Ann. I. H. Poincaré – PR 41 (2005) 349–373 www.elsevier.com/locate/anihpb
Decoherence of quantum Markov semigroups
Rolando Rebolledo 1
Pontificia Universidad Católica de Chile, Facultad de Matemáticas, Casilla 306, Santiago 22, Chile Received 3 February 2004; received in revised form 2 December 2004; accepted 9 December 2004 Available online 1 April 2005 To the memory of Paul-André Meyer
Abstract Quantum Decoherence consists in the appearance of classical dynamics in the evolution of a quantum system. This paper focuses on the probabilistic interpretation of this phenomenon, connected with the analysis of classical reductions of a quantum Markov semigroup. 2005 Elsevier SAS. All rights reserved. Résumé La décohérence quantique correspond à l’apparition d’une dynamique classique dans l’évolution d’un système quantique. Cet article fait une interprétation probabiliste de ce phénomène tout en étudiant les réductions classiques des semigroupes Markoviens quantiques. 2005 Elsevier SAS. All rights reserved.
1. Introduction
For a number of physicists decoherence consists of the dynamical loss of coherences due to the coupled dynam- ics of an open system and its environment. For a mathematician though, this statement contains numerous unclear concepts claiming for definition. When a quantum system is closed, a single complex separable Hilbert space h is used in its description together with a self-adjoint operator H – the Hamiltonian – which is the generator of a unitary group (Ut )t∈R of operators acting on h. Thus, Ut = exp(−itH) if we assume the Planck constant h¯ = 1 for simplicity, and given any element x in the von Neumann algebra M = L(h) of all bounded linear operators = ∗ ∈ R on h, its evolution is given by a group of automorphisms αt (x) Ut xUt , t (Heisenberg picture). The group α naturally determines a group of transformations α∗ on the predual space M∗ = I1(h) of trace-class operators by = = ∗ ∈ ∈ the equation tr(σ αt (x)) tr(α∗t (σ )x), that is, α∗t (σ ) Ut σUt ,(σ M∗, x M). In particular, the Schrödinger
E-mail address: [email protected] (R. Rebolledo). 1 Research partially supported by FONDECYT grant 1030552 and CONICYT/ECOS exchange program.
0246-0203/$ – see front matter 2005 Elsevier SAS. All rights reserved. doi:10.1016/j.anihpb.2004.12.003 350 R. Rebolledo / Ann. I. H. Poincaré Ð PR 41 (2005) 349Ð373 picture which describes the evolution of states, identified here with density matrices ρ ∈ M∗ which are positive elements with tr(ρ) = 1, is ρt = α∗t (ρ). For an open system, instead, two Hilbert spaces are needed. The previous h, which we term the system space and the environment space hE, so that the total dynamics is represented on the space HT = h ⊗ hE by a unitary group of generator HT = H + HE + HI , where HE (respectively HI ) denotes the Hamiltonian of the environment (respectively, the interaction Hamiltonian). If we want to analyze the reduced dynamics on the space h, we need to know in addition the state of the environment, that is, a density matrix ρE ∈ I1(hE) has to be given at the outset. With this density matrix in hands we obtain the Schrödinger picture on h by taking a partial trace on the environment variables (denoted trE): −itHT itHT T∗t (ρ) = trE e ρ ⊗ ρE e .
M Or, accordingly, we use ρE to determine the conditional expectation E (·) onto M to obtain M itHT −itHT Tt (x) = E e x ⊗ 1E e , for x ∈ M. What we now have in hands is a semigroup structure instead of a group. Suppose that an orthonormal basis (en)n∈N is given in h so that a density matrix ρ is characterized by its components ρ(m,n) =em,ρen.Ac- cordingly, ρt (m, n) denotes the components of ρt = T∗t (ρ),(n, m ∈ N). The off-diagonal terms ρt (m, n), n = m, are called the coherences. In a rough version, decoherence consists of the disapearance of these terms as time increases, that is ρt (m, n) → 0ast →∞. So that, for a large time, the evolution of states becomes essentially de- scribed by diagonal matrices which are commutative objects ruled by a classical dynamics. To paraphrase Giulini et al. (see [20]), decoherence is related to the appearance of a classical world in Quantum Theory. The subject, from the physical point of view, is certainly much more complex than the rough picture drawn before. Several authors pointed out that a most careful analysis of involved time scales should be considered (see [33]). Indeed in numerous physical models, the generator L of the semigroup T is obtained by different limit- ing procedures leading to the so-called master equations (a quantum version of Chapman–Kolmogorov equations), via adequate renormalizations on time and space. The recent book [1] contains a systematic study of these tech- niques synthesized in the concept of stochastic limit. Thus, for some physicists, decoherence is a phenomenon which precedes the derivation of the so-called Markov approximation of the dynamics. Others, have focused their research on algebraic properties leading to a definition and main properties of decoherence in the Heisenberg pic- ture (see [4]), while another group looks for the physical causes of decoherence, mainly attributed to interactions of the system with the boundaries of the cavity in which it is contained (see [13]). The debate has been reinforced by experimental results allowing to observe the decay of coherences for a given initial coherent superposition of two pure states (see the reports of the Haroche’s group at the ENS in Paris [7], and that of the Wineland’s group in Boulder [28]). The physical problem of decoherence is undoubtedly passionating and its discussion leads directly to philo- sophical arguments about the foundations of Quantum Mechanics. We do not pretend here to enter that arena. We address a much more modest problem which can be easily stated within a well known mathematical framework, we refer to Quantum Markov Theory. We separate two aspects in the analysis of decoherence. Firstly, the question of the existence of an Abelian subalgebra, generated by a given self-adjoint operator, which remains invariant under the action of a given quantum Markov semigroup (QMS). Thus, the restriction of the semigroup to that algebra will provide a classical Markov semigroup. Secondly, we focus on the asymptotic behavior of a QMS. Our goal in that part is to analyze structure properties of the semigroup leading to a classical limit behavior. Preferred by physicists to construct mathematical models of open quantum systems, the generator will be our main tool to analyze decoherence of a quantum Markov semigroup. R. Rebolledo / Ann. I. H. Poincaré Ð PR 41 (2005) 349Ð373 351
2. Quantum Markov Semigroups
A Quantum Markov Semigroup (QMS) arises as the natural non-commutative extension of the well known concept of Markov semigroup defined on a classical probability space and represents the loss-memory evolution of a microscopic system in accordance with the quantum uncertainty principle. The roots of the theory go back to the first researches on the so called open quantum systems (for an account see [2]), and have found its main non-commutative tools in much older abstract results like the characterization of complete positive maps due to Stinespring (see [30,31]). Indeed, complete positivity contains a deep probabilistic notion expressed in the language of operator algebras. In many respects it is the core of mathematical properties of (regular versions of) conditional expectations. Thus, complete positivity appears as a keystone in the definition of a QMS. Moreover, in classical Markov Theory, topology plays a fundamental role which goes from the basic setting of the space of states up to continuity properties of the semigroup. In particular, Feller property allows to obtain stronger results on the qualitative behavior of a Markov semigroup. In the non-commutative framework, Feller property is expressed as a topological and algebraic condition. Namely, a classical semigroup satisfying the Feller property on a locally compact state space leaves invariant the algebra of continuous functions with compact support, which is a particular example of a C∗-algebra. The basic ingredients to start with a non-commutative version of Markov semigroups are then two: firstly, a ∗-algebra A, that means an algebra endowed with an involution ∗ which satisfies (a∗)∗ = a, (ab)∗ = b∗a∗, for all a,b ∈ A, in addition we assume that the algebra contains a unit 1; and secondly, we need a semigroup of completely positive maps from A to A which preserves the unit. We will give a precise meaning to this below. We remind that positive elements of the ∗-algebra are of the form a∗a,(a ∈ A). A state ϕ is a linear map ϕ : A → C such that ϕ(1) = 1, and ϕ(a∗a) 0 for all a ∈ A.
Definition 1. Let A be a ∗-algebra and P : A → A a linear map. P is completely positive if for any finite collection a1,...,an,b1,...,bn of elements of A the element ∗ ∗ ai P(bi bj )aj i,j is positive.
Throughout this paper, we will restrict our ∗-algebras to the significant cases of C∗algebras and von Neumann algebras of operators on a complex separable Hilbert space h. The symbol M will be used to denote a generic von Neumann algebra, while B will be assigned to a C∗-algebra. Moreover, we always assume that our C∗-algebra B contains a unit 1. In this case states are elements of the dual B∗ of B.Astateϕ is pure if the only positive linear functionals majorized by ϕ are of the form λϕ with 0 λ 1. For an Abelian C∗-algebra, the set of pure states coincides with that of all characters, also called spectrum of the algebra (see [5], Proposition 2.3.27, p. 62). A character ϕ of an Abelian C∗-algebra A is a state which satisfies ϕ(ab) = ϕ(a)ϕ(b), for all a,b ∈ A;thesetof all these elements is usually denoted σ(A) (for spectrum)orPA (for pure states). If M is a von Neumann algebra, its predual is denoted M∗. The predual contains in particular all the normal states. As a rule, we will only deal with normal states ϕ for which there exists a density matrix ρ, that is, a positive trace-class operator of h with unit trace, such that ϕ(a) = tr(ρa) for all a ∈ A.
Definition 2. A quantum sub-Markov semigroup, or quantum dynamical semigroup (QDS) on a ∗-algebra A which T = T has a unit 1, is a one-parameter family ( t )t∈R+ of linear maps of A into itself satisfying
(M1) T0(x) = x, for all x ∈ A; (M2) Each Tt (·) is completely positive; (M3) Tt (Ts(x)) = Tt+s(x), for all t,s 0, x ∈ A; (M4) Tt (1) 1 for all t 0. 352 R. Rebolledo / Ann. I. H. Poincaré Ð PR 41 (2005) 349Ð373
A quantum dynamical semigroup is called quantum Markov (QMS) if Tt (1) = 1 for all t 0. If A is a C∗-algebra, then a quantum dynamical semigroup is uniformly (or norm) continuous if it additionally satisfies
T − = (M5) limt→0 sup x 1 t (x) x 0.
This is a very strong continuity condition which is sometimes replaced by the so called Feller continuity condition
(M5F) limt→0 Tt (x) − x =0, for all x ∈ A.
A quantum Markov semigroup satisfying (M5F) will be called quantum Feller. If A is a von Neumann algebra, (M5) is usually replaced by the weaker condition
(M5σ ) For each x ∈ A,themapt → Tt (x) is σ -weak continuous on A, and Tt (·) is normal or σ -weak continuous.
The generator L of the semigroup T is then defined in the w∗ or σ -weak sense. That is, its domain D(L) ∗ −1 consists of elements x of the algebra for which the w -limit of t (Tt (x) − x) exists as t → 0. This limit is denoted then L(x). The predual semigroup T∗ is defined on M∗ as T∗t (ϕ)(x) = ϕ(Tt (x)) for all t 0,x∈ M,ϕ∈ M∗. Its gener- ator is denoted L∗.
It is worth noticing that the generator L is often known indirectly through sesquilinear forms and the so called Master Equations in the case M = L(h). These equations are expressed in terms of density matrices ρ ∈ I1(h), the space of trace-class operators in L(h), and they correspond to a non-commutative version of Chapman– Kolmogorov’s equations: d v,ρt u=−L∗(ρt )(v, u), dt (1) ρ0 = ρ, u, v ∈ h, ρt corresponds to the action of the predual semigroup on ρ at time t and (u, v) →−L∗(ρ)(v, u) corresponds to a sesquilinear form which is linear in u ∈ h, and antilinear in v ∈ h.
2.1. The generator in the C∗-case
A quantum Markov semigroup is norm-continuous if and only if its generator L(·) is a bounded operator on B. In [10], Christensen and Evans provided an expression for the infinitesimal generator L of a norm-continuous quantum dynamical semigroup defined on a C∗-algebra, extending previous results obtained by Lindblad [25], Gorini, Kossakowski and Sudarshan [21]. We recall their result here below. Suppose that T is a norm-continuous quantum dynamical semigroup on B and denote B the σ -weak closure of the C∗-algebra B. Then there exists a completely positive map Ψ : B → B and an operator G ∈ B such that the generator L(·) of the semigroup is given by ∗ L(x) = G x + Ψ(x)+ xG (x ∈ B). (2)
The map Ψ can be represented by means of Stinespring Theorem [31] as follows. There exists a representation (k,π)of the algebra B and a bounded operator V from h to the Hilbert space k such that ∗ Ψ(x)= V π(x)V (x ∈ B). (3) R. Rebolledo / Ann. I. H. Poincaré Ð PR 41 (2005) 349Ð373 353
Notice that Ψ(1) = V ∗V =−(G∗ + G) =−2 (G) ∈ B , where (G) denotes the real part of G, since L(1) = 0. So that, if we call H the selfadjoint operator 2−1i(G − G∗) =− (G) ∈ B , where (G) stands for the imaginary part of G, then L(·) can also be written as 1 ∗ ∗ ∗ L(x) = i[H,x]− V Vx− 2V π(x)V − xV V (x ∈ B). (4) 2 The representation of L(x) in terms of G and Ψ is certainly not unique.
2.2. The generator in the von Neumann case
Consider a von Neumann algebra M on the Hilbert space h. The representation of the generator L(·) of a norm = ∗ continuous QMS on M is then improved as follows. There exists a set of operators (Lk)k∈N such that L k LkLk ∗ ∈ ∈ = ∗ ∈ is a bounded operator in M and k LkxLk M whenever x M and a selfadjoint operator H H M such that 1 ∗ ∗ ∗ L(x) = i[H,x]− (L L x − 2L xL + xL L ). (5) 2 k k k k k k k We recover the expression (2) if we put 1 ∗ ∗ G =−iH − L L ; Ψ(x)= L xL . (6) 2 k k k k k k
2.3. The case of a form-generator
In most of applications L(·) is not known as an operator directly but through a sesquilinear form. To discuss this case we restrict ourselves to the von Neumann algebra M = L(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 [14], Section 3.3, see also [9]. 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),wehave ∞ Gv, u+ Lv,Lu+v,Gu=0. =1
Under the above assumption (H-min), for each x ∈ L(h) let −L(x) be the sesquilinear form with domain D(G) × D(G) defined by ∞ −L(x)(v, u) =Gv, xu+ Lv,xLu+v,xGu. (7) =1 It is well-known (see e.g. [11], Section 3, [14], Section 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: t v,Tt (x)u =v,xu+ −L Ts(x) (v, u) ds, (8) 0 for u, v ∈ D. 354 R. Rebolledo / Ann. I. H. Poincaré Ð PR 41 (2005) 349Ð373
This equation, however, in spite of the hypothesis (H-min) and the fact that D is a core for G, does not nec- essarily determine a unique semigroup. The minimal QDS is characterized by the following property: for any ∗ T T (min) T w -continuous family ( t )t0 of positive maps on L(h) satisfying (8) we have t (x) t (x) for all positive x ∈ L(h) and all t 0 (see e.g. [11]; [14], Theorem 3.22; [22], Theorem 3.3.3, p. 86). If the minimal QDS is T (min) = Markov, that is t (1) 1, for all t 0, then it is the unique solution to (8). One may test that condition on the formal generator through the equation −L(1)(v, u) = 0 for all u, v ∈ D. Throughout this paper we will implicitly assume that formal generators satisfy the Markov property, and we will then refer to the unique minimal solution of (8) as the minimal quantum Markov semigroup. For a discussion on sufficient conditions for this property to hold the reader is addressed to [8,22]. The assumed density of D implies that the minimal quantum Markov semigroup – which we simply denote T from now on – possess a densely defined generator L, its domain D(L being given by all elements x ∈ L(h) for which the map (u, v) → −L(x)(u, v) is norm-continuous on the product Hilbert space h × h. And, for any x ∈ D(L), L(x) is given by v,L(x)u=−L(x)(u, v), u, v ∈ h, after extending −L(x) from the dense set D × D to all of h × h by continuity. Also, Proposition 3.1.6 in [5], shows that L is w∗-closed. Moreover, the predual semigroup T∗ is now defined on the predual space of L(h) which is I1(h), the Banach space of trace-class operators: tr(T∗t (ρ)x) = tr(ρTt (x)),(ρ ∈ I1(h), x ∈ L(h)). So that, L∗(D(L∗)) ⊂ I1(h) and, as Davies proved in [11] (see also [15]), the dense set {|uv|; u, v ∈ D} is included in the domain D(L∗) of the predual semigroup generator. This predual semigroup T∗ is weakly-continuous. This property, together with complete positivity, imply that it is also a strongly continuous semigroup of bounded linear operators (a so called C0-semigroup) since I1(h) is a Banach space and Corollary 2.5 in [29] holds. As a result, L∗ is a closed operator too.
2.4. Examples of generators
2.4.1. The quantum damped harmonic oscillator In this case the Hilbert space used to represent the system is h = 2(N) with its canonical orthonormal basis † (en)n∈N ; M = L(h). We use the customary notations for annihilation (a), creation (a ) and number (N) operators. The physical model corresponds to an atom which traverses an ideal resonator (a high quality cavity), its energy can be in two levels only (a so called two-level atom). Excitations of a mode of the quantized radiation field in the resonator correspond to photons which stay in the cavity, they have a finite life-time and they interact with the incident atom. The physical description of the dynamics have been obtained by different approximation procedures (weak coupling limit, coarse graining) which end in a Master Equation containing the (formal) generator of a quan- tum Markov semigroup. Here we start from that formal generator, the reader interested in its physical derivation is addressed to any textbook on Quantum Optics, here we use the presentation of B.E. Englert and G. Morigi in page 55 of the collective book [33]. Let introduce the physical parameters: A denotes the energy decay rate in the cavity; ν, the number of thermal excitations; ω, the natural (circular) frequency. The form-generator of the semigroup is given by the (formal) expression
1 1 L(x) = i[ωN,x]− A(ν + 1)(a†ax − 2a†xa + xa†a) − Aν(aa†x − 2axa† + xaa†), (9) 2 2 for x in a dense subset of M, which is the common domain of a and a†.
2.4.2. The quantum Brownian motion Let h = L2(Rd ; C) and M = L(h). We consider here another version of the harmonic oscillator like in [27], Chapter III (see also [3]). Though this is an extension of the previous example, the dimension d plays here an important role in the analysis of ergodic properties of the semigroup. A Quantum Brownian Motion means for us a R. Rebolledo / Ann. I. H. Poincaré Ð PR 41 (2005) 349Ð373 355 quantum Markov process with associated semigroup T on M which is the minimal semigroup (see [8,14] and the references therein) with form-generator
1 d 1 d −L(x) =− (a a†x − 2a xa† + xa a†) − (a†a x − 2a†xa + xa†a ), 2 j j j j j j 2 j j j j j j j=1 j=1 † where aj ,aj are the creation and annihilation operators √ √ = + † = − aj (qj ∂j )/ 2,aj (qj ∂j )/ 2,
∂j being the partial derivative with respect to the jth coordinate qj .
2.4.3. The quantum exclusion semigroup The generator of this example is constructed via a second quantization procedure. Consider first a self-adjoint bounded operator H0 defined on a separable complex Hilbert space h0. H0 will be thought of as describing the dynamics of a single fermionic particle. We assume that there is an orthonormal basis (ψn)n∈N of eigenvectors of H0, and denote En the eigenvalue of ψn (n ∈ N). The set of all finite subsets of N is denoted Pf (N) and for any ∈ N Λ ; ∈ Λ Pf ( ), we denote h0 the finite-dimensional Hilbert subspace of h0 generated by the vectors (ψn n Λ).To deal with a system of infinite particles we introduce the fermionic Fock space h = Γf (h0) associated to h0 whose construction we recall briefly (see [6] for full detail). The Fock space associated to h0 is the direct sum = ⊗n Γ(h0) h0 , n∈N ⊗n ⊗0 = C where h0 is the n-fold tensor product of h0, with the convention h0 . Define an operator Pa on the Fock space as follows, 1 P (f ⊗ f ⊗···⊗f ) = ε f ⊗···⊗f . a 1 2 n n! π π1 πn π
The sum is over all permutations π : {1,...,n}→{π1,...,πn} of the indices and επ is 1 if π is even and −1ifπ is odd. Define the anti-symmetric tensor product on the Fock space as f1 ∧···∧fn = Pa(f1 ⊗ f2 ⊗···⊗fn).In this manner, the Fermi–Fock space h is obtained as = = ⊗n = ∧n h Γf (h0) Pa h0 h0 . n∈N n∈N We follow [6] to introduce the so-called fermionic Creation b†(f ) and Annihilation b(f ) operators on h, asso- † (0) ciated to a given element f of h0. Firstly, on Γ(h0) we define a(f) and a (f ) by initially setting a(f)ψ = 0, † (0) (0) (1) (j) a (f )ψ = f ,forψ = (ψ ,ψ ,...)∈ Γ(h0) with ψ = 0 for all j 1, and √ a†(f )(f ⊗···⊗f ) = n + 1f ⊗ f ⊗···⊗f , (10) 1 n √ 1 n a(f )(f1 ⊗···⊗fn) = nf,f1f2 ⊗ f3 ⊗···⊗fn. (11) † † Finally, define annihilation and creation on Γf (h0) as b(f ) = Paa(f)Pa and b (f ) = Paa (f )Pa. These operators satisfy the Canonical Anti-commutation Relations (CAR) on the Fermi–Fock space: b(f ),b(g) = 0 = b†(f ), b†(g) , (12) b(f ),b†(g) =f,g1, (13) for all f,g ∈ h0, where we use the notation {A,B}=AB + BA for two operators A and B. 356 R. Rebolledo / Ann. I. H. Poincaré Ð PR 41 (2005) 349Ð373
Moreover, b(f ) and b†(g) have bounded extensions to the whole space h since b(f ) = f = b†(f ) . † † To simplify notations, we write bn = b (ψm) (respectively bn = b(ψn)) the creation (respectively annihilation) operator associated with ψn in the space h0, (n ∈ N). ∗ The C -algebra generated by 1 and all the b(f ), f ∈ h0, is denoted A(h0) and it is known as the canonical CAR algebra.
∗ ∗ Remark 1. The algebra A(h0) is the unique, up to -isomorphism, C -algebra generated by elements b(f ) satis- fying the anti-commutation relations over h0 (see e.g. [6], Theorem 5.2.5).
† Remark 2. It is worth mentioning that the family (b(f ), b (g); f,g ∈ h0) is irreducible on h, that is, the only operators which commute with this family are the scalar multiples of the identity ([6], Proposition 5.2.2). Clearly, † the same property is satisfied by the family (bn,bn; n ∈ N), since (ψn)n∈N is an orthonormal basis of h0. Remark 3. The algebra A(h ) is the strong closure of D = A(hΛ) (see [6], Proposition 5.2.6), this is 0 Λ∈Pf (N) 0 Λ the quasi-local property. Moreover, the finite dimensional algebras A(h0 ) are isomorphic to algebras of matrices with complex components.
An element η of {0, 1}N will be called a configuration of particles. For each n, η(n) will take the value 1 or 0 depending on whether the n-th site has been occupied by a particle in the configuration η. In other terms, we say that the site n is occupied by the configuration η if η(n) = 1. We denote S the set of configurations η with a finite ∞ ∈ number of 1’s, that is n η(n) < . Each η S is then identifiable to the characteristic function 1{s1,...,sm} of a finite subset of N, which, in addition, we will suppose ordered as 0 s1