Quantum Markov Semigroups and Their Stationary States Franco Fagnola ∗ Rolando Rebolledo∗

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Quantum Markov Semigroups and their stationary states Franco Fagnola ∗ Rolando Rebolledo∗

1 Introduction

The notion of a Quantum Markov Semigroup (QMS) is an extension of classical Markov semigroups motivated by the study of open quantum systems. Indeed, a number of physicists interested in open quantum systems introduced the notion of Quantum Dynamical Semigroups in the seventies. Since then the subject has been extensively discussed and improved by both physicists and mathematicians, especially by the founders of Quantum Stochastic Analysis. The current paper is a synthesis of recent results obtained by the authors on the qualitative analysis of Quantum Markov Semigroups. We focus our attention on the existence of invariant states for a given quantum Markov semigroup. Moreover, necessary and sufficient conditions for the faithfulness of the invariant state are derived. It is worth noticing several peculiarities of this mathematical study. To have a notion of semigroup broad enough to include both, quantum dynamics (“Master Equations”) as well as the classical Markov structure, one sacrifices strong topological properties. This leads to difficult problems like that of characterizing the semigroup from its generator, a problem which has been solved for an important class of Quantum Markov semigroups (see [34], [16], [17], [30]) but which is far from being closed. As we will see below, in most cases the generator of such a semigroup is given as a densely defined sesquilinear form on a Hilbert space. This forces to construct the Quantum Markov semigroup following the procedure used in Classical Probability to built up the minimal semigroup (see, for instance, [12]). In this case, the preservation of the identity, which is equivalent to the characterization of the domain of the generator, is often a non trivial problem. Tools for solving this problem have been developed by Chebotarev and Fagnola in [10] (see also [20], [11]). As a compensation of weak topological properties, a strong algebraic

∗This research has been partially supported by the “Cátedra Presidencial en Análisis Cualitativo de Sistemas Dinámicos Cuánticos”, FONDECYT grant 1990439 2 Franco Fagnola ∗ , Rolando Rebolledo∗ condition is assumed, namely the property of complete positivity. A linear map Φ defined on a ∗-algebra A is completely positive if for any finite collection ai, bi ∈ A,(i ∈ I), the element

X ∗ ∗ bi Φ(ai ai)bi, i is positive. This notion is the key feature which allowed the development of the current theory of Quantum Markov Semigroups. It is great time now to explain precisely what a Quantum Markov semigroup is. Definition 1.1. Let be given a von Neumann algebra A of operators over a complex separable Hilbert space h, endowed with a trace tr(·). A Quantum ∗ Markov Semigroup is a w -continuous semigroup T = (Tt)t≥0 of normal completely positive linear maps on A satisfying that T0(·) is the identity map of A and such that Tt(1) = 1, for all t ≥ 0, where 1 denotes the unit operator in A. In general these semigroups are termed as Quantum Dynamical Semi- groups too, however, we will reserve that name for the most general category of semigroups including both Markov and sub-Markov semigroups for which the weaker condition Tt(1) ≤ 1 holds. Definition 1.2. The infinitesimal generator of T is the operator L(·) with domain D(L) which is the vector space of all elements a ∈ A such that the ∗ −1 w –limit of t (Tt(a) − a) exists. For a ∈ D(L(·)), L(a) is defined as the limit above. Example 1. This definition provides a framework which includes quantum dynamics of closed systems defined via a group of linear unitary transformations Ut : h → h,(t ∈ R), with a generator H which is self-adjoint. Indeed, it suffices to consider the von Neumann algebra A = B(h) of all bounded linear operators acting on h, the quantum Markov semigroup being defined as ∗ itH −itH Tt(a) = Ut aUt = e ae , for all t ≥ 0. If H is bounded, the quantum Markov semigroup is uniformly strongly continuous, that is

lim sup kTt(a) − ak = 0. t→0 kak≤1 In this case the inﬁnitesimal generator is

L(a) = i[H, a], (a ∈ A). 1. Quantum Markov Semigroups and their stationary states 3

Example 2. A slight modiﬁcation of the above example consists in taking a general strongly continuous semigroup (Pt)t≥0 acting on a complex separable Hilbert space h, with generator G. Deﬁne ∗ Tt(a) = Pt aPt, for all t ≥ 0, a ∈ B(h). If G is bounded, the generator is L(a) = G∗a + aG, (a ∈ B(h)). If G is unbounded, the above expression needs to be interpreted as a sesquilinear form:

−L(a)(v, u) = hGv, aui + hv, aGui, for all u, v in the domain D(G) of G, a ∈ A.

Example 3. The connection with classical Markov semigroups works as follows. Consider a measurable space (E, E, µ), the state space, where µ is σ- ﬁnite, endowed with a semigroup of Markov transition kernel Qt : E × E → [0, 1] such that Qt(x, dy) = q(t, x, y)µ(dy). The associated semigroup is given by Z Z Ttf(x) = f(y)Qt(x, dy) = q(t, x, y)f(y)µ(dy), E E for all t ≥ 0 and any bounded measurable function f. We take h = L2 (E, E, µ) and denote by M the multiplication operator C f by a function f. Within this framework, the natural commutative von Neumann algebra to consider is A = {M : f ∈ L∞(E, E, µ)} . f C And the quantum Markov semigroup is simply

Tt(Mf ) = MTtf , for all t ≥ 0 and f ∈ A. In this case, if L denotes the generator of the semigroup (Tt)t≥0, the generator of T is L(Mf ) = MLf , (f ∈ D(L)).

Deﬁnition 1.3. A state ω on the given von Neumann algebra A is normal W W if it is σ–weakly continuous, or equivalently if ω( aα) = ω(aα) for α W α any increasing net (aα)α of positive elements in A, where α is the symbol for the least upper bound of a net. ω is faithful if ω(a) > 0 for all non-zero positive element a ∈ A. Given a quantum Markov semigroup T in A, a state ω is invariant with respect to the semigroup if ω(Tt(a)) = ω(a), for any a ∈ A, t ≥ 0. 4 Franco Fagnola ∗ , Rolando Rebolledo∗

A von Neumann algebra A is σ-ﬁnite if and only if there exists a normal faithful state on A (see [8], Proposition 2.5.6). In particular, any von Neumann algebra on a separable Hilbert space is σ-ﬁnite. Within this framework we will be placed throughout this paper.

To summarize, the guidelines of our current research are:

−1 R t 1. Under which conditions the Ces`aromean (t 0 Ts(a)ds)t≥0 do converge? 2. When does there exists an invariant state for a given QMS?

3. When does that state is faithful?

4. Is the semigroup returning to equilibrium?

The paper is organized as follows: ﬁrst we introduce some preliminary notations and concepts. We then analyze ergodic type theorems. Next, we establish a criterion on the existence of stationary states, depending on conditions on the generator of the semigroup. We continue analyzing faithfulness of stationary states and we ﬁnish by giving a result on the convergence towards the equilibrium of the QMS. Several examples inspired from open quantum systems and classical Markov processes are considered throughout the text.

1.1 Preliminaries We start by fixing notations which will be used throughout the remains of the current article. We write h a complex separable Hilbert space, endowed with a scalar product h·, ·i antilinear in the first variable, linear in the second. B(h) denotes the von Neumann algebra of all the bounded linear operators in h. The w∗ or σ–weak topology of B(h) is the weaker topology for which all maps x 7→ tr(ρx) are continuous, where ρ ∈ I1(h) and tr(·) denotes the trace. The predual algebra of a von Neumann algebra A, which is the space of all σ-weakly continuous linear functionals on A, is denoted A∗, in particular, B(h)∗ = I1(h), the algebra of trace-class operators. Any quantum Markov semigroup T on A induces a predual semigroup T∗ on A∗ defined by T∗t(ω)(a) = ω (Tt(a)) , (1.1) for all ω ∈ A∗, a ∈ A, t ≥ 0. The cone of positive elements in the algebra A is denoted A+. The space + of normal states is S = {ω ∈ A∗ : ω(1) = 1}, where 1 denotes the identity in A. 1. Quantum Markov Semigroups and their stationary states 5

A function t : D(t) × D(t) → C, where D(t) is a subspace of h, is a sesquilinear form over a Hilbert space h if t(v, u) is antilinear in v and linear in u. The set of all sesquilinear forms over h is denoted F (h). The form is said to be densely deﬁned if its domain D(t) is dense, symmetric if t(v, u) = t(u, v), for all u, v ∈ D(t), and positive if t(u, u) ≥ 0, for all u ∈ D(t). We follow Bratteli and Robinson [9] to recall some useful properties of forms. A quadratic form u 7→ t(u, u) is associated to each sesquilinear form t. This quadratic form determines t by polarization. A positive quadratic form is said to be closed whenever the conditions

1. un ∈ D(t),

2. kun − uk → 0, imply that u ∈ D(t) and t(un − u, un − u) → 0. Moreover, the quadratic form u 7→ t(u, u) is densely deﬁned, positive and closed if and only if there exists a unique positive selfadjoint operator T such that D(t) = D(T 1/2) and t(v, u) = hT 1/2v, T 1/2ui, for all u, v ∈ D(t). In that case it holds in particular that t(v, u) = hv, T ui, for any u ∈ D(T ), v ∈ D(t).

Deﬁnition 1.4. Given a quantum Markov semigroup T on a von Neumann algebra A, a dilation of the semigroup is a system

(A∞, ω, At, jt, Et)t≥0, where

•A∞ is a von Neumann algebra endowed with a normal state ω;

• (At)t≥0 is an increasing family of von Neumann sub-algebras of A∞;

∗ • Each jt : A → At is a normal -homomorphism of algebras, which preserves the identity, (t ≥ 0). (jt)t≥0 is called the quantum ﬂow of the dilation;

• For all t ≥ 0, Et is a normal conditional expectation on A∞ with range At, and Et ◦ Es = Et∧s,(s, t ≥ 0); • For all 0 ≤ s ≤ t, the Markov property is satisﬁed:

Es ◦ jt = js ◦ Tt−s. (1.2)

Thus, in particular,

Tt(a) = E0(jt (a)), (t ≥ 0, a ∈ A). (1.3) 6 Franco Fagnola ∗ , Rolando Rebolledo∗ 2 Ergodic theorems

We start analyzing the simpler of our problems which is the convergence of Ces`aromeans of a given quantum Markov semigroup. Indeed this problem is simpler due to the following well known fact.

Proposition 2.1. The unit ball of B(h) is w∗-compact.

Proof. Indeed this follows from Alaoglu-Bourbaki theorem and the fact that B(h) is the topological dual of the Banach space I1(h) endowed with the norm T 7→ tr(|T |) (see [8], Proposition 2.4.3, p.68).

∗ Thus, given a ∈ A, the ball Ba = {x ∈ B(h): kxk ≤ kak} is w -compact. Moreover, given any quantum Markov semigroup T , the orbit of a ∈ A is

T (a) = {Tt(a) ∈ A : t ≥ 0} .

We call co(T (a)) the convex hull of the orbit and we denote co(T (a)) its w∗-closure. As a result we have

Corollary 2.2. Given any quantum Markov semigroup on A, and any a ∈ A, the set co(T (a)) is compact in the w∗-topology.

Proof. Indeed, since kTt(a)k ≤ kak we have that co(T (a)) ⊆ Ba, is a closed subset of a compact set.

Here however, we are looking for more applicable results, thus, the use of sequences (or, say, criteria for sequential w∗-compactness) will be suﬃcient for our purposes.

Proposition 2.3. For any a ∈ A the w∗-limit of any sequence

1 Z tn Ts(a)ds; n ≥ 0 , tn 0

, with tn → ∞, is Tt-invariant, for all t ≥ 0.

Proof. Call b one of these limit points, i.e.

Z tn ∗ 1 b = w − lim Ts(a)ds. n tn 0 1. Quantum Markov Semigroups and their stationary states 7

Then

Z tn ∗ 1 Tr(b) = w − lim Ts+r(a)ds n tn 0 Z tn+r ∗ 1 = w − lim Ts(a)ds n tn r Z tn Z tn+r Z r ∗ 1 = w − lim Ts(a)ds + Ts(a)ds − Ts(a)ds n tn 0 tn 0 Z tn ∗ 1 = w − lim Ts(a)ds. n tn 0

Now, we would like to say a word about the convergence of states, which requires some additional notions.

Deﬁnition 2.4. A sequence of states (ωn)n is said to converge narrowly to ω ∈ S if it converges in the weak topology of the Banach space A∗ i.e.

lim ωn(x) = ω(x) n→∞ for all x ∈ A. A sequence of states (ωn)n is tight if for any > 0 there exists a ﬁnite rank projection p ∈ A and n0 ∈ N such that

ωn(p) ≥ 1 − , for all n ≥ n0. Theorem 2.5. Any tight sequence of states on B(h) admits a narrowly convergent subsequence. The reader is referred to [15] Lemma 4.3 p.291 or [40] Theorem 2 p.27 (see also [35] Appendix 1.4) for the proof. A detailed exposition of this kind of results is contained in [13]. Corollary 2.6. Suppose that A = B(h). If for each state ω the family

1 Z t T∗s(ω)ds, t > 0 t 0 is tight, then each sequential limit point of the family is invariant under T∗. The proof of this corollary is completely similar to Corollary 2.3 and it is omitted.

We denote A(T ) the set of fixed points of T in A. A straightforward generalization of Proposition 2.3, shows that any limit point of the family 8 Franco Fagnola ∗ , Rolando Rebolledo∗ of Cesàromeans of the orbit T (a) (indeed any limit point of co(T (a))) belongs to A(T ), for any a ∈ A. To prove the convergence of the whole family it suffices to prove that co(T (a))∩A(T ) is reduced to a single point. If the existence of a faithful, normal, stationary state ω is assumed, then A(T ) becomes a von Neumann subalgebra of A. Moreover, this subalgebra ω is globally invariant under the modular automorphism σt introduced in ω the theory of Tomita and Takesaki (see [8, 9], [29]), since σt (·) and Tt(·) commute. Therefore, there exists a faithful normal conditional expectation EA(T ) which satisfies CE1 EA(T ) : A → A(T ) is linear, w∗–continuous, completely positive, CE2 EA(T )(1) = 1, CE3 ω ◦ EA(T ) = ω, and EA(T )(aEA(T )(b)) = EA(T )(a)EA(T )(b), for all a, b ∈ A. The above characterization contains the Ergodic Theorem for QMS. Indeed EA(T ) is unique since, given any other map E which satisfies CE1, CE2, CE3, it follows that EA(T ) = E ◦ EA(T ) = EA(T ) ◦ E = E. More precisely, Theorem 2.7. If ω is a faithful, normal state which is invariant under T , then there exists a unique normal conditional expectation EA(T ) onto A(T ) A(T ) A(T ). In addition, E ◦ Tt(·) = E (·) for all t ≥ 0; for any element a ∈ A, EA(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 A(T ) the predual semigroup (T∗t)t≥0, are elements of the form ϕ ◦ E of the predual A∗, where ϕ runs over all states defined on the algebra A(T ). A(T ) A(T ) Proof. To prove the equality E ◦ Tt(·) = E , it suffices to notice A(T ) that E ◦ Tt(·) satisfies CE1, CE2 and CE3 as well. This means that A(T ) A(T ) both , E ◦ Tt(·) and E , are conditional expectations onto A(T ), so that they coincide. Moreover, given any a ∈ A, we have already proved that co(T (a)) ⊆ A(T ). From the generalization of Proposition 2.3, there exists a net (tα)α≥0, such that the limit Z tα ∗ 1 E(a) = w − lim Tt(a)dt, α tα 0 satisfies CE1, CE2 and CE3. Thus, E(a) = EA(T )(a), and co(T (a)) ∩ A(T ) is reduced to a single element EA(T )(a) . Finally, if ϕ is an invariant state defined on A(T ), it follows straightforward that ϕ ◦ EA(T ) is an invariant state on A. Let us prove the reciprocal property. Indeed, given any invariant state ψ on A and an element a ∈ A, write EA(T )(a) as an element of co(T (a)):

Z tα A(T ) ∗ 1 E (a) = w − lim Tt(a)dt. α tα 0 1. Quantum Markov Semigroups and their stationary states 9

Therefore,

Z tα A(T ) ∗ 1 ψ(E (a)) = w − lim ψ(Tt(a))dt = ψ(a). α tα 0 So that it suﬃces to take ϕ as the state-restriction of ψ to A(T ) to conclude the proof. It is worth noticing that co(T (a))∩A(T ) = {EA(T )(a)} and EA(T )(a) can be computed, for instance through the equivalent formulae:

Z T A(T ) ∗ 1 E (a) = w − lim Tt(a)dt (2.1) T →∞ T 0 Z ∞ ∗ −λt = w − lim λ e Tt(a)dt. (2.2) λ→0 0 Definition 2.8. According to the classification of non commutative dynamical systems developed by Dang Ngoc [14], [29], the QMS T is finite if there is enough normal invariant states, that is, if for all non-zero positive element x of A, there exists a normal state ω in the space of fixed points A∗(T∗) of the predual semigroup, such that ω(x) > 0. The QMS is infinite if it is not finite. Moreover, a given a projection p ∈ A(T ) is said to be finite if the induced p semigroup Tt (a) = pTt(a)p, for all a ∈ pAp is finite, where pAp is the algebra of elements of the form a = pa0p, a0 ∈ A. We recall here for easier reference the following result of Dang Ngoc [14]. For a given state ϕ, we denote S(ϕ) its support projection. Theorem 2.9. Let denote e the least upper bound of S(ω), when ω runs over the set of all normal invariant states. Then e belongs to the center of A(T ) and coincides with the least upper bound of finite projections in A(T ). The semigroup has no invariant state if e = 0. On the other hand, the following theorem summarizes the characterization of finite QMS. Theorem 2.10. The following conditions are equivalent: 1. The Quantum Dynamical Semigroup T is finite; 2. e = 1; 3. EA(T )(A) = A(T ); 4. EA(T ) is faithful; 5. There exists an invariant conditional expectation from A to A(T ). Moreover there exists at most one conditional expectation from A to A(T ) and if it exists, it coincides with EA(T ). 10 Franco Fagnola ∗ , Rolando Rebolledo∗

6. The orbits of T∗ in A∗ are relatively weakly compact. The equivalence of condition 1 with 2, 3 and 4 is a straightforward consequence of Dang Ngoc’s theorem [14]. Kovácsand Szücs [33] proved that condition 5 is necessary and sufficient for a semigroup to be finite. Finally, the characterization 6 is due to Störmer [46]. Since the paper of S.Ch.Moy [36], several authors have studied the construction of conditional expectations in a non-commutative framework, in particular Umegaki [48], Takesaki [47], Accardi and Cechini [1] (see eg. the survey included in the work of Petz [39]). 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.

3 The minimal quantum dynamical semigroup

Throughout this chapter, which is merely expository, we consider the von Neumann algebra A = B(h) and we rephrase, for easier reference the crucial result which allows to construct a quantum dynamical semigroup starting from a generator given as a sesquilinear form. For further details on this matter we refer to [20], section 3.3, see also [11]. Let G and L`,(` ≥ 1) be operators in h which satisfy the following hypothesis:

• (H-min) G is the inﬁnitesimal generator of a strongly continuous contraction semigroup in h, D(G) is contained in D(L`), for all ` ≥ 1, and, for all u, v ∈ D(G), we have

∞ X hGv, ui + hL`v, L`ui + hv, Gui = 0. `=1

Under the above assumption (H-min), for each x ∈ B(h) let−L(x) ∈ F (h) be the sesquilinear form with domain D(G) × D(G) deﬁned by

∞ X −L(x)(v, u) = hGv, xui + hL`v, xL`ui + hv, xGui. (3.1) `=1

It is well-known (see e.g. [16] Sect.3, [20] Sect. 3.3) that, given a domain D ⊆ D(G), which is a core for G, it is possible to built up a quantum dynamical semigroup, called the minimal QDS, satisfying the equation:

Z t hv, Tt(x)ui = hv, xui + −L(Ts(x))(v, u)ds, (3.2) 0 for u, v ∈ D. 1. Quantum Markov Semigroups and their stationary states 11

This equation, however, in spite of the hypothesis (H-min) and the fact that D is a core for G, does not necessarily determine a unique semigroup. The minimal QDS is characterized by the following property: for any w∗- continuous family (Tt)t≥0 of positive maps on B(h) satisfying (3.2) we have (min) Tt (x) ≤ Tt(x) for all positive x ∈ B(h) and all t ≥ 0 (see e.g. [20] Th. 3.21). (min) Let T∗ denote the predual semigroup on I1(h) with inﬁnitesimal (min) (min) generator L∗ . It is worth noticing here that T∗ is a weakly continuous semigroup on the Banach space I1(h), hence it is strongly continuous. The linear span V of elements of I1(h) of the form |uihv| is contained in the (min) domain of L∗ . Thus we can write the equation (3.2) as follows Z t (min) tr(|uihv|Tt(x)) = tr(|uihv|x) + tr(L∗ (|uihv|)Ts(x))ds. 0 This equation reveals that the solution to (3.2) is unique whenever the (min) linear manifold L∗ (V) is big enough. Indeed, the following characterization holds. Proposition 3.1. Under the assumption (H-min) the following conditions are equivalent:

(min) (i) the minimal QDS is Markov (i.e. Tt (1) = 1),

(min) ∗ (ii) (Tt )t≥0 is the unique w -continuous family of positive contractive maps on B(h) satisfying (3.2) for all positive x ∈ B(h) and all t ≥ 0,

(min) (iii) the domain V is a core for L∗ . We refer to [16] Th. 3.2 or [20] Prop. 3.31 (resp. [20] Th. 3.21) for the proof of the equivalence of (i) and (iii) (resp. (i) and (ii)). Assume that the minimal QDS is Markov and call (Pt)t≥0 the semigroup of contractions generated by G, then the equation (3.2) may be written in an equivalent form as

X Z t hv, Tt(x)ui = hPtv, Ptui + hL`Pt−sv, Tt(x)L`Pt−suids, (3.3) `≥1 0 Moreover, the solution to (3.3) is obtained as the supremum of an approximating (n) sequence (T )n∈N deﬁned recursively as follows on positive elements x ∈ B(h):

(0) ∗ Tt (x) = Pt xPt (3.4) (n+1) hu, Tt (x)ui = hPtu, xPtui (3.5) Z t X (n) + hL`Pt−su, Tt (x)L`Pt−suids, (u ∈ D(G)). `≥1 0 12 Franco Fagnola ∗ , Rolando Rebolledo∗

For any positive element x ∈ B(h) the above sequence is increasing with (n) n. This allows to deﬁne Tt(x) = supn Tt (x). T is the Minimal Quantum Markov semigroup, (see for instance [20]).

The above discussion has shown the importance of getting a minimal quantum dynamical semigroup which preserves the identity, that is, a quantum Markov semigroup. Recently A.M. Chebotarev and the ﬁrst author have obtained easier criteria to verify the Markov property. For instance, the following result ([10] Th. 4.4 p.394) which will be good enough to be applied in our framework. Proposition 3.2. Under the hypothesis (H-min) suppose that there exists a self-adjoint operator C in h with the following properties:

(a) the domain of G is contained in the domain of C1/2 and is a core for C1/2,

2 1/2 (b) the linear manifold L`(D(G )) is contained in the domain of C ,

X 2 1/2 2 −2

(d) for all u ∈ D(C1/2) we have kΦ1/2uk ≤ kC1/2uk,

(e) for all u ∈ D(G2) the following inequality holds

∞ 1/2 1/2 X 1/2 2 1/2 2 2

where b is a positive constant depending only on G, L`, C.

Then the minimal QDS is Markov. As shown in [20] the domain of G2 can be replaced by a linear manifold D which is dense in h, is a core for C1/2, is invariant under the operators Pt of the contraction semigroup generated by G, and enjoys the properties:

1/2 1/2 R(λ; G)(D) ⊆ D(C ),L` (R(λ; G)(D)) ⊆ D(C ) where R(λ; G)(λ > 0) are the resolvent operators. Moreover the inequality (3.6) must be satisﬁed for all u ∈ R(λ; G)(D). 1. Quantum Markov Semigroups and their stationary states 13 4 The existence of Stationary States

This section is aimed at ﬁnding a criterion for the existence of stationary states for a quantum Markov semigroup whose generator is unbounded and known as a sesquilinear form. We will be concerned with the case A = B(h) again.

4.1 A general result We begin by introducing a notation for truncated operators in B(h). Deﬁnition 4.1. For each self-adjoint operator Y , bounded from below, we denote by Y ∧ r the truncated operator

⊥ Y ∧ r = YEr + rEr (4.1) where Er denotes the spectral projection of Y associated with the interval ] − ∞, r]. We are now in a position to prove our ﬁrst result on the existence of normal stationary states. Theorem 4.2. Let T be a quantum Markov semigroup. Suppose that there exist two self-adjoint operators X and Y with X positive and Y bounded from below and with ﬁnite dimensional spectral projections associated with bounded intervals such that Z t hu, Ts(Y ∧ r)uids ≤ hu, Xui (4.2) 0 for all t, r ≥ 0 and all u ∈ D(X). Then the QMS T has a normal stationary state. Proof. Let −b (with b > 0) be a lower bound for Y . Note that, for each r ≥ 0 we have

⊥ Y ∧ r ≥ −bEr + rEr = −(b + r)Er + r1 so that (4.2) yields

Z t 2 −(b + r) hu, Ts(Er)uids + rtkuk ≤ hu, Xui 0 for all u ∈ D(X). Normalize u and denote by |uihu| the pure state with unit vector u. Dividing by t(b + r), for all t, r > 0 we have then

1 Z t r hu, Xui tr(T∗s(|uihu|)Er)ds ≥ − . t 0 b + r t(b + r) 14 Franco Fagnola ∗ , Rolando Rebolledo∗

It follows that the family of states

1 Z t T∗s(|uihu|) ds, t > 0 t 0 is tight. The conclusion follows then from Theorem 2.5 and Corollary 2.6.

Remark. It is worth noticing that we wrote the inequality (4.2) truncated (integral on [0, t], and Y ∧ r) to cope with two diﬃculties: the divergence of the integral and the unboundedness of Y . Deﬁning appropriately the supremum of a family of self-adjoint operators and then the potential U for positive self-adjoint operators, the formula (4.2) can be written as

U(Y ) ≤ X.

This also throws light on the classical potential-theoretic meaning of our condition which is currently under investigation. In the applications, however, the inequality (4.2) is hard to verify since very frequently the QMS is not explicitly given. Therefore we shall look for conditions involving the inﬁnitesimal generator. To this end we introduce now the class of QMS with possibly unbounded generators that concerns our research. This is suﬃciently general to cover a wide class of applications.

4.2 Conditions on the generator Here we use the notations and hypotheses of the previous chapter yielding to the construction of the minimal quantum dynamical semigroup associated to a given form-generator. Deﬁnition 4.3. Given two selfadjoint operators X,Y , with X positive and Y bounded from below, we write −L(X) ≤ −Y on D, whenever

∞ X 1/2 1/2 hGu, Xui + hX Lù, X Lùi + hXu, Gui ≤ −hu, Y ui, (4.3) `=1 for all u in a linear manifold D dense in h, contained in the domains of 1/2 G, X and Y , which is a core for X and G, such that L`(D) ⊆ D(X ), (` ≥ 1). Theorem 4.4. Assume that the hypothesis (H-min) of the previous chapter holds and that the minimal QDS associated with G, (L`)`≥1 is Markov. Suppose that there exist two self-adjoint operators X and Y , with X positive and Y bounded from below and with finite dimensional spectral projections associated with bounded intervals, such that

(i) −L(X) ≤ −Y on D; 1. Quantum Markov Semigroups and their stationary states 15

(ii) G is relatively bounded with respect to X;

−1 1/2 (iii) L`(n + X) (D) ⊆ D(X ),(n, ` ≥ 1).

Then the minimal quantum dynamical semigroup associated with G, (L`)`≥1 has a stationary state. It is worth noticing that the above sufficient conditions always hold for a finite dimensional space h. Indeed, by the hypothesis (H-min), it suffices to take X = 1, Y = 0 and D = h.

We begin the proof by building up approximations T (n) of T (min). Lemma 4.5. Under the hypotheses of Theorem 4.4, for all integer n ≥ 1 (n) (n) the operators G and L` with domain D deﬁned by

(n) −1 (n) −1 G = nG(n + X) ,L` = nL`(n + X) , admit a unique bounded extension. The operator on B(h) deﬁned by

(n) (n)∗ X (n)∗ (n) (n) L (x) = G x + L` xL` + xG (4.4) ` (n ≥ 1) generates a uniformly continuous quantum dynamical semigroup T (n). (n) (n) Proof. First notice that G and the L` ’s are bounded. Indeed, by the hypothesis (ii), the resolvent (n + X)−1 maps h into the domain of (n) (n) the operators G and L`, therefore, G and L` are everywhere deﬁned. Moreover, since G is relatively bounded with respect to X, there exist two constants c1, c2 > 0 such that, for each u ∈ h we have

−1 −1 −1 knG(n + X) uk ≤ c1knX(n + X) uk + c2kn(n + X) uk. By well known properties of the Yosida approximation the right hand side is bounded by (nc1 + c2)kuk. On the other hand, by (H-min), for each u ∈ h we also have

∞ X −1 2 −1 (n) 2 knL`(n + X) uk = −2

(n) Thus the L` ’s are bounded. Moreover, replacing u, v in condition (H- min) by n(n + X)−1u, u ∈ h, leads to

∞ (n) (n) X (n) 2 hu, L (1)ui = 2

We recall the following well-known result on the convergence of semigroups Proposition 4.6. Let A, A(n) ( n ≥ 1) be inﬁnitesimal generators of (n) strongly continuous contraction semigroups (Tt)t≥0, (Tt )t≥0 on a Banach space and let D0 be a core for A. Suppose that each element x of D0 (n) (n) belongs to the domain of A for n big enough and the sequence (A x)n≥1 (n) converges strongly to Ax. Then the operators Tt converge strongly to Tt uniformly for t in bounded intervals. We refer to [32] Th. 1.5 p.429, Th. 2.16 p. 504 for the proof. We shall need also the following elementary lemma.

(n) Lemma 4.7. Let (r`)`≥1, (s` )`≥1 ( n ≥ 1) be square-summable sequences (n) of positive real numbers. Suppose that, for every ` ≥ 1, s` is an inﬁnitesimum as n tends to inﬁnity and that there exists a positive constant c such that

2 X (n) s` ≤ c `≥1 for every n ≥ 1. Then X (n) lim r`s = 0. n→∞ ` `≥1 Proof. Suppose that our conclusion is false. Then, by extracting a subsequence (in n) if necessary, we ﬁnd an ε > 0 such that, for every n,

X (n) r`s` > ε. (4.5) `≥1

The sequences s(n) can be looked at as vectors in l2(N) uniformly bounded in norm by c. Therefore we can extract a subsequence (nm)m≥1 such (nm) (n) that (s )m≥1 converges weakly as m tends to infinity. Since s` is an infinitesimum as n tends to infinity for each ` ≥ 1, it follows that the weak limit must be the vector 0. This contradicts (4.5).

(n) (n) Lemma 4.8. Let G , L` the operators on h deﬁned in Lemma 4.5. Then, under the hypotheses of Theorem 4.4, for all u ∈ D(X), we have

(n) (n) lim G u = Gu, lim L u = L`u. n→∞ n→∞ `

(n) Moreover the operators T∗t on I1(h) converge strongly, as n tends to inﬁnity, to T∗t, uniformly for t in bounded intervals. 1. Quantum Markov Semigroups and their stationary states 17

Proof. For all u ∈ D(X), we have

(n) −1 G u − Gu = G(n(n + X) − 1)u −1 −1 ≤ c1 (n(n + X) − 1)Xu + c2 (n(n + X) − 1)u .

(n) Therefore the sequence G u n≥1 converges strongly to Gu as n tends to inﬁnity by well-known properties of Yosida approximations. Moreover, by (H-min), for u ∈ D(X), we have also

∞ 2 X (n) −1 −1 L` u − L`u = −2< (n(n + X) − 1)u, G(n(n + X) − 1)u . `=1 (4.6) (n) This shows the convergence of sequences L` u to L`u for all ` ≥ 1. n≥1 For all u, v ∈ D(X) we have

(n) (min) (n) (n) L∗ (|uihv|) − L∗ (|uihv|) = |(G − G)uihv| + |uih(G − G)v| ∞ X (n) + |(L` − L`)uihv| `=1 ∞ X (n) + |uih(L` − L`)v|. `=1

(n) (min) Therefore the trace norm of L∗ (|uihv|) − L∗ (|uihv|) can be estimated by

kvk · k(G(n) − G)uk + kuk · k(G(n) − G)vk ∞ ∞ X (n) X (n) + kLùk · k(L` − L`)vk + kL`vk · k(L` − L`)uk. `=1 `=1 Clearly the first two terms vanish as n tends to infinity. Moreover, by the inequality (4.6), since the operators X(n + X)−1 are contractive, we have

∞ 2 X (n) −1 (n) L` u − L`u ≤ 2k(n + X) Xuk · k(G − G)uk `=1 −1 −1 ≤ 2kuk c1 (n(n + X) − 1)Xu + c2 (n(n + X) − 1)u −1 −1 = 2kuk c1 (X(n + X) )Xu + c2 (X(n + X) )u

≤ 2kuk (c1 kXuk + c2 kuk) .

(n) An application of Lemma 4.7 shows then that the trace norm of L∗ (|uihv|)− (min) L∗ (|uihv|) converges to 0 as n tends to inﬁnity. Since the minimal QDS associated with G,(L`)`≥1 is Markov and D(X) is a core for G (it contains D), the linear manifold generated by |uihv| with 18 Franco Fagnola ∗ , Rolando Rebolledo∗

(min) u ∈ D(X) is a core for L∗ . The conclusion follows then from Proposition 4.6.

Lemma 4.9. Let Y ∧r the operator deﬁned by (4.1) and let X(n) = nX(n+ −1 (n) 2 −1 −1 X) ,(n ≥ 1). Deﬁne Yr = n (n + X) (Y ∧ r)(n + X) . Then, under the hypotheses of Theorem 4.4, the operator Z t (n) (n) (n) X − Ts (Yr )ds, 0 is positive for each t ≥ 0.

Proof. Notice that Yr ≤ Y . Therefore, by the hypothesis (i) of Theorem 4.4, we have the inequality ∞ X 1/2 1/2 hGu, Xui + hX Lù, X Lùi + hXu, Gui ≤ −hu, (Y ∧ r)ui, (4.7) `=1 for all u ∈ D. The domain D being a core for X and G being relatively bounded with respect to X, for every u ∈ D(X) we can find a sequence (un)n≥1 in D such that (Xun)n≥1 converges to Xu and (Gun)n≥1 converges to Gu. Then the convergence of (Lùn)n≥1 to Lù (for all ` ≥ 1) follows readily from the hypothesis (H-min). Moreover, for every n, m ≥ 1, the inequality (4.7) yields ∞ X 1/2 2 kX L`(un − um)k ≤ −2

− h(un − um), (Y ∧ r)(un − um)i.

Therefore, replacing u by un, and letting n tend to inﬁnity we show that (4.7) holds for all u ∈ D(X). −1 Since n(n + X) is a contraction and Yr ≤ Y , under the hypotheses of Theorem 4.4, for all u ∈ h we have ∞ (n) (n) (n) (n) X 1/2 (n) 1/2 (n) hu, L (X )ui ≤ 2

Therefore, Z t (n) (n) (n) (n) (n) X − Ts (Yr )ds ≥ Tt (X ) ≥ 0 0 for all t ≥ 0.

Proof. (of Theorem 4.4). By Lemma 4.9 for each u ∈ D(X), t, r ≥ 0 and n ≥ 1, we have

Z t (n) (n) (n) tr(T∗s (|uihu|)Yr )ds ≤ hu, X ui. 0

(n) The sequence (Yr )n≥1 converges strongly to Y ∧ r. Thus, by Lemma 4.8, we can let n tend to inﬁnity to obtain

Z t tr(T∗s(|uihu|)(Y ∧ r))ds ≤ hu, Xui. 0

This inequality coincides with (4.2). Therefore Theorem 4.4 follows from Theorem 4.2.

4.3 Applications 4.4 A multimode Dicke laser model We follow Alli and Sewell [4] where a model is proposed for a Dicke laser or maser. We begin by establishing the corresponding notations. The system consists of N identical two-level atoms coupled with a radiation ﬁeld corresponding to n modes. Therefore, one can choose the Hilbert space h which consists of the tensor product of N copies of C2 and n copies of l2(N). To simplify notations we simply identify any operator acting on a factor of the above tensor product with its canonical extension to h. Let σ1, σ2, σ3 be the Pauli matrices and deﬁne the spin raising and lowering operators σ± = (σ1 ± iσ2)/2. The atoms are located on the sites r = 1,...,N of a one dimensional lattice, so that we denote by σ,r ( = 1, 2, 3, +, −) the spin component of the atom at the site r. The free evolution of the atoms is described by a generator Lmat which is bounded and given in Lindblad form as

1 X L (x) = i[H, x] − (V ∗V x − 2V ∗xV + xV ∗V ), (4.8) mat 2 j j j j j j j where the sum contains a ﬁnite number of elements, H is bounded self- adjoint and the Vj’s are bounded operators. 20 Franco Fagnola ∗ , Rolando Rebolledo∗

∗ Moreover, we denote by aj , aj, the creation and annihilation operators corresponding to the j-th mode of the radiation, (j = 1, . . . , n). These operators satisfy the canonical commutation relations:

∗ [aj, ak] = δjk1, [aj, ak] = 0. The free evolution of the radiation is given by the formal generator n X ∗ ∗ ∗ ∗ Lrad(x) = (κ`(−a` a`x + 2a` xa` − xa` a`) + iω`[a` a`, x]) , (4.9) `=1 where κ` > 0 are the damping and ω` ∈ R are the frequencies corresponding to the `-th mode of the radiation. The coupling between the matter and the radiation corresponds to a Hamiltonian interaction of the form:

N n i X X H = λ (σ a∗e−2πik`r − σ a e2πik`r), (4.10) int N 1/2 ` −,r ` +,r ` r=1 `=1 where k` is the wave number of the `-th mode and the λ’s are real valued, N independent coupling constants. With the above notations, the formal generator of the whole dynamics is given by L(x) = Lmat(x) + Lrad(x) + i[Hint, x] (4.11)

To identify L` and G in our notations, we use in force the convention on the abridged version of tensor products with the identity. That is, here we ﬁnd √ L` = κ` a`, (` = 1, . . . , n) (4.12)

All the remaining L`’s are bounded operators. Among them a ﬁnite number (indeed at most 3N) coincides with some of the Vj’s appearing in (4.8) and the other vanish. So that the operator G becomes formally: 1 X X G = − L∗L − i ω a†a − iH − iH , (4.13) 2 ` ` ` ` ` int ` ` where the sum contains only a ﬁnite number of non zero terms. To make the above expression rigorous some preliminary work is needed. 2 Call (fm)m≥0 the canonical orthonormal basis on the space l (N). In the radiation space, which consists of the tensor product of n copies of l2(N), we denote f = f (1) ⊗ ... ⊗ f (n), α α1 αn (`) where α = (α1, . . . , αn) and fα` is an element of the canonical basis of the 2 n ` copy of l (N). Thus, (fα)α∈N is the canonical orthonormal basis of the radiation space. 1. Quantum Markov Semigroups and their stationary states 21

With these notations we have ( √ √ ∗ α`fα−1` if α` > 0 a` fα = α` + 1fα+1` , a`fα = , (4.14) 0 if α` = 0 where 1` is the vector with a 1 at the `th coordinate and zero elsewhere. Thus, the operator G is well defined over vectors of the form ufα where u ∈ C2N and the symbol of tensor product is dropped. It is well known (see [32], Thm. 2.7 p.499) that a perturbation of a negative selfadjoint operator, relatively bounded with relative bound less than 1, is the infinitesimal generator of a contraction semigroup. Therefore, Pn ∗ we choose X formally given by X = `=1 a` a`. That is, Xufα = |α|ufα, where, |α| = α1 + ... + αn. k k Since X ufα = |α| ufα it follows that the linear span of vectors of the form ufα is a dense subset of the analytic vectors for X. Therefore, by a theorem of Nelson (see e.g. [43]), X is essentially self-adjoint on the referred domain. From now on we identify X with its closure which is selfadjoint. We show now that Hint is relatively bounded with respect to X. Let ξ be a finite linear combination of elements of the form ufα. By Schwarz’ √ √ √ p inequality,√ and elementary inequalities like t + s ≤ t+ s ≤ 2(t + s), 2 ts ≤ t + −1s, we obtain

N n 1 X X kH ξk ≤ |λ |(ka∗ξk + ka ξk) int N 1/2 ` ` ` r=1 `=1 N n 1 X X 1/2 ≤ |λ | 4hξ, a∗a ξi + 2kξk2 N 1/2 ` ` ` r=1 `=1 " N n N n # 1 X X √ X X ≤ 2(|λ |2hξ, a∗a ξi)1/2 + 2 |λ |kξk N 1/2 ` ` ` ` r=1 `=1 r=1 `=1 " N n N n # 1 X X √ X X ≤ (ka∗a ξk + −1|λ |2kξk) + 2 |λ |kξk N 1/2 ` ` ` ` r=1 `=1 r=1 `=1 Pn √ Pn Finally, by the elementary inequality `=1 ks`k ≤ nk `=1 s`k, it follows

N n N n n1/2 X X 1 X X √ kH ξk ≤ k a∗a ξk + ( 2 + −1|λ |)|λ |kξk int N 1/2 ` ` N 1/2 ` ` r=1 `=1 r=1 `=1 N n 1 X X √ ≤ N 1/2n1/2kXξk + ( 2 + −1|λ |)|λ |kξk, N 1/2 ` ` r=1 `=1 thus, choosing < (Nn)−1/2, the above inequality yields the required relative boundedness of Hint with respect to X. 22 Franco Fagnola ∗ , Rolando Rebolledo∗

As a result, the operator G appears as a dissipative perturbation of 1 − 2 X, relatively bounded with respect to X, with bound strictly less than 1. Therefore, G is the generator of a contraction semigroup. Moreover, the domain of G coincides with that of X and hypothesis (H-min) easily checked. To apply our main result, we fix the domain D as the space of vectors ξ which are finite linear combinations of the form ufα. Notice that this is an invariant for X, G, and all the L`’s. To identify an appropriate operator Y to have −L(X) ≤ −Y , we first perform the computation of −L(X). For the sake of clarity, we avoid handling forms in the computations below. However, the reader may easily notice that all the expressions are well defined since the domain D is invariant under the action of the operators X, G and L`. Firstly, it holds Lmat(X) = 0, since the Vj’s act on the tensor product of N copies of C2 and leave the domain D invariant. Secondly, a straightforward computation using the canonical commutation relations, yields

n X † Lrad(X) = −2 κ`a`a`. `=1

Another easy computation leads us to

N n −1/2 X X † −2πik`r 2πik`r i[Hint,X] = −iN λ`(σ−,ra`e + σ+,ra`e ). r=1 `=1

Summing up,

n N n X i X X −L(X) = −2 κ a†a − λ (σ a†e−2πik`r + σ a e2πik`r). ` ` ` N 1/2 ` −,r ` +,r ` `=1 r=1 `=1

To identify Y it suﬃces to control the term i[Hint,X]. For each ξ ∈ D, it follows

−1/2 So that choosing 0 < < 2N min κ` the required operator Y may be taken as 1/2 Y = (2 min κ` − N )(X + c), where c > 0 is a suitable constant. The spectrum of X coincides with N. For each m ∈ N, the corresponding eigenspace is generated by the fα with |α| = m. Therefore, it follows that all spectral projections of X and Y associated with bounded intervals are ﬁnite dimensional. Similar arguments allow to check the hypotheses of Proposition 3.2 (with C = X) to show that the minimal QDS associated to the operators G and L`, (1 ≤ ` ≤ n), is Markov (this was also proved in [4] by another method). To summarize, our main theorem implies the following Corollary 4.10. There exists an invariant state for the multimode Dicke model.

4.5 A quantum model of absorption and stimulated emission This example corresponds to a family of models introduced by Gisin and Percival in [28]. The framework is given by the Hilbert space h = l2(N) where, as usual, we call (en)n≥0 the canonical orthonormal basis. The operators deﬁning the form-generator −L(·) are

† † L1 = νa a, L2 = µa, H = ξ(a + a), where µ, ν > 0 and ξ ∈ R. Thus, 1 G = −iξ(a† + a) − ν2(a†a)2 + µ2a†a , 2 Let us check the existence of an invariant state by means of Theorem 4.4. Take, for instance X = (N + 1)2, where N = a†a is the number operator. A straightforwrd computation yields −L(X) = iξ (a − a†)(N + 1) + (N + 1)(a − a†) + µ2N(1 − 2N). (4.15)

We ﬁrst study the term iξ (a − a†)(N + 1) + (N + 1)(a − a†), which corresponds to i[H,X]. Call D the linear manifold generated by (en)n≥0. For any u ∈ D, we have |hu, i[H,X]i| = |ξ| h(a − a†)u, (N + 1)ui

1 † ≤ (a − a )u k(N + 1)uk 2 ≤ N 1/2u k(N + 1)uk 2 2 ≤ hu, Nui + hu, (N + 1)2i. 2 2 24 Franco Fagnola ∗ , Rolando Rebolledo∗

Thus,

2 2 −L(X) ≤ −2µ2N 2 + µ2N + (N + 1)2 + N. 2 2 Finally, if we call

2 2 2 Y = 2µ2 − N 2 − µ2 + 2 + N − 1, 2 2 2 we obtain

−L(X) ≤ −Y. In a similar way one can verify as well the inequalities which show that the minimal quantum dynamical semigroup is Markov. To summarize, Corollary 4.11. The quantum semigroup which corresponds to the model of absorption and stimulated emission here before is Markov and has a stationary state.

4.6 The Jaynes-Cummings model We follow our article [21] to introduce the Jaynes-Cummings model in Quantum Optics. To this aim we use the same space h = l2(N), since here n = 1, we drop the index ` from the notations of creation and annihilation operators and S denotes the right-shift operator. In this framework the formal generator is given by

µ2 λ2 L(x) = − (a∗ax − 2a∗xa + xa∗a) − (aa∗x − 2axa∗ + xaa∗) 2 √ √ 2 √ √ + R2(cos(φ aa∗)x cos(φ aa∗) − x) + R2 sin(φ aa∗)S∗xS sin(φ aa∗), where λ, µ, R and φ are positive constants. In [21] the rigorous construction of the minimal QDS was done showing also that it is identity preserving. The above Jaynes-Cummings generator has a faithful invariant state if and only if µ2 > λ2. This state can be computed explicitly. The interested reader is referred to [21]. To check conditions of Theorem 4.4, one can take X = a∗a, the value of L(X) becoming √ L(X) = −(µ2 − λ2)a∗a + R2 sin2(φ aa∗).

Thus, it suﬃces to take Y = (µ2 − λ2)a∗a − R2 to prove the existence of a stationnary state via our main result. To prove the necessity, one can follow the argument explained in [21] inspired from classical probability. Thus, in this case, our result 4.4 turns out to be sharp. 1. Quantum Markov Semigroups and their stationary states 25

4.7 A quantum particle model We consider a quantum particle model based on the interaction of particles with a given reservoir. Assuming a localization hypothesis, the system consists of particles which are distributed on a sub-lattice of Zd. The free evolution of particles is described by a Hamiltonian H. Particles are supposed to jump from one position x ∈ Zd to another y ∈ Zd due to the action of the reservoir. As a result, a suitable choice of initial space h should be the tensor product of infinite copies of C2, however this needs some technical details. 2 Let denote (e0, e1) the canonical basis of C . We then consider the space    d X  Ξ = ξ ∈ {0, 1}Z : ξ(x) < ∞ .  d  x∈Z For each ξ ∈ Ξ, define O |ξi = eξ(x). d x∈Z Finally, h will be the Hilbert space generated by finite linear combinations of the above vectors |ξi and will be referred to as the electron space. We denote L = (Zd ∪ {?}) × (Zd ∪ {?}), where ? 6∈ Zd. To describe jumps of particles due to interactions with the reservoir, we consider a collision kernel C(x, y), which is a family of operators acting on h, and a collision rate γ(x, y) ≥ 0, (x, y ∈ Zd). The collision rate is a function of the sites which depends also on the chemical potential and the temperature, giving a measure of the electron distribution on the lattice (they are supposed to be localized). We will assume that the collision rate will eventually vanish with the distance between the sites x and y, more precisely, we suppose that there exist constants c1, δ > 0 such that c γ(x, y) ≤ 1 (x, y ∈ d). (4.16) 1 + |x − y|d+δ Z In addition, we need to include conditions for γ(?, y) and γ(x, ?): c γ(?, y) ≤ 2 , (y ∈ d). (4.17) 1 + |y|d+δ Z lim inf γ(x, ?) > 0. (4.18) |x|→∞ sup γ(x, ?) < ∞. (4.19) d x∈Z σ1, σ2, σ3 denote the Pauli matrices as in the first example before, and define the spin raising and lowering operators σ± = (σ1 ±iσ2)/2. Moreover, we introduce the operators σ,x ( = +, −) on each site x as follows

∗ σ+,x|ξi = (1 − ξ(x))|ξ + 1xi; σ−,x = σ+,x. (4.20) 26 Franco Fagnola ∗ , Rolando Rebolledo∗

Thus, the collision kernel is now written as:

C(x, y) = σ+,yσ−,x, ((x, y) ∈ L), (4.21) where we adopt the convention C(x, ?) = σ−,x,C(?, y) = σ+,y, and C(?, ?) = 0, for any x, y ∈ Zd. Moreover, the rate γ may be arbitrarily extended to all elements ` ∈ L. We will introduce later some adequate conditions on those coeﬃcients. The free evolution of the particles is described, formally, by a Hamiltonian given by X H = (ωxσ+,xσ−,x + zxσ+,x +z ¯xσ−,x) , (4.22) d x∈Z d where ωx ∈ R, zx ∈ C for all x ∈ Z and we assume,

sup |ωx| < ∞, (4.23) d x∈Z c3 |zx| ≤ d+δ , (4.24) 1 + |x| 2 for all x ∈ Zd. Finally, the formal Lindblad generator which describes the dynamics, is 1 X L(X) = i[H,X] − γ(`)(C(`)∗C(`)X − 2C(`)∗XC(`) + XC(`)∗C(`)). 2 `∈L (4.25) We turn now to the application of our criterion on the existence of invariant states. This will be accomplished in several steps, where additional hypotheses on the model will be introduced. First step: Constructing the operators G and L`. Here we define p L` = γ(`)C(`), for all ` ∈ L. Introduce the domain D as the set of finite linear combinations of elements of Ξ. The operator H will be well defined on D if and only if

X 2 |zx| < ∞, (4.26) d x∈Z which is implied by our assumption (4.24). P Indeed, given an element u = ξ cξ|ξi ∈ D (the sum is taken over a ﬁnite set of ξ’s omitted in the notations for the sake of simplicity), then X Hu = cξωxξ(x)|ξi x,ξ X + cξzx(1 − ξ(x))|ξ + 1xi x,ξ X + cξz¯xξ(x)|ξ − 1xi x,ξ 1. Quantum Markov Semigroups and their stationary states 27

Notice that the first and third terms contain a finite number of non-zero summands. Thus, Hu will be well defined if and only if the second term is a convergent series. This is clearly the case if and only if (4.26) holds. To identify the operator G of the main result, it suffices to compute 1 P ∗ −iH − 2 ` L` L`. Notice first that (4.16) and (4.17) imply X γ(x, y) < ∞, (4.27) d y∈Z for any x ∈ Zd ∪ {?}. This follows easily from the inequality:

k + d − 1 #{y ∈ d : |y| = k} = ≤ (k + 1)d−1. Z d − 1

1 P ∗ P Now computing − 2 ` L` L`u for u = ξ cξ|ξi ∈ D, one obtains

1 X 1 X X − L∗L u = − c γ(x, y)n (1 − n )|ξi 2 ` ` 2 ξ x y d ` ξ x,y∈Z , x6=y 1 X X 1 X X − c γ(x, x)n |ξi − c γ(x, ?)n |ξi 2 ξ x 2 ξ x ξ x ξ x 1 X X − c γ(?, y)(1 − n )|ξi, (4.28) 2 ξ y ξ y where nx denotes the number operator at the site x ∈ Z. Notice that nx|ξi = ξ(x)|ξi, thus the second and third sums on the left of the above equation, contain a finite number of non-zero terms. Therefore, to have 1 P ∗ − 2 ` L` Lù well defined, it suffices to assume conditions (4.16), (4.17). To summarize, G is well defined over D under the conditions (4.23), (4.24), (4.16), (4.17). Second step: G generates a strongly continuous contraction semigroup. This step is founded on a remarkable criterion of Jørgensen [31], Thm.2, p.398, for identifying the generators of strongly continuous contraction semigroups. This needs some additional notations. Consider an increasing sequence (Dk)k∈N of closed subspaces of h. Let denote Ek the orthogonal projection on Dk,(k ≥ 0). As it is proved in [31], G is the generator of a strongly continuous contraction semigroup whenever it satisfies

(i) Dk ⊆ D, GDk ⊆ Dk+1,(k ≥ 0);

(ii) kGEk − EkGEkk ≤ αk,(k ≥ 0);

P −1 (iii) k≥0 αk = ∞; (iv) G is dissipative on D. 28 Franco Fagnola ∗ , Rolando Rebolledo∗

To check the above conditions, we ﬁx k and deﬁne    d  Z X Dk = ξ ∈ { 0, 1 } : ξ(x) ≤ k .  d  x∈Z P Take u = ξ cξ|ξi ∈ Dk. On D, G becomes, X G = −i (ωxσ+,xσ−,x + zxσ+,x +z ¯xσ−,x) x 1 X − γ(x, y)n (1 − n ) (4.29) 2 x y d x,y∈Z ,x6=y 1 X 1 X − γ(x, x)n − γ(x, ?)n 2 x 2 x x x 1 X − γ(?, y)(1 − n ). 2 y y

0 0 d 0 Denote Fk(ξ, x) = {(ξ , x ) ∈ Dk × Z : hξ + 1x, ξ + 1x0 i= 6 0}. Now, the 2 expression k(GEk − EkGEk)uk is bounded above by

X 0 0 0 0 0 0 |zx ||cξ |1Fk(ξ,x)(ξ , x )|zx||cξ|1Fk(ξ ,x )(ξ, x) ξ,x,ξ0,x0  1/2  1/2 X 0 0 2 2 X 2 2 0 0 0 0 ≤  1Fk(ξ,x)(ξ , x )|zx | |cξ |   1Fk(ξ ,x )(ξ, x)|zx| |cξ|  . ξ,x,ξ0,x0 ξ,x,ξ0,x0

Notice the symmetry of the sums in the product before, which turns into

X 2 2 0 0 1Fk(ξ ,x )(ξ, x)|zx| |cξ| , (4.30) ξ,x,ξ0,x0 by an exchange of (ξ, x) with (ξ0, x0). P 0 0 ⊥ Moreover 1F (ξ,x)(ξ , x ) ≤ k. Indeed, E GEk is 0 on the |ξi’s for P ξ,x k P k which x ξ(x) 6= k. Take ξ such that x ξ(x) = k. The scalar product 0 0 hξ + 1x, ξ + 1x0 i is either 0 or 1. When x = x , the scalar product is 1 if and only if ξ = ξ0. If x 6= x0, the scalar product is 1 if and only if

ξ(x) = 0, and ξ0(x0) = 0, and ξ(x0) = 1, and ξ0(x) = 1, and ξ(y) = ξ0(y), 1. Quantum Markov Semigroups and their stationary states 29 for any y 6= x, x0. If (ξ, x) is ﬁxed then there is at most k elements x0 ∈ Zd such that ξ(x0) = 1. For those elements x0, there exists a unique ξ0 such 0 0 0 0 P that hξ + 1x, ξ + 1x i = 1. Thus, ξ,x 1Fk(ξ,x)(ξ , x ) ≤ k. We return to the expression (4.30).

X 2 2 X 2 2 X 0 0 0 0 1Fk(ξ ,x )(ξ, x)|zx| |cξ| = |zx| |cξ| 1Fk(ξ ,x )(ξ, x) ξ,x,ξ0,x0 ξ,x ξ0,x0

X 2 2 ≤ k |zx| |cξ| . ξ,x 2 P 2 2 Therefore, k(GEk − EkGEk)uk ≤ k x |zx| kuk . Thus, Jørgensen’s Theorem applies and G is the generator of a strongly continuous contraction semigroup. Third step: Stationarity To deal with the above properties we introduce a family of operators X Xϕ = ϕ(x)nx, (4.31) d x∈Z δ where ϕ(x) = |x| 2 . Firstly, Xϕ is a self-adjoint operator, since it reduces to a multiplication operator on the elements of the canonical basis. Moreover, notice that the choice of ϕ and the hypotheses (4.16), (4.17), (4.23) and (4.24) imply

X 2 ϕ(x)|zx| < ∞. (4.32) x X γ(?, y)ϕ(y) < ∞. (4.33) y

X d γ(x, y)ϕ(y) < ∞, for all x ∈ Z . (4.34) y

We now turn to the computation of L(Xϕ), which should be written rigourously by means of quadratic forms. However, since the domain D is invariant under the action of all the operators involved in these computations, there is no harm in keeping them purely algebraic. We ﬁrst rewrite L(Xϕ) as 1 X L(X ) = i[H,X ] + γ(`)(C∗(`)[X ,C(`)] + [C∗(`),X ]C(`)) . ϕ ϕ 2 ϕ ϕ `∈L (4.35) We now detail the computation of each component of the above expression, based on the commutation relations:

d [nx, σ+,x] = σ+,x, [σ−,x, nx] = σ−,x, (x ∈ Z ). 30 Franco Fagnola ∗ , Rolando Rebolledo∗

(i) The term i[H,Xϕ]. X i[H,Xϕ] = i ϕ(x)(¯zxσ−,x − zxσ+,x). (4.36) d x∈Z

P 2 Moreover, hu, Xϕui = xξ |cξ| ϕ(x)ξ(x), and

X 0 hu, i[H,Xϕ]ui = 2= c¯ξcξ0 ϕ(x)zxξ(x)hξ − 1x, ξ i x,ξ,ξ0  1/2 1/2 X 2 0 X 2 2 0 ≤ 2  |cξ| ϕ(x)ξ(x)hξ − 1x, ξ i  |cξ0 | ϕ(x)|zx| ξ(x)hξ − 1x, ξ i x,ξ,ξ0 x,ξ,ξ0  1/2 1/2 X 2 X 2 2 = 2  |cξ| ϕ(x)ξ(x)  |cξ| ϕ(x)|zx| ξ(x) , x,ξ x,ξ

that is, ! −1 X 2 2 hu, i[H,Xϕ]ui ≤ hu, Xϕui + ϕ(x)|zx| kuk . (4.37) x

(ii) Dissipative terms with x0, y0 ∈ Zd. A straightforward computation yields

0 0 X X [Xϕ,C(x , y )] = ϕ(x)σ+,y0 [nx, σ−,x0 ] + ϕ(x)[nx, σ+,y0 ]σ−,x0 . x x

Now, for each x we have,

 0, if x 6= x0 and x 6= y0   −ϕ(x)C(y0, x), if x = x0 and x 6= y0 0 0 0 0 ϕ(x)(σ+,y [nx, σ−,x ]+[nx, σ+,y ]σ−,x ) = 0 0 0  ϕ(x)C(x, x ), if x 6= x and x = y   0, if x = x0 and x = y0.

The above equalities yield

X 0 0 ∗ 0 0 0 0 ∗ 0 0 0 0 γ(x , y )(C (x , y )[Xϕ,C(x , y )] + [C (x , y ),Xϕ]C(x , y )) = 0 0 d x ,y ∈Z X ϕ(x)(γ(x0, x)C∗(x, x0)C(x0, x) − γ(x, x0)C∗(x0, x)C(x, x0)). 0 d x,x Z Notice that for any x, x0 both terms C∗(x, x0)C(x0, x) and C∗(x0, x)C(x, x0) vanish. 1. Quantum Markov Semigroups and their stationary states 31

(iii) The dissipative terms with x0 ∈ Zd and y0 = ?. Here we obtain 1 X γ(x0,?)(C∗(x0,?)[X ,C(x0,?)] + [C∗(x0,?),X ]C(x0,?)) = 2 ϕ ϕ x0

1 X 0 = γ(x ,?)ϕ(x)(σ 0 [n , σ 0 ] + [σ 0 , n ]σ 0 ). 2 +,x x −,x +,x x −,x x,x0

It is obvious that in the last sum only survive diagonal terms, which yields X − γ(x, ?)ϕ(x)nx. x

(iv) The dissipative terms with x0 = ?, y0 ∈ Zd. By an analogous computation we obtain

1 X X γ(?, y0)(C∗(?, y0)[X ,C(?, y0)]+[C∗(?, y0),X ]C(?, y0)) = ϕ(x)(1−n ), 2 ϕ ϕ x y0 x

which makes sense as a densely deﬁned operator due to (4.33).

To summarise, the dissipative part is reduced to X X − γ(x, ?)ϕ(x)nx + γ(?, y)ϕ(y)(1 − ny). x y

The above estimates are connected with both, the proof of conservativity via Proposition 3.2 and that of the existence of stationary states through our main result. We postpone for the moment (see proposition below) the discussion of conservativity, and we go into the proof of the existence of stationary states. For this we show a choice of X and Y as they appear in Deﬁnition 4.3. Indeed, X may be chosen as Xϕ with an adequate ϕ. The above estimates lead to Proposition 4.12. Suppose that (4.16), (4.17), (4.18), (4.23) and (4.24) hold. Then Xϕ and Y given by ! X X −1 X 2 Y = (γ(x, ?) − ε)ϕ(x)nx − γ(?, y)ϕ(y) + ε ϕ(x)|zx| 1, x y x (4.38) satisfy −L(Xϕ) ≤ −Y . Fourth step: Conservativity 32 Franco Fagnola ∗ , Rolando Rebolledo∗

We verify that the hypotheses of Proposition 3.2 hold, choosing an operator C of the form C = αXϕ +β1 with ϕ ≡ 1 and α, β > 0 two constants which will be appropriately chosen below. P ∗ First, we have that D(G) = D( ` L` L`) = D(X1). We start proving P ∗ that D( ` L` L`) = D(X1), where X1 is deﬁned through (4.31). Notice P ∗ that ` L` L` and X1 are functions of nx therefore they are commuting operators essentially self-adjoint on D. Moreover, choose 0 < < lim inf|x|→∞ γ(x, ?) , then

X ∗ X X L` L` ≥ nx + nx ≥ X1 + B, (4.39) ` x:γ(x,?)> x:γ(x,?)≤ P P ∗ where B = x:γ(x,?)≤ nx is a bounded operator. Thus D( ` L` L`) ⊆ D(X1). To prove the converse inclusion, (4.28) yields to

X ∗ X X X 0 L` L` ≤ γ(x, y)nx(1−ny)+ γ(x, x)nx+ γ(x, ?)nx+B , (4.40) ` x,y x x 0 where B is another bounded operator. Notice that γ(x, ?) ≤ c1, therefore, the above expression is bounded above by X X X (γ(x, ?) + c1)nx + nx γ(x, y)(1 − ny) ≤ c4X1 + c51, x x y P ∗ where c4, c5 are constants. As a result we obtain that D( ` L` L`) and D(X1) coincide. Furthermore, we prove that D(H) ⊇ D(X1). The first piece of H, that is P x ωxnx has a domain including that of X1, since ωx is uniformly bounded. For the second part of H we use inequality (4.37), with ϕ ≡ 1 and zx P P 1/2 replaced by izx. This yields D( x zxσ+,x + x z¯xσ−,x) ⊇ D(X1 ) ⊇ D(X1). Therefore, D(G) = D(X1). Now we review the hypotheses of Proposition 3.2. Hypothesis (a) is obviously satisfied. Moreover, one can check readily that the domain of X1 is invariant under the action of all the operators L`. Thus, condition P ∗ (b) is also satisfied as well as (c), for which it suffices to take Φ = ` L` L`. Equations (4.39), (4.40) show that condition (d) also holds whenever we choose the constants α > c4 and β > c5. To check hypothesis (e), since all domain problems have been cleared out, it suffices to compute −L(C), which lead to use the formula giving −L(Xϕ). It turns out that

X X X −L(X1) = − γ(x, ?)nx + γ(?, y)(1 − ny) + i (¯zxσ−,x − zxσ+,x). x y x The ﬁrst term is negative and the second is bounded, thus they are trivially majorized. The third term can be estimated as (4.36) by (4.37) with ε = 1. This completes the veriﬁcation of the hypotheses of Proposition 3.2, thus our semigroup is Markov and ends the study of the current model. 1. Quantum Markov Semigroups and their stationary states 33 5 Faithful Stationary States and Irreducibility

5.1 Introduction Consider a probability space (Ω, F, P), a measurable space (E, E) and a Markov process

(Ω, F, P, (Ft)t∈R+ ,E, E, (Xt)t≥0), defined on (Ω, F, P) with states in E. The Markov semigroup generated by this process is given by Ttf(x) = E (f(Xt)|X0 = x), for all t ≥ 0, x ∈ E and any bounded and measurable function f defined on E. A function f of the above class is subharmonic (resp. superharmonic, resp. harmonic) for the given semigroup if Ttf ≥ f (resp. Ttf ≤ f, Ttf = f) for all t ≥ 0. Now take a positive measure µ on (E, E) and consider the von Neumann algebra A = L∞(E, E, µ). A state ν over this von Neumann algebra is given by a probability measure ν absolutely continuous with respect to µ. The state ν is faithful whenever ν(f) = 0, for a positive f ∈ A, implies that f = 0, (i.e. f(x) = 0, µ-almost surely). One important question then arises: when does (Tt)t≥0 has a faithful stationary state in A? This question appears in the non-commutative theory of Markov semigroups as well. Indeed, the existence of a faithful stationary state is a crucial hypothesis in most of the ergodic studies developed in this field (see for instance Chapter 2 before and [26], [27], [22], [23]). In a previous paper (see [25] and Chapter 4) we obtained sufficient conditions for the existence of a stationary state of a given quantum Markov semigroup.

5.2 The support of an invariant state Deﬁnition 5.1. Given a semiﬁnite von Neumann algebra A of operators over a complex separable Hilbert space h, endowed with a trace tr(·), and a Quantum Markov Semigroup (Tt)t≥0 on A, a positive operator a ∈ A is subharmonic (resp. superharmonic, resp. harmonic) for the semigroup if Tt(a) ≥ a, (resp. Tt(a) ≤ a, resp. Tt(a) = a), for all t ≥ 0. Subharmonic events play a fundamental role in the Potential Theory of classical Markov semigroups. They are related to stationarity, recurrence, supermartingale properties. In our framework, we will start by showing a relation between invariant states and subharmonics projections. Lemma 5.2. Let p be a projection of the von Neumann algebra A, and x ∈ A a positive element. If pxp = 0, then p⊥xp = pxp⊥ = 0 Proof. Suppose A ⊆ B(h) and let u, v ∈ h with pu = u, pv = 0. Since x is positive, hzu + v, x(zu + v)i is positive for every z ∈ C. Then, since 34 Franco Fagnola ∗ , Rolando Rebolledo∗ pxp = 0, for every z ∈ C, we have also 2< (z hv, xui) + hv, xvi ≥ 0.

Therefore hv, xui must vanish and the conclusion readily follows.

Theorem 5.3. The support projection of a stationary state for a quantum Markov semigroup is subharmonic. Proof. Let p be the support projection of a given stationary state ρ of (Tt)t≥0. That is, p is the projection on the closure of the rank of ρ, thus ρp = pρ = ρ, and T∗t(ρ) = ρ, (for all t ≥ 0). Let be given an arbitrary t ≥ 0. We ﬁrst notice that pTt(p)p ≤ p, since p ≤ 1. Therefore,

tr(ρ(p − pTt(p)p)) = tr(ρ(p − Tt(p))) = 0, and, since ρ is faithful on the subalgebra pAp, it follows

pTt(p)p = p.

On the other hand,

⊥ pTt(p )p = pTt(1)p − pTt(p)p = p − p = 0.

⊥ Moreover, since Tt(p ) is positive, the previous lemma yields

⊥ ⊥ ⊥ ⊥ pTt(p )p = 0 = p Tt(p )p.

To summarize, ⊥ ⊥ Tt(p) = p + p Tt(p)p , and the projection p is subharmonic.

Proposition 5.4. For any Quantum Markov semigroup T the following propositions are equivalent: a) p is subharmonic for T . b) The subalgebra p⊥Ap⊥ is invariant for T .

⊥ c) For any normal state ρ such that pρp = ρ, it holds tr(ρTt(p )) = 0, for all t ≥ 0.

Proof. Assume that condition a) holds. By hypothesis, we have that pTt(p)p = p, thus

⊥ pTt(p )p = pTt(1)p − pTt(p)p = 0. 1. Quantum Markov Semigroups and their stationary states 35

⊥ ⊥ Therefore, for any positive x ∈ p Ap it follows pTt(x)p = 0 since ⊥ 0 ≤ pTt(x)p ≤ kxk pTt(p )p = 0. ⊥ ⊥ From Lemma 5.2, pTt(x)p = p Tt(x)p = 0, since Tt(x) is positive. ⊥ ⊥ ⊥ ⊥ Thus Tt(x) = p Tt(x)p ∈ p Ap . The same conclusion holds for any arbitrary x ∈ p⊥Ap⊥ since all those elements may be decomposed as a linear combination of four positive elements of p⊥Ap⊥. Now we prove that b) implies c). By hypothesis, for any x ∈ p⊥Ap⊥, it ⊥ ⊥ holds Tt(x) = p yp for some y ∈ A. Clearly, y = Tt(x) since hv, Tt(x)ui = hv, yui for all vectors u and v for which u = p⊥u, v = p⊥v. ⊥ ⊥ Therefore, Tt(x) = p Tt(x)p . In particular, ⊥ ⊥ ⊥ ⊥ Tt(p ) = p Tt(p )p . From this it follows that, given a state ρ such that pρ = ρp = ρ, we have ⊥ ⊥ ⊥ p ρ = ρp = 0 which yields tr(ρTt(p )) = 0. We ﬁnally prove that c) implies a). From c) we obtain that

⊥ tr(ρpTt(p )p) = 0, ⊥ thus pTt(p )p = 0. As a result, by Lemma 5.2, ⊥ ⊥ ⊥ ⊥ ⊥ Tt(p ) = p Tt(p )p ≤ p , ⊥ which gives Tt(p) = Tt(1 − p ) ≥ p.

Deﬁnition 5.5. We say that a quantum Markov semigroup is irreducible if there is no non-trivial subharmonic projection.

5.3 Subharmonic projections. The case A = B(h) Now we concentrate on the case of A = B(h). The quantum Markov semigroup is the minimal obtained from an unbounded generator given as a sesquilinear form X −L(x)(v, u) = hGv, xui + hL`v, xLùi + hv, xGui, `≥1 under the hypotheses of Chapter 3, that is, (H-min) is supposed satisfied and the associated minimal semigroup is assumed to be Markov. Thus, for all x ∈ B(h), the quadratic form−L(x) is defined over the domain D(G) × D(G). We refer the reader to Chapter 3 for the notations and concepts connected with the minimal quantum Markov semigroup associated to −L(·).

In which follows, we use the same notation p for both, a closed subspace and the projection determined by this subspace. 36 Franco Fagnola ∗ , Rolando Rebolledo∗

Lemma 5.6. Let (Pt)t≥0 be the semigroup generated by G. Then a closed subspace p is invariant for (Pt)t≥0 if and only if for any u ∈ D(G) ∩ Rk(p) it holds Gpu = pGpu.

Proof. If p is invariant for the semigroup, Ptp = pPtp, then it is also invariant for the resolvent:

R(λ; G)p = pR(λ; G)p.

As a result for any u ∈ D(G) such that pu = u, if we deﬁne uλ = λR(λ; G)u, then we obtain puλ = uλ too. Therefore D(G) ∩ Rk(p) is dense in Rk(p). Moreover, for any u ∈ D(G) ∩ Rk(p) we have 1 1 1 p (P u − u) = (P pu − pu) = (P u − u) . t t t t t t Therefore, letting t → ∞, we obtain Gpu = pGpu. Conversely, if Gpu = pGpu for all u ∈ D(G) ∩ Rk(p), then p⊥Gp is zero on D(G) and d p⊥P pu = p⊥GP pu = (p⊥Gp)P u = 0. dt t t t ⊥ It follows that p Ptpu = 0 for all u ∈ D(G) ∩ Rk(p) and t ≥ 0, hence ⊥ p Ptp = 0 since D(G) ∩ Rk(p) is dense in Rk(p).

Theorem 5.7. Under the previous hypotheses, a projection p is subharmonic for T if and only if the following conditions are satisﬁed:

Gpu = pGpu, L`pu = pL`pu, (5.1) for all u ∈ D(G) ∩ Rk(p), ` ≥ 1.

Proof. We start assuming that p is subharmonic, thus Tt(p) ≥ p for all t ≥ 0. From equation (3.3) we obtain

⊥ ⊥ ∗ ⊥ p ≥ Tt(p ) ≥ Pt p Pt. Therefore, for all u ∈ Rk(p),

∗ ⊥ ⊥ 2 hu, Pt p Ptui = p Ptu = 0,

⊥ that is p Ptp = 0. Thus Ptp = pPtp, for all t ≥ 0. Then Lemma 5.6 implies that Gpu = pGu for all u ∈ D(G) ∩ Rk(p). In addition, the equation satisfied by the minimal quantum semigroup yields   Z t ⊥ X ⊥ ⊥ hGu, Ts(p )ui + hLù, Ts(p )Lùi + hu, Ts(p )Gui ds ≤ 0, 0 `≥1 1. Quantum Markov Semigroups and their stationary states 37 for all t ≥ 0 and all u ∈ D(G). As a result, computing the derivative in 0 of the above equation, we obtain:

⊥ X ⊥ ⊥ hGu, p ui + hL`u, p L`ui + hu, p Gui ≤ 0. `≥1 Now, if u ∈ D(G) ∩ Rk(p) the above inequality gives

X ⊥ 2 p L`pu ≤ 0, `≥1

⊥ that is p L`pu = 0 or, equivalently,

pL`pu = pL`u, for all ` ≥ 1 and u ∈ D(G) ∩ Rk(p). Conversely, we assume condition (5.1). We will prove that p is subharmonic (n) by an induction argument which relays on the sequence (T )n∈N used in the construction of T . Firstly, p is subharmonic for T (0) since

(0) ⊥ ∗ ⊥ ⊥ ∗ ⊥ ⊥ ⊥ Tt (p ) = Pt p Pt = p Pt p Ptp ≤ p . Secondly, assume that p is subharmonic for T (n), we prove that it is subharmonic for T (n+1) too. Indeed, for all u ∈ D(G)∩Rk(p), the deﬁnition of T (n+1) and the induction hypothesis yield Z t (n+1) ⊥ ∗ ⊥ X ⊥ hu, Tt (p )ui ≤ hu, Pt p Ptui + hL`Pt−su, p L`Pt−suids = 0, `≥1 0 for any t ≥ 0. (n+1) ⊥ ⊥ (n+1) ⊥ It follows that pTt (p )p = 0 and Lemma 5.2 implies p Tt (p )p = (n+1) ⊥ ⊥ pTt (p )p = 0. Therefore, (n+1) ⊥ ⊥ Tt (p ) ≤ p , for all t ≥ 0 and p is subharmonic for T (n). Hence, p is subharmonic for the minimal semigroup T and the proof is complete.

5.4 Examples 5.5 A classical Markov semigroup Assume E to be a metric space and E its Borel σ–ﬁeld. Given a positive ∞ R measure µ on (E, E) call A = L (E, E, µ). Let M(x, A) = A m(x, y)µ(dy) be a transition function on E × E. Then Z Lf(x) = (f(y) − f(x)) M(x, dy), (5.2) E 38 Franco Fagnola ∗ , Rolando Rebolledo∗ deﬁnes a bounded operator on A which is the generator of a (classical) Markov semigroup (Tt)t≥0 (see for instance [18]). 1 In this case, the predual von Neumann algebra A∗ is L (E, E, µ) and any state is represented by a probability measure ν absolutely continuous with respect to µ.

Proposition 5.8. A projection 1A,(A ∈ E), is subharmonic for the given T semigroup if and only if µ(A \ x∈E {y ∈ E : m(x, y) > 0}) = 0. In particular, if an invariant measure ν exists, it is faithful if and only if its support coincides with that of µ. T If µ x∈E {y ∈ E : m(x, y) > 0} = 0, there is no invariant measure for the given semigroup.

It is worth noticing that the Markov process, starting from a point x ∈ A, visits the points inside the support of M(x, ·). To say that 1A is subharmonic means that µ almost-surely A is included in the support of any M(x, ·), for x ∈ A. That is, for any x ∈ A, the process cannot jump outside of A. Proof. Take a projection p = 1A ∈ A. This projection is subharmonic if and only if L1A ≥ 0. However, for x ∈ A,

L1A(x) = (M(x, A) − M(x, E)) ≤ 0, and

L1A(x) = M(x, A), for x ∈ Ac. Therefore, 1A is subharmonic if and only if for µ–almost all x ∈ A, c T M(x, A ) = 0. Call S = x∈E {y ∈ E : m(x, y) > 0}. Obviously M(x, S) = M(x, E) for all x ∈ E and, in particular we obtain that M(x, S) = M(x, E) = M(x, A) for all x ∈ A. Thus, for any x ∈ A, m(x, y) > 0 µ–almost surely R c and since Ac m(x, y)µ(dy) = 0, it follows that µ(A ) = 0. Therefore, \ µ(A \ {y ∈ E : m(x, y) > 0}) = 0. x∈E

Given a stationary probability ν, its support Sν is subharmonic by Theorem T T 5.3. Thus, µ(Sν \ x∈E {y ∈ E : m(x, y) > 0}) = 0. Then if µ x∈E {y ∈ E : m(x, y) > 0} = 0, there is no invariant measure. On the other hand, ν is faithful if and only if its Radon-Nikodym derivative dν/dµ is strictly positive µ–almost surely. This means that ν and µ have the same support. 1. Quantum Markov Semigroups and their stationary states 39

5.6 Gisin-Percival model of absorption and stimulated emission, continued We continue the analysis of the model introduced in 4.5, and refer to the notations therein. We want to characterize all common invariant closed subspaces for G, L1,L2. Since L1 is a multiple of the number operator N, its invariant subspaces IK are spanned by {ek : k ∈ K}, where K ⊆ N. However, notice that if K is ﬁnite and k0 = max K, to have IK invariant also for L2, one needs to have all of the ek for k ≤ k0 inside IK , since L2 is a multiple of the annihilation operator. Deﬁne h0 = {0}; hk, the space generated by {e0, . . . , ek−1} for any k ≥ 1 and h∞ = h. The unique collection of invariant subspaces for L1 and L2 is (hk)k∈N∪{∞}. As a result, if ξ = 0, there is a full collection of non trivial invariant subspaces. Suppose ξ 6= 0. In this case there is no hk invariant under G for k > 0. Therefore, the only common invariant spaces for G, L1,L2 are trivial: h0 and h. As a result, the QMS has a faithful stationary state, say ρ∞. Moreover, in the next chapter we will see that given any other state ρ, the semigroup T associated to (G, L1,L2) satisfy that

tr(T∗t(ρ)X) → tr(ρ∞X),

∗ 0 for all X ∈ B(h). This means that any other w -limit of T∗t(ρ), ρ say, has 0 0 to satisfy tr(ρ X) = tr(ρ∞X) for any X ∈ B(h), so that ρ = ρ∞ due to the faithfulness of ρ∞.

6 The convergence towards the equilibrium

Within this chapter we assume again A = B(h) and the hypotheses leading to the construction of a minimal quantum Markov semigroup introduced in Chapter 3. Moreover, also make the hypothesis that the quantum Markov semigroup has a faithful normal stationary state ρ. Our aim here is to reinforce the ergodic results obtained in Chapter 2. Indeed, we want to derive conditions under which T∗t(σ) converges in the w∗ topology towards ρ as t → ∞, for any initial state σ. We start giving a useful criterion to characterize the domain of the generator, assuming two hypothesis: (H-min) of Chapter 3 and (H-Markov) that the minimal QDS is Markov. We recall that, as in Chapter 3, D is a core for G. Moreover, V the linear manifold generated by the rank-one operators |uihv|, where u, v ∈ D, is a core for L∗. Lemma 6.1. Under the above hypotheses the domain of L is given by all the elements X ∈ B(h) for which the application (v, u) 7→ −L(X)(v, u) is norm–continuous in the product Hilbert space. 40 Franco Fagnola ∗ , Rolando Rebolledo∗

Proof. We remark that X ∈ D(L) if and only if the linear form ρ 7→ tr(L∗(ρ)X), deﬁned on D(L∗), is continuous for the norm k · k1 of I1(h), ∗ since L = (L∗) . So that the essential of the proof consists in establishing the equivalence of the above property with the continuity of (v, u) 7→ −L(X)(v, u) as stated. Moreover, the reader will agree that the latter is a necessary condition for X being an element of D(L), so that it remains to prove the suﬃciency. Indeed, if L(X) is bounded then it is represented by Y = L(X) ∈ B(h). Then for any ρ ∈ V, the computation of tr(()L∗(ρ)X) yields

|tr(L∗(ρ)X)| = |tr(ρY )| ≤ kρk1 kY k . The proof is then completed by a standard argument based on the core property of V.

6.1 A result due to Frigerio and Verri Suppose that the semigroup T has a faithful stationary state ρ. Then there exists a conditional expectation X 7→ T∞(X) in the sense of Umegaki, deﬁned over the von Neumann algebra of invariant elements under the action of T . We recall an early result of Frigerio and Verri ([27], Theorem 3.3, p.281) proved in this framework. We denote N (T ) the set of elements ∗ ∗ ∗ ∗ X ∈ B(h) for which Tt(X X) = Tt(X )Tt(X) and Tt(XX ) = Tt(X)Tt(X ), for all t ≥ 0 Theorem 6.2 (Frigerio–Verri). If the semigroup T has a faithful stationary state ρ and the set of ﬁxed points of T coincides with N (T ), then

∗ w − lim Tt(X) = T∞(X), (6.1) t→∞ for all X ∈ B(h). The basic idea of the proof consists in associating with T a strongly continuous contraction semigroup on the Hilbert space of the GNS representation based on the state ρ. To look for a more friendly criterion based on the generator of the semigroup we need to introduce ﬁrst some auxiliary notions and results on Quantum Markovian Cocycles.

6.2 Quantum Markovian Cocycles Within this section we will built up a concrete dilation of the QMS, as 2 follows. Our basic Hilbert space is the tensor product H = h⊗Γ(L (R+; K)) where the ﬁrst factor is the initial space and the second, the Fock space 2 associated to L (R+; K), where K is a complex separable Hilbert space with an orthonormal basis denoted by (zk; k ≥ 1). 1. Quantum Markov Semigroups and their stationary states 41

2 The exponential vector on Γ(L (R+; K)), associated to a function f ∈ 2 L (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 the vector u ⊗ e(f) in the form ue(f), and we identify any operator on h with its canonical extension to the whole space H. A Quantum Markovian Cocycle (QMC) can be associated to our quantum dynamical semigroup. This notion was first introduced by Accardi and was later used in the Quantum Stochastic Calculus developed by Hudson and Parthasarathy to dilate Quantum Dynamical Semigroups with bounded generators. In the case of unbounded generators, there is a need for some additional hypotheses (see [19]) which are listed below: • (H-D) There exists a domain D which is a core for both G and G∗; • (H-R) For all u ∈ D, the image R(n; G)u by the resolvent of G, ∗ ∗ belongs to D(G ) and the sequence (nG R(n; G)u)n≥1 strongly converges. Under the hypotheses (H-D) and (H-R) it follows that the minimal quantum dynamical semigroup is connected with a unique contractive cocycle (see e.g. [19]) V = (Vt; t ≥ 0) through the relationship ∗ Tt(X) = EVtXVt E (6.2) and the equation: ∞ ! X ∗ † ∗ dVt = Vt [LkdAk(t) − LkdAk(t)] + G dt , (6.3) k=1 † in D, where (Ak; k ≥ 1), (Ak; k ≥ 1) denote annihilation and creation operators (see e.g. [38]) defined by Z t Ak(t)e(f) = ( fk(s)ds)e(f), (6.4) 0 d A† (t) = e(f + z 1 )| , (6.5) k d k [0,t] =0 2 for all f ∈ L (R+, K), for which fk denotes its k–th component, (t ≥ 0, k ≥ 1). ∗ The dual cocycle Ve = (Vet; t ≥ 0), is given by Vet = RtVt Rt , where Rt denotes the unitary time reversal operator on the Fock space, (t ≥ 0), (see eg. [35], ch.VI, 4.9). In addition, the equation satisfied by Ve is obtained by ∗ replacing Lk by −Lk and G by G in equation (6.3). This dual cocycle is associated to another semigroup Te as we did before: ∗ Tet(X) = EVetXVet E, (t ≥ 0). (6.6) The property of preservation of the identity by the semigroup and that of being an isometry for the cocycle, are related as it follows from a known result ([19], Th.5.3) quoted below for further easy reference. 42 Franco Fagnola ∗ , Rolando Rebolledo∗

Proposition 6.3. Under the hypotheses (H-min), (H-D), (H-R), the cocycle V (respectively the dual cocycle Ve) is an isometry if and only if the semigroup Te (resp. T ) preserves the identity. From now on we further assume,

• (H-Markov2) Both semigroups, T and Te, preserve the identity. Deﬁnition 6.4. We say that a quantum dynamical semigroup is natural if it satisﬁes hypotheses (H-min), (H-D), (H-R), (H-Markov2).

6.3 Main results We begin by establishing a property of the space N (T ). Proposition 6.5. Given a natural quantum Markov semigroup, the space N (T ) is a von Neumann algebra contained in the generalized commutator ∗ 0 algebra of L = (Lk,Lk; k ≥ 1), denoted by L . Proof. Let X be any element in N (T ). Then for all ﬁxed t ≥ 0,

∗ ∗ Tt(X X) = Tt(X )Tt(X). (6.7)

We introduce the quantum ﬂow jt associated to the cocycle V through ∗ the relationship jt(X) = VtXVt . This is an homomorphism and we obtain ∗ ∗ jt(X X) = jt(X )jt(X), which yields to

∗ ∗ Ejt(X X)E = Ejt(X )jt(X)E. (6.8)

The left hand side of both (6.8) and (6.7) are equivalent. Moreover, the ∗ right–hand side of (6.7) can be written Ejt(X )Ejt(X)E, and it follows

∗ ∗ Ejt(X )Ejt(X) = Ejt(X )jt(X)E. (6.9)

Let E⊥ denote the projection orthogonal to E, since E + E⊥ is equal to the identity of H, it yields

∗ ∗ ⊥ Ejt(X )Ejt(X) = Ejt(X )(E + E )jt(X)E, so that ∗ ⊥ Ejt(X )E jt(X)E = 0, (6.10) ⊥ that is, the operator E jt(X)E is null. Replacing X by X∗ in (6.7), (6.8), (6.9), (6.10), one obtains similarly ⊥ ∗ that E jt(X )E is null. 2 Hence, for any f in L (R+; K), u, v ∈ h, and all t ≥ 0:

0 = hvRtf, jt(X)ue(0)i = hVetvf, XVetue(0)i, 1. Quantum Markov Semigroups and their stationary states 43 and the equation for Ve yields Z t 0 = [hVesGvf, XVesue(0)i + hVesvf, XVesGue(0)i 0 d X + hVesLkvf, XVesLkue(0)i]ds k=1 d Z t X ∗ ¯ + [hVesve(0),XVesLkue(0)i − hVesLkve(0),XVesue(0)i]fk(s)ds, k=1 0 where we have adopted the notation fk for the eventually zero components of f according to the basis (zk; k ≥ 1). In particular, if a continuous f is chosen and the derivative of both members of the above equation is performed:

0 = hVetGvf, XVetue(0)i + hVetvf, XVetGue(0)i d X + hVetLkvf, XVetLkue(0)i k=1 d X ∗ ¯ + hVetve(0),XVetLkue(0)i − hVetLkve(0),XVetue(0)i]fk(t). k=1

Now, for all k ≥ 1 ﬁxed, we choose a function f such that fk(0) 6= 0, and make t → 0. Then the previous equation becomes

∗ hv, XLkui − hLkv, Xui = 0 (6.11)

From (6.11) we ﬁrst deduce that Xu ∈ D(Lk) for all k ≥ 1, u ∈ D, thus, for all u ∈ D(G). In addition,

hv, XLkui = hv, LkXui, (6.12) from which we obtain for all k ≥ 1,

XLk ⊆ LkX, since Lk is the closure of its restriction to the domain D(G). ∗ ∗ Similarly, X Lk ⊆ LkX is proven and it follows (see e.g. [49], Th.3, p.195): ∗ ∗ ∗ ∗ ∗ ∗ LkX = (X Lk) ⊇ (LkX ) ⊇ XLk for all k ≥ 1 and the proof is over.

It is an interesting problem to determine under which conditions the equality of N (T ) and L0 holds. To answer this question, we ﬁrst notice 44 Franco Fagnola ∗ , Rolando Rebolledo∗ that from the equality L(I) = 0 it follows that G can be decomposed as a sum ∞ 1 X G = − L∗L − iH, 2 k k k=1 where the series weakly converges in D and H is a symmetric operator deﬁned on that domain. Theorem 6.6. A natural quantum Markov semigroup converges in the sense that ∗ w − lim Tt(X) = T∞(X), (6.13) t→∞ for all X ∈ B(h), whenever the generalized commutator L0 is reduced to the trivial algebra CI. Proof. This result follows straightforward from the Theorem of Frigerio and Verri, since N (T ) is contained in L0.

Proposition 6.7. Under the above hypotheses, if in addition the closure of H is self–adjoint, then the set A(T ) of ﬁxed elements for the semigroup is given by ∗ 0 A(T ) = {Lk,Lk,H; k ≥ 1} (6.14) Proof. Since A(T ) ⊂ N (T ), and N (T ) is contained in L0, therefore A(T ) ⊂ L0. So that, for all X ∈ A(T ) and any u, v ∈ D it holds: 0 = −L(X)(v, u) ∞ ∞ 1 X 1 X = − hL∗L v, Xui + hL v, XL ui 2 k k 2 k k k=1 k=1 ∞ ∞ 1 X 1 X + hL v, XL ui − hv, XL∗L ui 2 k k 2 k k k=1 k=1 − hiHv, Xui − hv, iXHui.

We now study the right hand side of the above equation. Since XLk ⊆ LkX, the ﬁrst two terms cancel and the computation

∗ ∗ ∗ ∗ hLkv, XLkui = hX Lkv, Lkui = hLkX v, Lkui = hX v, LkLkui, shows that the third and fourth terms cancel as well. From the above we deduce XH ⊆ HX. Therefore, X belongs to

∗ 0 {Lk,Lk,H; k ≥ 1} .

∗ 0 Reciprocally, if X ∈ {Lk,Lk,H; k ≥ 1} , the equation for −L(X)(v, u) gives 0 and from Lemma 6.1 we obtain that X is a ﬁxed point of T . 1. Quantum Markov Semigroups and their stationary states 45

Proposition 6.8. For any natural quantum Markov semigroup semigroup which satisﬁes either

(a) H is bounded; or

(b) H is selfadjoint and eitH (D) ⊂ D(G), it holds, ∗ 0 N (T ) = {Lk,Lk; k ≥ 1} . (6.15) Proof. Consider ﬁrst the case (a) of a bounded operator H. Then X ∈ D(L), and −L(X)(v, u) = hv, i(H∗X − XH)ui, (6.16) so that, iHt −iHt Tt(X) = e Xe . (6.17)

Since L0 is a ∗–algebra, it contains both X∗ and X∗X. Then (6.17) holds as well for those elements, and

∗ iHt ∗ −iHt ∗ Tt(X )Tt(X) = e X Xe = Tt(X X), for all t ≥ 0. Therefore, X ∈ N (T ). Now, in case we have hypothesis (b), H being self-adjoint and exp(itH)(D) ⊆ D(G), take u, v ∈ D. The algebra N (T ) is invariant under the action of the semigroup by its own deﬁnition. Then,

d hv, e−isH T (X)eisH ui = hiHeisH v, T (X)eisH ui ds t−s t−s isH isH + he v, Tt−s(X)(iH)e i isH isH − −L(Tt−s(X))(e v, e u) = 0.

It is worth noticing that the hypothesis (b) on the core D implies that eitH u ∈ D(H). From the above equations it holds that

−isH isH hv, e Tt−s(X)e ui is constant in s. Therefore X ∈ N (T ).

The corollary which follows is easily derived from the propositions and theorem before. Corollary 6.9. For any natural quantum Markov semigroup, the convergence ∗ 0 ∗ 0 towards the equilibrium holds if {Lk,Lk,H; k ≥ 1} = {Lk,Lk; k ≥ 1} and either 46 Franco Fagnola ∗ , Rolando Rebolledo∗

(a) H is bounded; or (b) H is selfadjoint and eitH (D) ⊆ D(G). Remark. The suﬃcient condition obtained for proving the convergence towards the equilibrium is necessary, at least for a wide class of operators H, as we state in the following theorem. Theorem 6.10. Under the above notations, given a natural quantum Markov semigroup for which H is a self–adjoint operator with pure point spectrum and either (a) H is bounded; or (b) H is selfadjoint and eitH (D) ⊆ D(G).

Then Tt(·) converges towards the equilibrium if and only if ∗ 0 ∗ 0 {Lk,Lk,H; k ≥ 1} = {Lk,Lk; k ≥ 1} . (6.18) Proof. From the corollary before, (6.18) is a suﬃcient condition for the convergence towards the equilibrium. We will prove below that it is a necessary condition as well. Indeed, the hypotheses assumed imply that itH −itH Tt(X) = e Xe , for all X ∈ N (T ). For any two diﬀerent eigenvalues λ and µ of H, choose corresponding eigenvector v, u ∈ h. Then, he−itH v, Xe−itH ui = eit(µ−λ)hv, Xui, converges when t → ∞. Therefore, hv, Xui = 0 and X commute with H. Consequently,

∗ 0 ∗ 0 {Lk,Lk,H; k ≥ 1} = {Lk,Lk; k ≥ 1} .

6.4 Applications 6.5 The asymptotic behavior of the Jaynes–Cummings model in quantum optics Here we consider again the model introduced in subsection 4.6, (see [21] and [22]), which is the quantum Markov semigroup associated to master 1. Quantum Markov Semigroups and their stationary states 47 equations in Quantum Optics. 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 coefficients G and Lk (k = 1,..., 4) are given by the expressions √ † † L1 = µa, L2 = λa ,L3 = R cos(φ aa ), √ † 4 † sin(φ aa ) 1 X ∗ L4 = Ra √ ,G = − L Lk, † 2 k aa k=1 where the parameters φ, R ≥ 0, λ < µ, specify the physical model. In this case the natural set D to choose is the domain of the number operator. Moreover, the existence of a stationary state for T has been proved in our previous article [21]. Indeed if λ < µ, then T has a stationary state given by ∞ X ρ∞ = πn|enihen| n=0 where (πn)n≥0 is the sequence defined by n √ Y λ2k + R2 sin2(φ k) π = c, π = c (n ≥ 1). 0 n µ2k k=1 where c is a suitable normalization constant. The remaining hypotheses showing that the semigroup is natural have been checked in [21] as well. Now, to verify the hypotheses of Theorem 6.6, it suffices to study the action of operators on the canonical basis (em; m ≥ 0) of h. In particular, it brings about a recurrence relationship among the elements of the basis from which it follows that her, Xemi = 0 for all element X of the generalized ∗ commutator algebra of Lk,Lk,(k = 1,..., 4). Corollary 6.11. The quantum Markov semigroup introduced before approaches the equilibrium in the sense of the w∗ topology, as t → ∞. As a trivial consequence of the above corollary, the Cesàromean of the semigroup converge in the w∗ topology. This result had been stated in [22] with a different direct proof.

6.6 A class of examples with a non-trivial ﬁxed point algebra Keeping the notations on spaces and operators of the above example, consider a quantum Markov semigroup with generator given by

L1 = α(N),Lk = 0, (k 6= 1),H = β(N), with α and β given functions, α assumed to be injective and β real–valued. So that, any faithful state which is a function of the number operator 48 Franco Fagnola ∗ , Rolando Rebolledo∗

∗ 0 is an invariant state. In addition the algebras {Lk,Lk,H; k ≥ 1} and ∗ 0 0 {Lk,Lk; k ≥ 1} coincide with {N} if and only if the support of β is included in that of α. Therefore, the hypotheses of the Corollary are satisﬁed, whenever the support of β is included in that of α and the semigroup converges towards the equilibrium.

6.7 Simple absorption and stimulated emission Now we complete the example 4.5. In this case, the reader can easily verify that ∗ 0 ∗ 0 {H,Lk,Lk; k = 1, 2} = {Lk,Lk; k = 1, 2} . Therefore, Corollary 6.12. If ξ 6= 0, the quantum model of simple absorption and stimulated emission introduced in 4.5 has a unique faithful stationary state and the quantum Markov semigroup converges towards the equilibrium, that is w∗ T∗t(ρ) −−→ ρ∞, for any state ρ where ρ∞ denotes the stationary state.

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