψ and H, it holds (1) ψ ∈ hpp (H) if and only if for every → 0 there is a finite rank projection π such that (8)

sup k(I − π )e−itH ψk < . t∈R

(2) ψ ∈ hcont (H) if and only if for any finite rank projection π, Z 1 τ (9) lim kπe−itH ψk2 dt = 0. τ →∞ τ 0 Equation (8) in the first part of the above theorem yields to tr (T∗t (|ψihψ|)π ) > 1 − , for all t ≥ 0, > 0. This implies that the orbit (T∗t (|ψihψ|))t≥0 is weakly relatively compact in M∗ . So that, if ψ ∈ hpp , then |ψihψ| is a bound state in our sense. On the other hand, equation (9), implies that T (|ψihψ|, π) = 0 for all π < I. Therefore, if ψ ∈ hcont , it follows that |ψihψ| is a scattering state in our sense.

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If hcont (H) is trivial, there exists an orthonormal basis of h consisting of eigenvectors en of H. A state is then faithful and invariant if and onlyP if it is associated to a positive trace–class P operator ρ of the form ρ = n pn |en ihen | with pn > 0 for all n and n pn = 1. In that case a simple computation shows that w∗ − lim supt eitH πe−itH is a finite rank projection if π < I, so that π cannot be recurrent unless π = I. Acknowledgments This research has been supported by the“Cátedra Presidencial en Análisis Estocástico” and FONDECYT grant 1960917. References [1] L. Accardi and C. Cecchini. Conditional expectations in von Neumann algebras and a theorem of Takesaki, J.Functional Anal., vol. 45, 245–273, (1982). [2] L. Accardi, C. Fern´ andez, H. Prado and R. Rebolledo. On Mean Quantum Sojourn Time, submitted to IDAQP, (1997). [3] O. Bratteli and D. Robinson. Operator Algebras and Quantum Statistical Mechanics, vol. 1, 2nd. edition. Springer–Verlag, Texts and Monographs in Physics, 505p., (1987). [4] Ph. Biane. Calcul Stochastique non–commutatif. In Lectures on Probability Theory, Lect.Notes in Maths. Springer, vol. 1608,1–96, (1995). [5] A. Chebotarev and F. Fagnola. Sufficient conditions for conservativity of quantum dynamical semigroups. J.Funct.Anal., vol. 118, 131–153, (1993). [6] N. Dang Ngoc. Classification des systèmes dynamiques non commutatifs. J.Funct.Anal., vol.15, 188–201, (1974). [7] E.B. Davies. Quantum dynamical semigroups and the neutron diffusion equation. Rep.Math.Phys., 11:169–188, (1977). [8] E.B. Davies. Generators of Dynamical Semigroups. J. of Funct. Anal. 34: 421– 432,(1979). [9] D.E. Evans. Irreducible quantum dynamical semigroups. Comm. Math. Phys. 55, 293–297, (1977). [10] F. Fagnola. Unitarity of solutions to quantum stochastic differential equations and conservativity of the associated quantum dynamical semigroup. Quantum Probability and Related Topics VII, 139–148,(1992). [11] F. Fagnola. Characterization of isometric and unitary weakly differentiable cocycles in fock space. Quantum Probabability and Related Topics, VIII:143– 164, (1993). [12] F. Fagnola, R. Rebolledo, and C. Saavedra. Quantum flows associated to master equations in quantum optics. J. Math.Phys., 35:1–12, (1994). [13] F. Fagnola and R. Rebolledo. An ergodic theorem in Quantum Optics. Proceedings of the Univ. of Udine Conference in honour of A. Frigerio, (1996). [14] F. Fagnola and R. Rebolledo. Sur l’approche de l’équilibre au moyen des flots quantiques , C.R.Acad.Sci.Paris, sér. I,t.321, 473–476,(1995). [15] F. Fagnola and R. Rebolledo. The approach to equilibrium of a class of Quantum Dynamical Semigroups. Submitted to the Journal on Inf.Dim. Analysis and Q. Probability.


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