This is page 1 Printer: Opaque this 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 semi- groups (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 “C´atedra Presidencial en An´alisis Cualitativo de Sistemas Din´amicos Cu´anticos”, 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.