DECOHERENCE OF QUANTUM MARKOVSEMIGROUPS

ROLANDO REBOLLEDO

To the memory of Paul-Andre Meyer

Abstract. Quantum Decoherence consists in the appearance ofclassical dynamics in the evolution of a quantum system.

This paper focuses on the probabilistic interpretation of thisphenomenon which leads to the analysis of classical reductions ofa Quantum Markov semigroup.

1. Introduction

For a number of physicists decoherence consists of the dynamicalloss of coherences due to the coupled dynamics of an open system andits environment. For a mathematician though, this statement containsnumerous unclear concepts claiming for definition. When a quantumsystem is closed, a single complex separable Hilbert space h is used in itsdescription together with a self-adjoint operator H -the Hamiltonian-which is the generator of a unitary group (Ut)t∈R of operators actingon h. Thus, Ut = exp(−itH) if we assume the Planck constant ~ = 1for simplicity, and given any element x in the von Neumann algebraM = L(h) of all bounded linear operators on h, its evolution is given bya group of automorphisms αt(x) = U∗

t xUt, t ∈ R (Heisenberg picture).The group α naturally determines a group of transformations α∗ onthe predual space M∗ = I1(h) of trace-class operators by the equationtr(σαt(x)) = tr(α∗t(σ)x), that is, α∗t(σ) = UtσU

∗t , (σ ∈ M∗, x ∈ M).

In particular, the Schrodinger picture which describes the evolution ofstates, identified here with density matrices ρ ∈ M∗ which are positiveelements with tr(ρ) = 1, is ρt = α∗t(ρ).

For an open system, instead, two Hilbert spaces are needed. The pre-vious 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 aunitary group of generator HT = H+HE +HI , where HE (respectively

Research partially supported by FONDECYT grant 1030552 and CONI-CYT/ECOS exchange program.

1

2 ROLANDO REBOLLEDO

HI) denotes the Hamiltonian of the environment (respectively, the in-teraction Hamiltonian). If we want to analyze the reduced dynamics onthe 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 Schrodinger pictureon h by taking a partial trace on the environment variables (denotedtrE):

T∗t(ρ) = trE

(e−itHT ρ⊗ ρEe

itHT).

Or, accordingly, we use ρE to determine the conditional expectationEM (·) onto M to obtain

Tt(x) = EM(eitHTx⊗ 1Ee

−itHT),

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 sothat a density matrix ρ is characterized by its components ρ(m,n) =〈em, ρen〉. Accordingly, ρt(m,n) denotes the components of ρt = T∗t(ρ),(n,m ∈ N). The off-diagonal terms ρt(m,n), n 6= m, are called the co-herences. In a rough version, decoherence consists of the dissapearanceof these terms as time increases, that is ρt(m,n) → 0 as t → ∞. Sothat, for a large time, the evolution of states becomes essentially de-scribed by diagonal matrices which are commutative objects ruled by aclassical dynamics. To paraphrase Giulini et al. (see [23]), decoherenceis related to the appearance of a classical world in Quantum Theory.

The subject, from the physical point of view, is certainly much morecomplex than the rough picture drawn before. Several authors pointedout that a most careful analysis of involved time scales should be con-sidered (see [34]). Indeed in numerous physical models, the generator Lof the semigroup T is obtained by different limiting procedures leadingto the so-called master equations (a quantum version of Chapman-Kolmogorov equations), via adequate renormalizations on time andspace. The recent book [1] contains a systematic study of these tech-niques synthesized in the concept of stochastic limit. Thus, for somephysicists, decoherence is a phenomenon which precedes the deriva-tion of the so-called Markov approximation of the dynamics. Others,have focused their research on algebraic properties leading to a defini-tion and main properties of decoherence in the Heisenberg picture (see[6]), while another group looks for the physical causes of decoherence,mainly attributed to interactions of the system with the boundariesof the cavity in which it is contained (see [15]). The debate has beenreinforced by experimental results allowing to observe the decay of co-herences for a given initial coherent superposition of two pure states

DECOHERENCE OF QUANTUM MARKOV SEMIGROUPS 3

(see the reports of the Haroche’s group at the ENS in Paris [9], andthat of the Wineland’s group in Boulder [29]).

The physical problem of decoherence is undoubtedly passionatingand its discussion leads directly to philosophical arguments about thefoundations of Quantum Mechanics. We do not pretend here to enterthat arena. We address a much more modest problem which can beeasily stated within a well known mathematical framework, we refer toQuantum Markov Theory.

We separate two apects in the analysis of decoherence. Firstly, thequestion of the existence of an abelian subalgebra, generated by a givenself-adjoint operator, which remains invariant under the action of agiven quantum Markov semigroup (QMS). Thus, the restriction of thesemigroup to that algebra will provide a classical Markov semigroup.Secondly, we focus on the asymptotic behavior of a QMS. Our goal inthat part is to analyze structure properties of the semigroup leading toa classical limit behavior. Preferred by physicists to construct math-ematical models of open quantum systems, the generator will be ourmain tool to analyze decoherence of a quantum Markov semigroup.

2. Quantum Markov Semigroups

A Quantum Markov Semigroup (QMS) arises as the natural noncommutative extension of the well known concept of Markov semigroupdefined on a classical probability space and represents the loss-memoryevolution of a microscopic system in accordance with the quantumuncertainty principle. The roots of the theory go back to the first re-searches on the so called open quantum systems (for an account see[2]), and have found its main non commutative tools in much olderabstract results like the characterization of complete positive maps dueto Stinespring (see [31]). Indeed, complete positivity contains a deepprobabilistic notion expressed in the language of operator algebras. Inmany respects it is the core of mathematical properties of (regular ver-sions of) conditional expectations. Thus, complete positivity appearsas a keystone in the definition of a QMS. Moreover, in classical MarkovTheory, topology plays a fundamental role which goes from the basicsetting of the space of states up to continuity properties of the semi-group. In particular, Feller property allows to obtain stronger resultson the qualitative behavior of a Markov semigroup. In the non com-mutative framework, Feller property is expressed as a topological andalgebraic condition. Namely, a classical semigroup satisfying the Fellerproperty on a locally compact state space leaves invariant the algebraof continuous functions with compact support, which is a particular

4 ROLANDO REBOLLEDO

example of a C∗-algebra. The basic ingredients to start with a noncommutative version of Markov semigroups are then two: firstly, a ∗-algebra A, that means an algebra endowed with an involution ∗ whichsatisfies (a∗)∗ = a, (ab)∗ = b∗a∗, for all a, b ∈ A, in addition we assumethat the algebra contains a unit 1; and secondly, we need a semigroupof completely positive maps from A to A which preserves the unit. Wewill give a precise meaning to this below. We remind that positiveelements of the ∗-algebra are of the form a∗a, (a ∈ A). A state ϕ isa linear map ϕ : A → C such that ϕ(1) = 1, and ϕ(a∗a) ≥ 0 for alla ∈ A.

Definition 1. Let A be a ∗-algebra and P : A → A a linear map. P iscompletely positive if for any finite collection a1, . . . , an, b1, . . . , bn ofelements of A the element ∑

i,j

ai∗P(bi

∗bj)aj

is positive.

Throughout this paper, we will restrict our ∗-algebras to the signif-icant cases of C∗algebras and von Neumann algebras of operators ona complex separable Hilbert space h. The symbol M will be used todenote a generic von Neumann algebra, while B will be assigned to aC∗-algebra. Moreover, we always assume that our C∗-algebra B con-tains a unit 1. In this case states are elements of the dual B∗ of B.A state ϕ is pure if the only positive linear functionals majorized byϕ 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 calledspectrum of the algebra (see [7], Prop. 2.3.27, p. 62). A character ϕof an abelian C∗-algebra A is a state which satisfies ϕ(ab) = ϕ(a)ϕ(b),for all a, b ∈ A; the set of all these elements is usually denoted σ(A)(for spectrum) or PA (for pure states).

If M is a von Neumann algebra, its predual is denoted M∗. Thepredual contains in particular all the normal states. As a rule, we willonly deal with normal states ϕ for which there exists a density matrixρ, that is, a positive trace-class operator of h with unit trace, such thatϕ(a) = tr(ρ a) for all a ∈ A.

Definition 2. A quantum sub-Markov semigroup, or quantum dy-namical semigroup (QDS) on a ∗-algebra A which has a unit 1, is aone-parameter family T = (Tt)t∈R+

of linear maps of A into itself sat-isfying

(M1) T0(x) = x, for all x ∈ A;

DECOHERENCE OF QUANTUM MARKOV SEMIGROUPS 5

(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.

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 uni-formly (or norm) continuous if it additionally satisfies

(M5) limt→0 sup‖x‖≤1 ‖Tt(x)− x‖ = 0.

This is a very strong continuity condition which is sometimes replacedby 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 quantumFeller.

If A is a von Neumann algebra, (M5) is usually replaced by theweaker condition

(M5σ) For each x ∈ A, the map t 7→ 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∗ orσ-weak sense. That is, its domain D(L) consists of elements x of thealgebra for which the w∗-limit of t−1(Tt(x) − x) exists as t → 0. Thislimit 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 generator is denoted L∗.

It is worth noticing that the generator L is often known indirectlythrough sesquilinear forms and the so called Master Equations in thecase M = L(h). These equations are expressed in terms of densitymatrices ρ ∈ I1(h), the space of trace-class operators in L(h), and theycorrespond to a non-commutative version of Chapman-Kolmogorov’sequations:

(1)

{ddt〈v, ρtu〉 = L∗− (ρt)(v, u),

ρ0 = ρ,

u, v ∈ h, ρt corresponds to the action of the predual semigroup on ρat time t and (u, v) 7→ L∗− (ρ)(v, u) corresponds to a sesquilinear formwhich is linear in u ∈ h, and antilinear in v ∈ h.

2.1. The generator in the C∗-case. A quantum Markov semigroupis norm-continuous if and only if its generator L(·) is a bounded oper-ator on B.

6 ROLANDO REBOLLEDO

In [10], Christensen and Evans provided an expression for the infin-itesimal generator L of a norm-continuous quantum dynamical semi-group defined on a C∗-algebra, extending previous results obtained byLindblad, Gorini, Kossakowski, Sudarshan. We recall their result herebelow.

Suppose that T is a norm-continuous quantum dynamical semigroupon B and denote B the σ-weak closure of the C∗-algebra B. Then thereexists a completely positive map Ψ : B → B and an operator G ∈ Bsuch that the generator L(·) of the semigroup is given by

(2) L(x) = G∗x+ Ψ(x) + xG, (x ∈ B).

The map Ψ can be represented by means of Stinespring Theorem[31] as follows. There exists a representation (k, π) of the algebra Band a bounded operator V from h to the Hilbert space k such that

(3) Ψ(x) = V ∗π(x)V, (x ∈ B).

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 theselfadjoint operator 2−1i(G − G∗) = −=(G) ∈ B, where =(G) standsfor the imaginary part of G, then L(·) can also be written as

(4) L(x) = i[H, x]− 1

2(V ∗V x− 2V ∗π(x)V − xV ∗V ) , (x ∈ B).

The representation of L(x) in terms of G and Ψ is certainly notunique.

2.2. The generator in the von Neumann case. Consider a vonNeumann algebra M on the Hilbert space h. The representation of thegenerator L(·) of a norm continuous QMS on M is then improved asfollows. There exists a set of operators (Lk)k∈N such that L =

∑k L

∗kLk

is a bounded operator in M and∑

k L∗kxLk ∈ M whenever x ∈ M and

a selfadjoint operator H = H∗ ∈ M such that

(5) L(x) = i[H, x]− 1

2

∑k

(Lk∗Lkx− 2Lk

∗xLk + xLk∗Lk) .

We recover the expression (2) if we put

(6) G = −iH − 1

2

∑k

L∗kLk; Ψ(x) =∑

k

L∗kxLk.

DECOHERENCE OF QUANTUM MARKOV SEMIGROUPS 7

2.3. The case of a form-generator. In most of applications L(·) isnot known as an operator directly but through a sesquilinear form. Todiscuss this case we restrict ourselves to the von Neumann algebra M =L(h) and we rephrase, for easier reference the crucial result which allowsto construct a quantum dynamical semigroup starting from a generatorgiven as a sesquilinear form. For further details on this matter we referto [16], section 3.3, see also [12].

Let G and L`, (` ≥ 1) be operators in h which satisfy the followinghypothesis:

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

〈Gv, u〉+∞∑

`=1

〈L`v, L`u〉+ 〈v,Gu〉 = 0.

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

(7) L−(x)(v, u) = 〈Gv, xu〉+∞∑

`=1

〈L`v, xL`u〉+ 〈v, xGu〉.

It is well-known (see e.g. [13] Sect.3, [16] Sect. 3.3) that, given adomain D ⊆ D(G), which is a core for G, it is possible to built up aquantum dynamical semigroup, called the minimal QDS, satisfying theequation:

(8) 〈v, Tt(x)u〉 = 〈v, xu〉+

∫ t

0

L−(Ts(x))(v, u)ds,

for u, v ∈ D.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 uniquesemigroup. The minimal QDS is characterized by the following prop-erty: for any w∗-continuous family (Tt)t≥0 of positive maps on L(h)

satisfying (8) we have T (min)t (x) ≤ Tt(x) for all positive x ∈ L(h) and

all t ≥ 0 (see e.g. [16] Th. 3.22).

2.4. Examples of generators.

2.4.1. The quantum damped harmonic oscillator. In this case the Hilbertspace used to represent the system is h = `2(N) with its canonical or-thonormal basis (en)n∈N ; M = L(h). We use the customary notationsfor annihilation (a), creation (a†) and number (N) operators. The

8 ROLANDO REBOLLEDO

physical model corresponds to an atom which traverses an ideal res-onator (a high quality cavity), its energy can be in two levels only (aso called two-level atom). Excitations of a mode of the quantized ra-diation field in the resonator correspond to photons which stay in thecavity, they have a finite life-time and they interact with the incidentatom. The physical description of the dynamics have been obtained bydifferent approximation procedures (weak coupling limit, coarse grain-ing) which end in a Master Equation containing the (formal) generatorof a quantum Markov semigroup. Here we start from that formal gen-erator, the reader interested in its physical derivation is addressed toany textbook on Quantum Optics, here we use the presentation ofB.E. Englert and G. Morigi in page 55 of the collective book [34]. Letintroduce the physical parameters: A denotes the energy decay rate inthe cavity; ν, the number of thermal excitations; ω, the natural (cir-cular) frequency. The form-generator of the semigroup is given by the(formal) expression

L(x) = i[ωN, x]− 1

2A(ν + 1)

(a†ax− 2a†xa + xa†a

)− 1

2Aν(aa†x− 2axa† + xaa†

),(9)

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 likein [28], Ch. III. Though this is an extension of the previous example,the dimension d plays here an important role in the analysis of ergodicproperties of the semigroup. A Quantum Brownian Motion means forus a quantum Markov process with associated semigroup T on M whichis the minimal semigroup (see [11], [16] and the references therein) withform-generator

L−(x) = −1

2

d∑j=1

(aja

†jx− 2ajxa

†j + xaja

†j

)−1

2

d∑j=1

(a†jajx− 2a†jxaj + xa†jaj

),

where a†j, aj are the creation and annihilation operators

aj = (qj + ∂j) /√

2, a†j = (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 exam-ple is constructed via a second quantization procedure. Consider firsta self-adjoint bounded operator H0 defined on a separable complex

DECOHERENCE OF QUANTUM MARKOV SEMIGROUPS 9

Hilbert space h0. H0 will be thought of as describing the dynamics ofa single fermionic particle. We assume that there is an orthonormalbasis (ψn)n∈N of eigenvectors of H0, and denote En the eigenvalue ofψn (n ∈ N). The set of all finite subsets of N is denoted Pf (N) and forany Λ ∈ Pf (N), we denote hΛ

0 the finite-dimensional Hilbert subspaceof h0 generated by the vectors (ψn; n ∈ Λ). To deal with a systemof infinite particles we introduce the fermionic Fock space h = Γf (h0)associated to h0 whose construction we recall briefly (see [8] for fulldetail).

The Fock space associated to h0 is the direct sum

Γ(h0) =⊕n∈N

h⊗n0 ,

where h⊗n0 is the n-fold tensor product of h0, with the convention h⊗0

0 =C. Define an operator Pa on the Fock space as follows,

Pa(f1 ⊗ f2 ⊗ . . .⊗ fn) =1

n!

∑π

επfπ1 ⊗ . . .⊗ fπn .

The sum is over all permutations π : {1, . . . , n} → {π1, . . . , πn} ofthe indices and επ is 1 if π is even and −1 if π is odd. Define theanti-symmetric tensor product on the Fock space as f1 ∧ . . . ∧ fn =Pa(f1 ⊗ f2 ⊗ . . . ⊗ fn). In this manner, the Fermi-Fock space h isobtained as

h = Γf (h0) = Pa

(⊕n∈N

h⊗n0

)=⊕n∈N

h∧n0 .

We follow [8] to introduce the so-called fermionic Creation b†(f) andAnnihilation b(f) operators on h, associated 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, a†(f)ψ(0) = f , for ψ = (ψ(0), ψ(1), . . .) ∈ Γ(h0) with ψ(j) = 0 for allj ≥ 1, and

a†(f)(f1 ⊗ . . .⊗ fn) =√n+ 1f ⊗ f1 ⊗ . . .⊗ fn.(10)

a(f)(f1 ⊗ . . .⊗ fn) =√n〈f, f1〉f2 ⊗ 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, g〉1,(13)

for all f, g ∈ h0, where we use the notation {A,B} = AB+BA for twooperators A and B.

10 ROLANDO REBOLLEDO

Moreover, b(f) and b†(g) have bounded extensions to the whole spaceh since ‖b(f)‖ = ‖f‖ =

∥∥b†(f)∥∥.

To simplify notations, we write b†n = b†(ψm) (respectively bn = b(ψn))the creation (respectively annihilation) operator associated with ψn inthe space h0, (n ∈ N).

The C∗-algebra generated by 1 and all the b(f), f ∈ h0, is denotedA(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) satisfying the anti-commutationrelations over h0 (see e.g. [8], 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 withthis family are the scalar multiples of the identity ([8], Prop.5.2.2).Clearly, the same property is satisfied by the family (bn, b

†n; n ∈ N),

since (ψn)n∈N is an orthonormal basis of h0.

Remark 3. The algebra A(h0) is the strong closure of D =⋃

Λ∈Pf (N) A(hΛ0 )

(see [8], Proposition 5.2.6), this is the quasi-local property. Moreover,the finite dimensional algebras A(hΛ

0 ) are isomorphic to algebras ofmatrices 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 then-th site has been occupied by a particle in the configuration η. Inother terms, we say that the site n is occupied by the configurationη if η(n) = 1. We denote S the set of configurations η with a finitenumber of 1’s, that is

∑n η(n) =

∑n η

2(n) < ∞. Each η ∈ S is thenidentifiable to the characteristic function 1{s1,...,sm} of a finite subsetof N, which, in addition, we will suppose ordered as 0 ≤ s1 < s2 <. . . < sm. For simplicity we write 1k the configuration 1{k}, (k ∈ N).Furthermore, we define

b† (η) = b†smb†sm−1

. . . b†s1(14)

b (η) = bsmbsm−1 . . . bs1 ,(15)

for all η = 1{s1,...,sm}. Clearly, b† (1k) = b†k, b (1k) = bk, (k ∈ N).To obtain a cyclic representation of A(h0) we call |0〉 the vacuum

vector in h, and |η〉 = b† (η) |0〉, (η ∈ S). Then (|η〉, η ∈ S) isan orthonormal basis of h. In this manner, any x ∈ A(h0) can berepresented as an operator in L(h). Moreover, call v the vector spacespanned by (|η〉, η ∈ S).

DECOHERENCE OF QUANTUM MARKOV SEMIGROUPS 11

An elementary computation based on the C.A.R. shows that for anyη, ζ ∈ S, it holds

b†k|η〉 = (1− η(k))|η + 1k〉,(16)

bk|η〉 = η(k)|η − 1k〉, (k ∈ N).(17)

We assume in addition that

(18)∑

n

|En| <∞.

Then, the second quantization of H0 becomes a self-adjoint operatorH acting on h, with domain D(H) which includes v and can formallybe written as

(19) H =∑

n

Enb†nbn.

It is worth mentioning that the restriction HΛ =∑

n∈ΛEnb†nbn of H

to each space Γf (hΛ0 ) is an element of the algebra A(hΛ

0 ), Λ ∈ Pf (N),so that HΛ is a bounded operator, due to condition (18).

The transport of a particle from a site i to a site j, at a rate γi,j isdescribed by an operator Li,j defined as

(20) Li,j =√γi,j b

†jbi.

This corresponds to the action of a reservoir on the system of fermionicparticles pushing them to jump between different sites. Each operatorLi,j is an element of A(h0) and ‖Li,j‖ =

√γi,j. We additionnally assume

that

(21) supi

∑j

γi,j <∞.

Now, for each Λ ∈ Pf (N) and x ∈ A(h0), define

ϕΛ(x) =∑i,j∈Λ

Li,j∗xLi,j.

12 ROLANDO REBOLLEDO

ϕΛ : A(h0) → A(h0) is a completely positive map. Moreover, for eachvector |η〉 of the orthonormal basis in h,

‖ϕΛ(x)|η〉‖ ≤∑i,j∈Λ

‖Li,j∗xLi,j|η〉‖

=∑i,j∈Λ

η(i)(1− η(j))√γi,j ‖Li,j

∗x|η − 1i + 1j〉‖

≤∑i,j∈Λ

η(i)(1− η(j))γi,j ‖x‖

≤

(sup

i

∑j

γi,j

)‖x‖ .

So that ‖ϕΛ(x)|η〉‖ is uniformly bounded as Λ run over Pf (N). More-over, ∑

i,j

‖Li,j∗xLi,j|η〉‖ <∞

so that ‖(ϕΛ(x)− ϕ(x))u‖ → 0 for all u ∈ v as Λ ↑ N, where theoperator

ϕ(x) =∑i,j

Li,j∗xLi,j,

is defined on the dense manifold v for all x ∈ A(h0).As a result, ϕΛ(1) converges in the same sense to ϕ(1) =

∑i,j Li,j

∗Li,j.To summarize, the generator of the QMS in this case is given in the

form (5):

(22) L(x) = i[H, x]− 1

2

∑i,j

(Li,j∗Li,jx− 2Li,j

∗xLi,j + xLi,j∗Li,j) ,

with H and Li,j introduced in (19), (20), x ∈ A(h0) and v is a coredomain for L(x).

3. The appearance of Classical Markov Semigroups

Given a quantum Markov semigroup T and a self-adjoint operatorK, this section studies different conditions on the generator of thesemigroup to leave invariant the abelian algebra generated by K.

Definition 3. We say that a completely positive map P defined on Bis reduced by an abelian ∗-subalgebra A if A ⊆ B is invariant underthe action of P.

DECOHERENCE OF QUANTUM MARKOV SEMIGROUPS 13

Analogously, a quantum dynamical (resp. Markov) semigroup (Tt)t∈R+

defined on B is reduced by A if A ⊆ B is invariant under the actionof Tt for all t ≥ 0.

We will simply say that the semigroup is reduced by a normal operatorK when it is reduced by the algebra generated by K.

Remark 4. Suppose that P is a completely positive map defined onB such that P(1) = 1. If P is reduced by an abelian sub C∗-algebra Aof the algebra B, then its restriction P to A defines a norm-continuouskernel.

Indeed, since A is an abelian C∗-algebra which contains the unit, itsset of characters σ(A) is a w∗-compact Hausdorff space (see [7], Thm.2.1.11A, p.62).

A is isomorphic to the algebra of continuous functions C(σ(A)) viathe Gelfand transform: a 7→ a, where a(γ) = γ(a), for all a ∈ A,γ ∈ σ(A).

Define Pa = P(a). Then P : C(σ(A)) → C(σ(A)) is linear, positiveand continuous in norm since for all a ∈ A:

‖Pa‖ =∥∥∥P(a)

∥∥∥ ≤ ‖a‖ = ‖a‖ .

Therefore, by the disintegration of measures property (see [14]), thereexists a kernel P : σ(A) × B(σ(A)) → [0, 1] such that P (ψ, ·) is a(Radon) probability measure for all ψ ∈ σ(A), P (·, A) is a continuousfunction, for all A ∈ B(A) and

Pa(ψ) =

∫σ(A)

P (ψ, dϕ)ϕ(a).

Remark 5. It follows immediately from the previous remark that if aFeller quantum Markov semigroup T (in particular, a norm-continuoussemigroup) defined on B is reduced by an abelian C∗-subalgebra A ofB, there exists a classical Feller semigroup which is isomorphic to therestriction of T to A, called the reduced semigroup.

Given a ∗- abelian subalgebra A ⊆ L(h), it is included in its commu-tant A′ which is a von Neumann algebra. A is maximal if A = A′, andin that case A becomes a von Neumann algebra too.

Remark 6. Suppose that T is a quantum Markov semigroup definedon L(h) which is reduced by a maximal abelian von Neumann subalge-bra A. Then there exists a compact Hausdorff space E endowed with aRadon measure µ such that the restriction of T to A is ∗-isomorphic toa classical Markov semigroup (Tt)t∈R+

on L∞(E, µ). Moreover, if the

semigroup T is quantum Feller, then (Tt)t∈R+is a Feller semigroup.

14 ROLANDO REBOLLEDO

The above remark follows directly from the Spectral Theorem: SinceA is maximal abelian, there exists a triple (E, µ, U), where E is acompact second countable Hausdorff space, µ a Radon measure on Eand U is an isometry from L2(E, µ) onto h. E is in fact the space ofcharacters of A which is w∗-compact.

So that U : L∞(E, µ) → A, defined by U(f) = UMfU∗, where

f ∈ L∞(E, µ) and Mf denotes the multiplication operator by f inL2(E, µ), is an isometric ∗-isomorphism of algebras.

A semigroup (Tt)t∈R+is defined on L∞(E, µ) through the relation

(23) MTtf = U∗Tt(UMfU∗)U.,

for all f ∈ L∞(E, µ).The semigroup (Tt)t∈R+

preserves the identity, since U is an isometry.

Moreover, ‖Tt(x)‖ ≤ ‖x‖ (x ∈ M) implies that Tt is a contraction.Therefore, (Tt)t∈R+

is a Markov semigroup on L∞(E, µ).If T is Feller, this property is inherited by the classical reduction

through Remark 5.The previous results motivate the study of classical reductions of

quantum Markov semigroups by abelian algebras generated by a self-adjoint operator. To this end we refine below the applications of theSpectral Theorem to reduce completely positive maps. Given a nor-mal operator K, we denote W ∗(K) its generated von Neumann algebrawhich coincides with the weak closure of the C∗-algebra C∗(K) gener-ated by the same operator.

The abelian algebraW ∗(K) is maximal if and only ifK is multiplicity-free or non-degenerate. If K is bounded, non-degeneracy means thatthere exists a cyclic vector for C∗(K), that is, {f(K)w : f ∈ C(Sp (K))}is dense in h for some vector w, where Sp (K) denotes the spectrumof K. If K is unbounded, it is non-degenerate if there is a vector w inthe intersection of all domains D(Kn), (n ≥ 1), such that the subspacespanned by the vectors (Knw; n ≥ 1) is dense in h.

If K is degenerate, one can decompose the Hilbert space in orthogo-nal subspaces on which K is multiplicity-free. Here below we will dealwith this more general case.

We denote ξ the spectral measure of K. In addition, given a Radonmeasure on the measurable space (Sp (K) ,B(Sp (K))) we denote L(Sp (K))the ∗–algebra which is obtained as the quotient of the set of Borel func-tions by null functions under the given Radon measure.

DECOHERENCE OF QUANTUM MARKOV SEMIGROUPS 15

Lemma 1. Assume that P is a normal linear completely positive mapdefined on M and such that P(1) = 1. Let K be an unbounded self-adjoint operator affiliated with M. Then the following propositions areequivalent:

(i) W ∗(K) is invariant under P.(ii) For any projection p ∈ W ∗(K), P(p) ∈ W ∗(K).(iii) For all A ∈ B(Sp (K)), P(ξ(A)) ∈ W ∗(K).(iv) There exists a kernel P : Sp (K)× B(Sp (K)) → R+ such that

P (x,Sp (K)) = 1, for all x ∈ Sp (K) and

P(ξ(A)) =

∫Sp(K)

ξ(dx)P (x,A),

for all A ∈ B(Sp (K)).

Proof. Clearly, (i) implies (ii) which in turn implies (iii). The equiv-alence of (i) and (iii) follows from a straightforward application of theSpectral Theorem for general self-adjoint operators, since P is linearand normal. So that (i), (ii), and (iii) are equivalent.

To prove that (iii) implies (iv), we first notice that P◦ξ is an operatorvalued measure. Indeed, since ξ is the spectral measure of K and P islinear and completely positive, the map P ◦ξ is additive on B(Sp (K)).Moreover, take any pairwise disjoint sequence (An)n∈N of Borel subsetsof Sp (K). The projection

∑n ξ(An) exists as a strong limit of the

partial sums∑

k≤n ξ(Ak). Moreover∑

n ξ(An) = l.u.b.∑

k≤n ξ(Ak) inthe order of positive operators. Thus, the normality of the map P yieldsP (∑

n ξ(An)) = l.u.b.P(∑

k≤n ξ(Ak)), and P ◦ ξ is an operator-valued

measure.If we assume (iii), given any A ∈ B(Sp (K)), the Spectral Theorem

implies that there exists P (·, A) ∈ L(Sp (K)) such that

(24) P ◦ ξ(A) =

∫Sp(K)

ξ(dx)P (x,A).

Denote (en)n∈N an orthonormal basis of h and define the positivemeasure µ =

∑n 2−n〈en, ξ(·)en〉. Since P ◦ ξ is an operator-valued

measure, it follows that A 7→ P (x,A) satisfies

P (x,⋃n

An) =∑

n

P (x,An),

for µ–almost all x ∈ Sp (K).Since µ is a probability measure, it is tight on Sp (K) ⊆ R. There-

fore, for each n ≥ 1, there exists a compact Kn such that µ(Kn) ≥1−2−n, so that J =

⋃nKn ⊆ Sp (K) satisfies µ(J) = 1. We imbed J in

[−∞,∞], and consider µ as a probability measure defined on [−∞,∞],

16 ROLANDO REBOLLEDO

supported by J . Let denote k a vector space over the field of rationalnumbers, closed for lattice operations ∨,∧, dense in C([−∞,∞]) andsuch that 1 ∈ k. Define

A = {x ∈ J : f 7→ P (x, f)is a positive Q-linear form on k and P (x, 1) = 1} .For all x ∈ A, P (x, ·) can be extended as a positive linear form to

all of C([−∞,∞]), and then to L([−∞,∞]). Moreover, µ(A) = 1 and

µ({x ∈ Sp (K) : P (x, J c) = 0}) = 1.

We can complete the definition of the kernel P choosing P (x, ·) = θ(·)for all x 6∈ A, where θ is an arbitrary probability measure.

Finally to prove that (iv) implies (iii), it suffices to apply the Spec-tral Theorem again which yields

∫ξ(dx)P (x,A) ∈ W ∗(K).

Remark 7. It is worth noticing that if K is degenerate, we haveW ∗(K) ⊂ W ∗(K)′ strictly and W ∗(K)′ is not abelian. Indeed, anexample borrowed to Pedersen [32] shows the first assertion. Supposethat Ku = λu and Kv = λv for two orthogonal unit vectors u and v.There exists a unitary operator W ′ such that W ′u = v, W ′v = −u andW ′ = 1 on the orthogonal complement of Cx⊕Cy. One can check eas-ily that all elements a ∈ W ∗(K) have to satisfy 〈(u+ v), a(u− v)〉 = 0,while 〈(u+ v),W ′(u− v)〉 = 2. So that W ′ 6∈ W ∗(K), however a directcomputation shows that W ′ ∈ W ∗(K)′. On the other hand, if one as-sume W ∗(K)′ to be commutative, then it has a separating vector. Thismeans that the von Neumann algebra W ∗(K) has a cyclic vector andthis property is equivalent to its maximality, that is W ∗(K) = W ∗(K)′,contradicting the degeneracy of K.

The above discussion shows that the reduction of a completely pos-itive map by the commutant W ∗(K)′ of the von Neumann algebragenerated by K leads to a classical kernel if and only if K is non de-generate.

3.1. The C∗-case. Consider first a semigroup T defined on a C∗-algebra B and let a bounded normal operator K be given. We thencharacterize the classical reduction of the semigroup as follows.

Theorem 1. Assume that K is a normal operator in the C∗–algebra Band call C∗(K) the abelian C∗–algebra generated by K. Then a norm-continuous quantum Markov semigroup T defined on B is reduced byC∗(K) if and only if L(Kn) ∈ C∗(K) for all n ∈ N.

In particular, suppose that the generator is implemented by (4), whereH and V satisfy:

DECOHERENCE OF QUANTUM MARKOV SEMIGROUPS 17

(i) [H,K] ∈ C∗(K);(ii) V ∗V ∈ C∗(K);(iii) For each n ∈ N, there exists a constant αn ∈ C such that

V Kn − π(Kn)V = αnV.

Then, the semigroup T is reduced by the algebra C∗(K).

Proof. Let K denote the ∗-subalgebra generated by the commutingvariables 1, K, K∗. K is strongly dense in C∗(K). On the other hand,T is norm-continuous, so that C∗(K) is invariant under T if and onlyif L(K) ⊆ C∗(K). Since L(1) = 0 and L(K∗n) = L(Kn)∗, it followseasily that L(K) ⊆ C∗(K) if and only if L(Kn) ∈ C∗(K), for all n ∈ N.

To prove the second part of the Theorem, from hypothesis (i) andthe derivation property of [H, ·] it follows that [H,Kn] ∈ C∗(K) for alln ∈ N. On the other hand, hypothesis (iii) yields

V ∗V Kn − V ∗π(Kn)V = αnV∗V,

so that V ∗π(Kn)V belongs to C∗(K) as well as V ∗V Kn and KnV ∗V ,applying (ii). As a result, L(Kn) ∈ C∗(K) for all n ∈ N, and the proofis complete.

3.2. Norm-continuous semigroups defined on von Neumannalgebras. Given a normal operator K, W ∗(K) reduces the norm-continuous quantum Markov semigroup T if and only if L(x) ∈ W ∗(K)for all x ∈ W ∗(K). This follows immediately from the definition of thegenerator. Here below a modification of this elementary result in caseof non-degeneracy.

Theorem 2. If a bounded self-adjoint operator K ∈ M is non-degenerate,W ∗(K) reduces the norm-continuous quantum Markov semigroup T ifand only if L(x) commutes with K for any x ∈ W ∗(K).

In particular, suppose that

(i) [H,K] ∈ W ∗(K), and(ii) [Lk, K] = ckLk, where ck = c∗k ∈ W ∗(K), for all k ∈ N.

Then W ∗(K) reduces the semigroup T .

Proof. If K is non-degenerate, then W ∗(K) is maximal abelian andcoincides with its commutator W ∗(K)′. Thus, L(W ∗(K)) ⊆ W ∗(K) ifand only if L(x) lies in W ∗(K)′ for any element x ∈ W ∗(K)′.

To prove the second part, consider an arbitrary x ∈ W ∗(K)′. Tocompute [L(x), K], we first observe that each L∗kLk ∈ W ∗(K), since

18 ROLANDO REBOLLEDO

[L∗kLk, K] = −L∗kckLk+L∗kckLk = 0. In addition, [[H, x], K] = [[H,K], x]+

[H, [x,K]] = 0. Therefore,

[L(x), K] =

[i[H, x]− 1

2

∑k

[L∗kLkx− 2L∗kxLk + xL∗kLk, K

]= i[[H, x], K]−

∑k

[L∗kxLk, K]

= −∑

k

(−L∗kckxLk + L∗k[x,K]Lk + L∗kxckLk) , (x, ck and K commute)

= 0

The previous theorem can be improved to consider an unboundedself-adjoint operator K affiliated with the von Neumann algebra M.

For any quantum Markov semigroup T there exists M > 0 andβ ∈ R such that ‖Tt‖ ≤ M exp(βt) for all t ≥ 0 (see [7], Prop. 3.1.6,p. 166). As a result, the resolvent Rλ (·) of the semigroup is given bythe Laplace transform

Rλ (x) = (λ1− L)−1(x) =

∫ ∞

0

dte−λtTt(x),

for all x ∈ M, whenever <λ > β.

Theorem 3. Let be T a quantum Markov semigroup on the von Neu-mann algebra M and K an unbounded self-adjoint operator affiliatedwith M. Then the following propositions are equivalent:

(i) The semigroup is reduced by W ∗(K).(ii) For all A ∈ B(Sp (K)) and any t ≥ 0, Tt(ξ(A)) ∈ W ∗(K).(iii) The manifold D(L) ∩ W ∗(K) is non trivial and for all x ∈

D(L) ∩W ∗(K), it holds that L(x) ∈ W ∗(K).(iv) There exists a classical Markov semigroup (Tt)t∈R+ on Sp (K)

such that for all f ∈ L(Sp (K)),

Tt(f(K)) =

∫Sp(K)

ξ(dx)Ttf(x).

(v) For all λ such that <λ > β and all A ∈ B(Sp (K)) Rλ (ξ(A)) ∈W ∗(K).

Proof. We clearly have the equivalence of (i), (ii) and (iii). Further-more, the equivalence of (i) with (iv) follows from Lemma 1. Proposi-tion (v) is equivalent to the existence of a family of kernels Rλ on thespectrum of K which defines a classical semigroup T. Thus (v) and

DECOHERENCE OF QUANTUM MARKOV SEMIGROUPS 19

(iv) are equivalent and this completes the proof.The particular case of norm-continuous semigroups enjoys a richer char-acterization in terms of the generator.

Corollary 1. Suppose that K is a non-degenerate self-adjoint operator.W ∗(K) reduces a norm-continuous quantum Markov semigroup T ifand only if one of the following equivalent conditions is satisfied:

(i) L(ξ(A)) ∈ W ∗(K) for all A ∈ B(Sp (K)).(ii) [L(ξ(A)), ξ(B)] = 0 for all A,B ∈ B(Sp (K)).(iii) There exists a dense domain D ⊆ L(Sp (K)) and an operator

L : D → L(Sp (K)), such that for all f ∈ D, f(K) ∈ D(L)and

L(f(K)) =

∫Sp(K)

ξ(dx)Lf(x).

In particular, suppose that the generator L(·) is given by (5) whichin addition satisfies the two conditions below:

(a) [H, ξ(A)] ∈ W ∗(K), and(b) [Lk, ξ(A)] = ck(A)Lk, where ck(A) is a self-adjoint element in

W ∗(K), for all k ∈ N and A ∈ B(Sp (K)).

Then W ∗(K) reduces the semigroup T .

Proof. The generator L(·) is everywhere defined since the semigroupis norm-continuous. Thus, the equivalence of (i), (ii), (iii) with (i) ofthe previous result is a simple consequence of the Spectral Theorem.

The last part follows from the first and Theorem 2 applied to ξ(A).

4. Semigroups with form-generators

Finally, if the generator is given as a form through a Master Equa-tion, for M = L(h), the above results have to be amended as follows.

Theorem 4. Assume that for all x ∈ W ∗(K), and all spectral projec-tion ξ(A), where A ∈ B(Sp (K)) is such that ξ(A)(D) ⊆ D it holds

(25) L−(x)(v, ξ(A)u) = L−(x)(ξ(A)v, u),

for all (u, v) ∈ D × D. Then the minimal semigroup T is reduced byK.

In particular, this is the case when the following two conditions hold:

(a) The operator G is affiliated with W ∗(K),(b) Given any x ∈ W ∗(K), (u, v) ∈ D ×D, ` ≥ 1,

(26) 〈L`v, xL`ξ(A)u〉 = 〈L`ξ(A)v, xL`u〉,for all A ∈ B(Sp (K)) such that ξ(A)(D) ⊆ D.

20 ROLANDO REBOLLEDO

Proof. The proof follows the construction of the minimal quantum dy-namical semigroup associated to the form L−(·), as presented by Cheb-otarev (see [12]) and extensively used by him and Fagnola in their jointresearch on the Markov property of this minimal semigroup (see [11]).

Define T (0)t (x) = x. Then, clearly 〈v, T (0)

t (x) pu〉 = 〈pv, T (0)t (x)u〉,

for all x ∈ W ∗(K), all projection p = ξ(A) leaving D invariant, (u, v) ∈D × D. We follow by defining T (1)

t (x) as follows: for each (u, v) ∈D ×D,

〈v, T (1)t (x)u〉 = 〈v, xu〉+

∫ t

0

L−(T (0)s (x))(v, u)ds.

Take x ∈ W ∗(K) a projection p = ξ(A) as before, and apply hypoth-esis (25). Then it follows that

〈v, T (1)t (x) pu〉 = 〈pv, T (1)

t (x)u〉.

This yields that T (1)t (x) ∈ W ∗(K) if x ∈ W ∗(K).

By induction, suppose T (0)t (·) , . . . , T (n)

t (·) constructed and reduced

by W ∗(K), then define T (n+1)t (·) through the relation

〈v, T (n+1)t (x)u〉 = 〈v, xu〉+

∫ t

0

L−(T (n)s (x))(v, u)ds.

By the induction hypothesis, and (25) again, it follows that

〈v, T (n+1)t (x) pu〉 = 〈pv, T (n+1)

t (x)u〉,

for all projection p = ξ(A) such that p(D) ⊆ D and (u, v) ∈ D × D,whenever x ∈ W ∗(K). Therefore, K reduces the whole sequence(T (n))n∈N . This sequence is used in the construction of the minimal

quantum dynamical semigroup as follows. It is proved that 〈u, T (n)t (x)u〉

is increasing with n and 〈u, Tt(x)u〉 is defined as its limit, for all u ∈ h,x ∈ L(h) (see [16]). Then by polarization 〈v, Tt(x)u〉 is obtained. Thus,the minimal quantum dynamical semigroup T satisfies Tt(1) ≤ 1, andgiven any other σ-weakly continuous family (St)t∈R+ satisfying (8) andevery positive operator x ∈ L(h), it holds Tt(x) ≤ St(x), for all t ≥ 0.

Moreover, since T (n)t (W ∗(K)) ⊆ W ∗(K), for all n ∈ N and t ≥ 0, it

follows that K reduces the minimal quantum dynamical semigroup.Assume now hyotheses (a) and (b). Condition (a) implies that

Gξ(A) = ξ(A)G for all projection ξ(A) leaving D invariant. Moreover,(b) yields

∑`〈L`v, xL`ξ(A)u〉 =

∑`〈L`ξ(A)v, xL`u〉 and this, together

with (a), clearly determine (25) and the proof is complete.

DECOHERENCE OF QUANTUM MARKOV SEMIGROUPS 21

Corollary 2. With the notations and assumptions previous to theabove theorem, suppose that in addition the two hypotheses below aresatisfied:

(a) G is affiliated with W ∗(K),(b) For all ` ≥ 1 and any A ∈ B(Sp (K)) such that ξ(A) leaves D

invariant, there exists a selfadjoint operator c`(A) ∈ W ∗(K),such that

(27) L`ξ(A) = (ξ(A) + c`(A))L`.

Then K reduces the minimal quantum dynamical semigroup T .

Proof. Hypothesis (a) is identical to condition (a) of the previoustheorem. On the other hand, if x ∈ W ∗(K), and A ∈ B(Sp (K)) issuch that ξ(A) leaves D invariant

〈L`ξ(A)v, xL`u〉 = 〈ξ(A)L`v, xL`u〉+ 〈c`(A)L`v, xL`u〉= 〈L`v, xξ(A)L`u〉+ 〈L`v, xc`(A)L`u〉= 〈L`v, x(L`ξ(A)− c`(A)L`)u〉+ 〈L`v, xc`(A)L`u〉= 〈L`v, xL`ξ(A)u〉,

for all (u, v) ∈ D × D. Thus, condition (b) of Theorem 4 is satisfiedand the proof is complete.

4.1. Returning to examples.

4.1.1. The harmonic oscillator. In Example 2.4.1 L(x) was indeed aform-generator which should be more rigourously written L−(x). Theexpression of the formal generator L−(·) suggest to consider the reduc-tion by W ∗(N). Indeed, Sp (N) = N, the elements en are the eigenvec-tors of N and for any bounded function f : N → C a straightforwardcomputation yields

L−(f(N))(v, |en〉〈en|u) = L−(f(N))(|en〉〈en|v, u) = Lf(n)〈v, en〉〈en, u〉,where,

(28) Lf(n) = λn(f(n+ 1)− f(n)) + µn(f(n− 1)− f(n)),

and

(29) λn = Aν(n+ 1), µn = A(ν + 1)n, (n ∈ N).

As it is easily seen, the expression (28) corresponds to the generatorof a classical birth and death Markov semigroup, with birth rate λn

and death rate µn.

22 ROLANDO REBOLLEDO

4.1.2. The quantum Brownian Motion. The commutative von Neu-mann subalgebra W ∗(q) of M whose elements are multiplication opera-torsMf by a function f ∈ L∞(Rd; C) is T -invariant and Tt(Mf ) = MTtf

where

(30) (Ttf)(x) =1

(2πt)d/2

∫Rd

f(y)e−|x−y|2/2tdy.

The same conclusion holds for the commutative algebra W ∗(p) =F ∗W ∗(q)F , where F denotes the Fourier transform (see [21]). There-fore, this QMS named quantum Brownian motion semigroup containsa couple of non commuting classical Brownian semigroups as classicalreductions.

Moreover, notice that the von Neumann algebraW ∗(N) generated by

the number operator N =∑

j a†jaj is also T invariant and the classical

semigroup obtained by restriction of T to W ∗(N) like in the previousexample is a birth and death on N with birth rates (n + 1)n≥0 anddeath rates (n)n≥0.

4.1.3. The quantum exclusion semigroup. Given η ∈ S, i, j ∈ N, defineci,j(η) = η(i)(1− η(j))γi,j.

Proposition 1. For each x ∈ A(h0) the unbounded operator

(31) L(x) = i[H, x]− 1

2

∑i,j

(L∗i,jLi,jx− 2L∗i,jxLi,j + xL∗i,jLi,j

),

whose domain contains the dense manifold v, is the generator of aquantum Feller semigroup T on the C∗-algebra A(h0). This semigroupis extended into a σ–weak continuous QMS defined on the whole algebraL(h).

Moreover, the semigroup is reduced by the algebra W ∗(H). The re-duced semigroup T corresponds to a classical exclusion process withgenerator

(32) Lf(η) =∑i,j

ci,j(η) (f(η + 1j − 1i)− f(η)) ,

for all bounded cylindrical function f : S → R.

Proof. We first notice that there exists the minimal quantum Markovsemigroup associated with the generator (31). Indeed this holds sincev is dense in h = Γf (h0), and it is a core for G = −iH− 1

2

∑i,j Li,j

∗Li,j,which is the generator of a contraction semigroup, and v is also con-tained in the domain of all the operators Li,j, and the Markovian prop-erty is guaranteed by L(1) = 0. Let denote T this minimal semigroup

DECOHERENCE OF QUANTUM MARKOV SEMIGROUPS 23

(see [13] for further details) which is defined through the use of theσ-weak topology in all of L(h). It satisfies the equation

(33) 〈v, Tt(x)u〉 = 〈v, xu〉+

∫ t

0

〈v,L(Ts(x))u〉ds,

where u, v ∈ v, x ∈ L(h).Take x ∈ A(h0) and Λ ∈ Pf (N). Call EΛ the projection of h =

Γf (h0) onto hΛ = Γf (hΛ0 ). Notice that for all a ∈ A(h0), the net

of projected operators EΛaEΛ ∈ A(hΛ0 ) converges strongly to a, as

Λ ↑ N since A(h0) is the strong closure of D =⋃

A(hΛ0 ). Call EΛ(a) =

EΛaEΛ the projection of an element of the algebra A(h0) and T Λt (x) =

EΛTt(x)EΛ, x ∈ A(hΛ0 ), t ≥ 0. This is a semigroup acting on A(hΛ

0 )whose generator, determined by (33), is LΛ(x) = EΛL(x)EΛ, for eachx ∈ A(hΛ

0 ). LΛ(x) is a matrix in a finite-dimensional space, so thatit is a bounded operator. As a result, each T Λ is a norm-continuoussemigroup.

To prove that the minimal semigroup T satisfies the Feller property(M5F) on the algebra A(h0), we first consider x ∈ D. So that thereis Λ0 ∈ Pf (N) such that x ∈ A(hΛ

0 ), which yields EΛ(x) = x for allΛ ∈ Pf (N) containing Λ0. Then, for all such Λ,

‖Tt(x)− x‖ ≤∥∥Tt(x)− T Λ

t (x)∥∥+

∥∥T Λt (x)− x

∥∥ .Since EΛ ◦ Tt(x) strongly converges to Tt(x) as Λ increases, given anyε > 0 we can choose Λ ∈ Pf (N) to have the first right-hand term in theprevious inequality less than ε/2. On the other hand, for this Λ we alsohave limt→0

∥∥T Λt − 1

∥∥ = 0 and t0 may be selected to have the secondright-hand term in the inequality less than ε/2 too for any t < t0. Thisproves (M5F) for x ∈ D.

If x ∈ A(h0), we pick a net xΛ ∈ A(hΛ0 ) which strongly converges to x,

use the fact that Tt(·) is a contraction and the property (M5F) provedfor elements in D to conclude. T is thus a quantum Feller semigroupon the algebra A(h0).

To study the classical reduction, it suffices to use the C.A.R. In-deed, [b†kbk, b

†jbi] = (δkj − δki)b

†jbi. Moreover, since each operator Li,j

is bounded and Li,jv ⊂ v ⊂ D(H), the commutator [H,Li,j] is welldefined on v and can be extended to all of h as a bounded operatorsince

[H,Li,j] =∑

k

Ek√γi,j[b

†kbk, b

†jbi] = (Ej − Ei)Li,j.

So that, adapting to this case the proof of Theorem 2, we conclude thatH reduces the semigroup.

24 ROLANDO REBOLLEDO

We now obtain the expression of the reduced generator. The spectraldecomposition of H may be written,

H =∑

η

E(η)|η〉〈η|,

where E(η) =∑

i η(i)Ei, for all configuration η. The algebra Cyl(H)of operators of the form

x =∑

η

f(η)|η〉〈η|,

where f : S → C is a bounded cylindrical function, form a densesubalgebra of C∗(H).

We compute L(x) for x ∈ Cyl(H).

The following additional notation will be used: iη−→ j, means that

η(i) = 1 and η(j) = 0 (under the configuration η a particle occupyingthe site i can move to the free site j).

An elementary computation yields

L∗i,jLi,j|η〉〈η| = γi,jη(i)(1− η(j))|η〉〈η||η〉〈η|L∗i,jLi,j = γi,jη(i)(1− η(j))|η〉〈η|L∗i,j|η〉〈η|Li,j = γi,j(1− η(i))η(j)|η − 1j + 1i〉〈η − 1j + 1i|.

From this it follows that

L(|η〉〈η|) = −1

2

∑i,j

(L∗i,jLi,j|η〉〈η| − 2L∗i,j|η〉〈η|Li,j + |η〉〈η|L∗i,jLi,j

)=

∑i,j:j

η−→i

γi,j|η − 1j + 1i〉〈η − 1j + 1i| −∑

i,j:iη−→j

γi,j|η〉〈η|.

Now, for any x =∑

η f(η)|η〉〈η| ∈ Cyl(H),

L(x) =∑

η

∑i,j:j

η−→i

γi,jf(η)|η − 1j + 1i〉〈η − 1j + 1i| −∑i,j

γi,jf(η)|η〉〈η|

,

and notice that a change of variables η 7→ η − 1j + 1i yields∑η

∑i,j:j

η−→i

γi,jf(η)|η−1j+1i〉〈η−1j+1i| =∑

η

∑i,j:i

η−→j

γi,jf(η−1i+1j)|η〉〈η|.

Therefore, we finally obtain

L(x) =∑

η

∑i,j:i

η−→j

γi,j (f(η − 1i + 1j)− f(η)) |η〉〈η|,

DECOHERENCE OF QUANTUM MARKOV SEMIGROUPS 25

from which (32) follows.The above expression gives the generator of the semigroup restricted

to W ∗(H). C∗(H) is isomorphic with the algebra C(Sp (K)) of contin-uous complex valued functions on the compact set Sp (H) and containscontinuous cylindrical functions as a uniformly dense sub-algebra. Itis clear that L given by (32) leaves the above dense subalgebra invari-ant, thus the reduced semigroup applies C(Sp (H) into itself, moreover,the Feller continuity property is inherited from the quantum Markovsemigroup.

5. Decoherence as a limit behavior

Let be given a quantum Markov semigroup T defined on M = L(h).We introduce two additional notations: F (T ) = {x ∈ M : Tt(x) = x, for all t ≥ 0}and N (T ) = {x ∈ M : Tt(x

∗x) = Tt(x∗)Tt(x), for all t ≥ 0}. The first

set corresponds to invariant elements under the action of the semi-group, whereas the second consists of elements for which equality holdsin Schwartz inequality for completely positive maps, a feature which ischaracteristic of an automorphism group. In general, F (T ) ⊆ N (T ).Assume further that a faithful normal stationary state ω∞ = tr(ρ∞ ·)exists for the semigroup T . Under this hypothesis, Frigerio and Verriproved in [22] that for a norm-continuous semigroup, F (T ), N (T )are von Neumann algebras, the conditional expectations EF(T ) (·) andEN(T ) (·) exist and any other stationary state can be represented asω = ω∞ ◦ EF(T ) (·). In addition they proved that if F (T ) = N (T ),then the semigroup is ergodic, that is, for any initial state ω and anyelement x of the algebra, T∗t(ω)(x) = ω(Tt(x)) → ω(EF(T ) (x)) = ω∞(x)as t → ∞. Since we are assuming all states to be normal, this meansthat for any initial density matrix ρ, T∗t(ρ) weakly converges to theinvariant faithful density matrix ρ∞ associated to ω∞ as t→∞. Thisresult has been extended by Fagnola and Rebolledo in [18] to a gen-eral class of QMS. Moreover, for a generator given by a form like (7),with G as in (6), where H is a self-adjoint operator with pure pointspectrum, one obtains the following nice characterization of ergodicity.

Theorem 5 (Fagnola-Rebolledo, Thm. II.2,[18]). Suppose that theminimal semigroup T associated to L−(·) is Markov and that it has anormal faithful stationary state.

Assume in addition that H is a self–adjoint operator with pure pointspectrum and either

26 ROLANDO REBOLLEDO

(a) H is bounded;or

(b) H is selfadjoint and eitH(D) ⊆ D(G), where D ⊆ D(G) is adense linear subspace.

Then T is ergodic if and only if

(34) {Lk, L∗k, H; k ≥ 1}′ = {Lk, L

∗k; k ≥ 1}′.

It is worth mentioning that sufficient conditions for the existence ofa T -stationary normal faithful state have been obtained in terms ofthe generator too in [19] and [20]. On the other hand, no stationarystate exist for a transient quantum Markov semigroup. Transience andrecurrence of quantum Markov semigroups have been studied in [21].

We now proceed with the definition of decoherence in our frame-work. To keep this notion close to the first approach of physicists,throughout this section we consider a self-adjoint operatorKwith pure point spectrum and denote (en)n∈N an orthonormalbasis of eigenvectors of K.

Definition 4. We say that K induces decoherence of the quantumMarkov semigroup T if there exists a faithful T -stationary density ma-trix which commutes with K.

Equivalently, K induces decoherence of T if there exists a com-mon faithful stationary density matrix for both T and the group αof automorphisms associated to K, αt(x) = exp(itK)x exp(−itK),(x ∈ M, t ∈ R).

Remark 8. Another equivalence of the above definition can be phrasedin terms of the generator L∗ of the predual semigroup: K inducesdecoherence of T if and only if there exists a faithful density matrixρ ∈ D(L∗) which commutes with K and satisfies L∗(ρ) = 0.

Remark 9. Notice that if K is non-degenerate, the definition impliesthat for all density matrix ρ there exists a sequence (tr)r∈N such thattr →∞ and for all n 6= m,

1

tr

∫ tr

0

〈em, T∗s(ρ)en〉ds→ 0.

Moreover, we have the following easy proposition improving theabove remark.

Proposition 2. Assume that a non-degenerate self-adjoint operator Kwith pure point spectrum as before induces decoherence of T and that

DECOHERENCE OF QUANTUM MARKOV SEMIGROUPS 27

the semigroup is ergodic. Then, given any density matrix ρ ∈ I1(h),and n 6= m, it holds

〈em, T∗t(ρ)en〉 → 0,

as t→∞.

Proof. Since the semigroup is ergodic and K induces decoherence, itholds that tr(T∗t(ρ)x) converges to tr(ρ∞x) for all x ∈ M, as t → ∞and any density matrix ρ, where ρ∞ is a faithful density matrix whichcommutes with K. If there is another density matrix ρ′ such thatlim tr(T∗t(ρ)x) = tr(ρ′x), then tr(ρ′x) = tr(ρ∞x) for all x ∈ M andfaithfulness of ρ∞ implies that ρ′ = ρ∞.

The last statement follows straightforward from the above since〈em, ρ∞en〉 = 0 for n 6= m.

Proposition 3. Suppose that the self-adjoint operator K is non-degenerateand that W ∗(K) reduces the quantum Markov semigroup T . If K in-duces decoherence of T , then the reduced semigroup has a faithful sta-tionary probability measure.

Proof. Call ρ a faithful stationary density matrix which commuteswith K. So that ρ can be written as

ρ =∑

n

p(n)|en〉〈en|,

where∑

n p(n) = tr(ρ∞) = 1, and each p(n) > 0 due to the faithfulnessof ρ. Given any bounded function f on the spectrum of K, it holds:

tr(ρTt(f(K))) =∑

n

p(n)Ttf(n) = tr(ρf(K)) =∑

n

p(n)f(n),

for all t ≥ 0. Thus, the density p(n) defines a probability measure onSp (K) which is stationary under the reduced semigroup (Tt)t∈R+ .

It is an important problem for applications to Physics to know whetherthe knowledge of a stationary probability for a reduced semigroup couldleads to decoherence. To give a partial answer for a wide class ofsemigroups, we previously need to take care of some technical mattersconcerning the domain of the generator. Suppose that we are given aform-generator L−(·) with which we construct the associated minimalquantum Markov semigroup (see [13], [11]). We will assume here thatthe orthonormal basis of eigenvectors of K can be chosen on the densesubset D which is a core for the operator G and all the operators Lk

defining L−(·). Then, as proved in [13], sections 2 and 3, the linear spacespanned by all the projections |en〉〈em|, (n,m ∈ N), is a core for the

28 ROLANDO REBOLLEDO

predual generator L∗. As a result, all the operators L∗(|en〉〈en|) arewell defined.

Theorem 6. Suppose that the self-adjoint operator K is non-degenerateand that W ∗(K) reduces the quantum Markov semigroup T . Assumethat there exists a faithful probability density (p(λ))λ∈Sp(K) on the spec-trum of K which is stationary for the reduced semigroup. If for alln ∈ N, L∗(|en〉〈en|) commutes with K, then K induces decoherence ofT .

Proof. Call ρ = p(K) =∑

n p(λn)|en〉〈en|. We will prove thatρ ∈ D(L∗) and that L∗(ρ) = 0. Given any bounded function f onSp (K), call L the reduction of the generator, that is 〈en,L(f(K))en〉 =Lf(λn). The hypothesis on the stationarity of p is then expressed as∑

n p(λn)Lf(λn) = 0.Define

ρN =∑n≤N

p(λn)|en〉〈en|, (N ∈ N).

Notice that ρN ∈ D(L∗), since each projection |en〉〈en| belongs toD(L∗). Moreover, L∗(ρN) is a trace-class operator as well. Indeed,given any x ∈ D(L), which is the domain where all the maps (u, v) 7→L−(x)(u, v) are continuous, it follows

|tr(L∗(ρN)x)| =

∣∣∣∣∣∑n≤N

p(λn)L−(x)(en, en)

∣∣∣∣∣ ≤ C(x)tr(ρN) ≤ C(x),

where C(x) is a positive constant. Therefore, by Schatten’s Theo-rem, L∗(ρN) is a trace-class operator for all N ∈ N. Moreover, sinceL∗(|en〉〈en|) commutes with K and K is non-degenerate, L∗(ρn) ∈W ∗(K) as well.

Taking N,M ∈ N, with N > M say, one obtains

|tr((L∗(ρN)− L∗(ρM))x)| ≤ C(x)N∑

n=M+1

p(λn),

for any fixed x ∈ D(L). Since∑

n p(λn) = 1, we obtain that L∗(ρN)weakly converges as N → ∞. On the other hand, ρN converges inthe norm of the trace to ρ. Since L∗(·) is weakly closed, then L∗(ρ) =limN L∗(ρN) and ρ ∈ D(L∗). Moreover, L∗(ρ) is a trace-class operatorwhich commutes with K.

DECOHERENCE OF QUANTUM MARKOV SEMIGROUPS 29

After the previous result, to prove that L∗(ρ) = 0 it suffices to showthat 〈ek,L∗(ρ)ek〉 = tr(L∗(ρ)|ek〉〈ek|) = 0 for all k ∈ N. Now,

tr(L∗(ρ)|ek〉〈ek|) = tr(ρL(|ek〉〈ek|))=

∑n

p(λn)〈en,L(|ek〉〈ek|)en〉

=∑

n

p(λn)L1{λk}(λn)

= 0, (since p is stationary for the reduced semigroup).

Thus, L∗(ρ) = 0 and ρ is a stationary state for T . As a result, Kinduces decoherence of the quantum Markov semigroup.

Corollary 3. Suppose that the semigroup T is norm-continuous withgenerator given by (5) and satisfies the following hypotheses:

(i) H has a pure point spectrum;(ii) [H,Lk] = αkLk, αk ∈ R, for all k ≥ 1;(iii) the generalized commutants {H,Lk, L

∗k, k ∈ N}′ and {Lk, L

∗k, k ∈ N}′

coincide;(iv) There is a faithful probability p on the spectrum of H solving

the balance equation

(35) Lkp(H)L∗k = L∗kLkp(H) = p(H)L∗kLk, (k ∈ N).

Then the semigroup is ergodic, it is reduced by H, and H induces de-coherence of T . As a result, if (en)n∈N is an orthonormal basis ofeigenvectors of H, then for any density matrix ρ it holds

〈en, T∗t(ρ)em〉 → 0,

as t→∞, for all n 6= m, n,m ∈ N.

Proof. Hypothesis (ii) implies that H reduces the semigroup. Then(iv) determines the existence of a faithful and normal stationary stategiven by a density matrix ρ∞ = p(H), since L∗(p(H)) = 0. Thus Hinduces decoherence of T . Condition (iii) implies the ergodicity of thesemigroup by Theorem II.2 in [18] which has been recalled here in The-orem 5. Finally Proposition 2 leads to the conclusion.

The above result can be improved to consider more general QMS asfollows

Corollary 4. Let K be as in Theorem 6. Assume that the semigroupT is reduced by W ∗(K) and that L∗(|en〉〈en|) ∈ W ∗(K) for all n ∈ N.

30 ROLANDO REBOLLEDO

Suppose in addition that the generalized commutant {Lk, Lk∗, k ∈ N}′

is reduced to C1.If there exists a faithful stationary probability on Sp (K) for the re-

duced semigroup, then the QMS is ergodic and K induces decoherenceof T

Proof. By Theorem 6, p(K) defines a faithful normal stationary statefor the semigroup T so that, from one hand, K induces decoherenceof T and in addition, the results on ergodicity of [18] can be applied.Since F (T ) ⊆ N (T ) ⊆ {Lk, Lk

∗, k ∈ N}′ and the latter is trivial, oneobtains F (T ) = N (T ) = C1, so that T is ergodic after Theorem II.1in [18].

5.1. Examples. We come back to our well-known examples.

5.1.1. The harmonic oscillator. Clearly the hypothesis of the last corol-lary apply here. The algebra generated by a, a† and 1 is topologicallyirreducible, that means that the commutant is C1. The birth anddeath semigroup has a faithful invariant probability measure p sinceλn < µn, for all n. So that H induces decoherence of the semigroup T .

5.1.2. The quantum exclusion process. Here, the elements of the or-thonormal basis are denoted |η〉 according to the notations introducedin the second quantization procedure. We summarize below the appli-cation of the previous corollaries to this model. Consider first a densitymatrix which is of the form p(H), that is:

(36) ρ =∑

η

p(η)|η〉〈η|,

where η 7→ p(η) is a summable function with∑

η p(η) = 1.

Proposition 4. Let assume that

(37) π(i)γi,j = π(j)γj,i, (i, j ∈ N)

where π : N → R+ is summable. Then a normal state ω with densitymatrix ρ given by (36) is stationary if

p(η) =∏i∈N

α(i)η(i),

for all η ∈ S, where α : N → [0, 1] is given by

(38) α(i) =π(i)

1 + π(i), (i ∈ N).

Moreover, H induces decoherence of the semigroup T .

DECOHERENCE OF QUANTUM MARKOV SEMIGROUPS 31

Proof. Let be given ρ by (36). Then L∗(ρ) = 0 if and only iftr(L∗(ρ)|η〉〈η|) = 0 for all η ∈ S.

Notice that ζ− 1j +1i = η if and only if ζ = η− 1i +1j, for η, ζ ∈ S,i, j ∈ N. Thus, if we write

ρ =∑

ζ

p(ζ)|ζ〉〈ζ|,

the previous theorem yields

L∗(ρ) =∑

ζ

∑i,j:i

ζ−→j

γi,jp(ζ) (|ζ − 1i + 1j〉〈ζ − 1i + 1j| − |ζ〉〈ζ|) .

Now, tr(|ζ−1i+1j〉〈ζ−1i+1j||η〉〈η|) = 1 if and only if ζ = η−1j+1i.Thus, tr(L∗(ρ)|η〉〈η|) = 0 if and only if∑

i,j:jη−→i

γi,j (p(η − 1j + 1i)− p(η)) = 0.

The last expression is Lp(η) = 0, and we can apply Theorem 2.1 inChapter VIII of [25]. Moreover the above computations yield

L∗(|η〉〈η|) =∑i,j

ci,j(η) (|η − 1i + 1j〉〈η − 1i + 1j| − |η〉〈η|) ,

which commutes with H, so that Theorem (6) applies and the proof iscomplete.

The above result can be rephrased in a slightly different frameworkto recover a unique faithful stationary state in the Gibbs form. Weadd a cemetery to the classical Markov chain by completing N with apoint ∞ 6∈ N. We assume that γi,∞, γ∞,j > 0 but γ∞,∞ = 0. On the

other hand, we put Li,∞ =√γi,∞ bi, L∞,j =

√γ∞,j b

†j, for all i, j ∈ N,

and L∞,∞ = 0. Configurations are now defined on N = N ∪ {∞}.Finally define E∞ = µ > 0, which we call the chemical potential. Thegenerator L(·) is naturally extended taking the sum in (31) runningover all indexes (i, j) ∈ N × N. Moreover, this time the set {Lk, L

∗k}

includes all the operators b†(k), b(k) generating the CAR algebra, sothat its generalized commutant algebra is trivial. Thus, in this case wehave F (T ) = N (T ) = C1 and the semigroup is ergodic as soon as weprovide a faithful stationary state.

In the following corollary we assume that H has a finite number ofeigenvalues Ei < µ, so that

∑i exp(−β(Ei − µ)) <∞ for any β > 0.

Corollary 5. Assume that H given by (19) is non degenerate, witha finite number of eigenvalues dominated by the chemical potential µ,

32 ROLANDO REBOLLEDO

and such that∑

i |Ei| < ∞. Moreover, suppose that (37) is satisfiedwith

(39) π(i) = e−β(Ei−µ), (i ∈ N),

where β > 0. Denote

Z(β, µ) = tr(e−β(H−µN)).

Then

ρ =1

Z(β, µ)e−β(H−µN),

is a faithful stationary state of the quantum Markov semigroup associ-ated to H and the operators L` considered in the previous results. Thesemigroup is ergodic and H induces decoherence of T .

Remark 10. As a final remark, it is worth noticing that if a quantumMarkov semigroup is transient, there is no stationary state. As a result,decoherence as defined here cannot take place. As an example of atransient QMS, the reader is referred to [21] where it is showed thatthe semigroup of a quantum Brownian motion for d ≥ 2 is transient.

Acknowledgements I want to express here my deep gratitude to theorganizers of the Colloquium in memoriam of Paul-Andre Meyer inStrasbourg for both, the high scientific level of the meeting as well astheir warm hospitality, which is one of the most striking features of theso called Strasbourg school. Several generations of probabilists havebeen touched by Meyer’s influential work and life. A number of themtook part in the colloquium, exchanging memories about Paul-Andrwhich I gratefully acknowledge.

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Pontificia Universidad Catolica de Chile, Facultad de Matematicas,Casilla 306, Santiago 22, Chile

E-mail address: rrebolle@mat.puc.cl