Quantum Dynamical Semigroups and Decoherence

Advances in Mathematical Physics Volume 2011, Article ID 625978,16pages doi:10.1155/2011/625978

Research Article

Quantum Dynamical Semigroups and Decoherence

Mario Hellmich

1Faculty of Physics, University of Bielefeld, Universit¨atsstraβe 25, 33615 Bielefeld, Germany

2Bundesamt f ¨ur Strahlenschutz (Federal Office for Radiation Protection), Willy-Brandt-Straße 5, 38226 Salzgitter, Germany

Correspondence should be addressed to Mario Hellmich,[email protected] Received 22 June 2011; Accepted 31 August 2011

Academic Editor: Christian Maes

Copyrightq2011 Mario Hellmich. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove a version of the Jacobs-de Leeuw-Glicksberg splitting theorem for weak^∗continuous one- parameter semigroups on dual Banach spaces. This result is applied to give sufficient conditions for a quantum dynamical semigroup to display decoherence. The underlying notion of decoherence is that introduced by Blanchard and Olkiewicz2003. We discuss this notion in some detail.

1. Introduction

The theory of environmental decoherence starts from the question of why macroscopic physical systems obey the laws of classical physics, despite the fact that our most fundamental physical theory—quantum theory—results in contradictions when directly applied to these objects. The infamous Schr ¨odinger cat is a well-known illustration of this problem. This is an embarrassing situation since, from its inception in the 1920s until today, quantum theory has seen a remarkable success and an ever increasing range of applicability.

Thus the question of how to reconcile quantum theory with classical physics is a fundamental one, and efforts to find answers to it persisted throughout its history. At present, the most promising and most widely discussed answer is the notion of environmental decoherence.

The starting point is the contention that quantum theory is universally valid, in particular in the macroscopic domain, but that one has to take into account the fact that macroscopic objects are strongly interacting with their environment, and that precisely this interaction is the origin of classicality in the physical world. Thus classicality is a dynamically emergent phenomenon due to the essential openness of macroscopic quantum systems, that is, their interaction with other quantum systems surrounding them leads to an effective restriction of the superposition principle and results in a state space with properties different from the pure quantum case.

In order to clarify the status of decoherence and to provide a rigorous definition, Ph. Blanchard and R. Olkiewicz suggested a notion of decoherence formulated in the algebraic framework1,2of quantum physics in3, drawing on earlier work in4. The algebraic framework is especially useful for the discussion of decoherence, since it is able to accommodate classical systems, provides an elegant formulation of superselection rules, and can even describe systems with infinitely many degrees of freedom in a rigorous way. This is why it is becoming increasingly popular in the discussion of foundational and philosophical problems of quantum physics5,6.

In the present paper, we assume that the algebra of observables of the system under study is a von Neumann algebra, and due to its openness the time evolution is irreversible and hence given by a family{Tt}_t≥0of normal completely positive and unital linear maps on the von Neumann algebra7,8. In the Markovian approximation, the family{Tt}_t≥0becomes a so-called quantum dynamical semigroup. It is our purpose to discuss the Blanchard- Olkiewicz notion of decoherence for quantum dynamical semigroups. To this end we study a weak^∗ version of the so-called Jacobs-de Leeuw-Glicksberg splitting for one-parameter semigroups on dual Banach spaces. In the Markovian case, the Blanchard-Olkiewicz notion of decoherence relies on the so-called isometric-sweeping splitting, which is similar to the Jacobs-de Leeuw-Glicksberg splitting, and we will be able to prove a new criterion for the appearance of decoherence in the case of uniformly continuous quantum dynamical semigroups by examining the connection between the two asymptotic splittings.

The paper is organized as follows. In Section 2 we establish the Jacobs-de Leeuw- Glicksberg splitting for weak^∗ continuous contractive one-parameter semigroups on dual Banach spaces. We provide a sufficient condition which ensures that the semigroup is weak^∗ stable on the stable subspace of the splitting Proposition 2.3. InSection 3we turn to the study of quantum dynamical semigroups on von Neumann algebras. We begin by applying the results of Section 2in the von Neumann algebra settingProposition 3.3. As a complement toProposition 2.3, we prove Proposition 3.6, which gives another condition for weak^∗ stability on the stable subspace of the splitting. InSection 4, we discuss a notion of decoherence which is very close to that given in 3 and establish some mathematical results related to it. In the finalSection 4.2, we use the previous results to give a sufficient condition that a uniformly continuous quantum dynamical semigroup having a faithful normal invariant state displays decoherence.

2. The Jacobs-de Leeuw-Glicksberg Splitting

Suppose thatXis a Banach space and assume that it has a predual space denoted byX∗, that is, X_∗^∗ ∼ X. If x ∈ Xand ϕ ∈ X_∗, we will denote the evaluation ofxat ϕbyx, ϕand consider this as a dual pairing betweenX andX_∗. The set of all bounded linear operators fromXtoX, endowed with the operator norm, will be denoted by LX, and its unit ball by LX₁ {T ∈LX: T ≤1}. Operators from LX₁ are called contractive. We consider the algebraic tensor productXX_∗and endow it with the projective cross normγ; the completion ofXX∗with respect toγis a Banach space which will be denoted byX⊗γX∗. Then its dual spaceX⊗γX∗^∗ is isometrically isomorphic in a canonical way with LX,X∗^∗ LX: If ψ∈X⊗γX_∗^∗, we defineΦψ∈LXby

x⊗ϕ, ψ

Φ ψ

x, ϕ

2.1 for all x ∈ X and ϕ ∈ X∗. It can now be shown that ψ → Φψ extends to an isometric isomorphism, and we can thus writeX⊗γX_∗^∗∼LX.

We now introduce the pointwise weak^∗ topology on LX. Let x ∈ X,ϕ ∈ X∗, and define the seminorm LXT →px,ϕT |Tx, ϕ|. The pointwise weak^∗topology is the locally convex topology on LXinduced by the family{px,ϕ:x∈X, ϕ∈X∗}of seminorms.

IfT Φψ∈LX, we see thatpx,ϕT |Tx, ϕ||x⊗ϕ, ψ|; thus the pointwise weak^∗ topology coalesces with theσLX,X⊗γX_∗topology on LX, that is, the pointwise weak^∗ topology is a weak^∗ topology as well. Thus we can conclude from Alaoglu’s theorem that LX₁is compact in the pointwise weak^∗topology.

A linear operator T ∈ LX will be called normal provided it is a continuous map fromXtoXwhenXis endowed with the weak^∗topology. We denote the set of all normal operators by LnX. We can consider the set of all normal contractive operators LnX₁ as a semigroup under multiplication of operators, that is, if T1, T2 ∈ LnX₁, then T1 ◦ T2 is normal and contractive; moreover, the multiplication is associative. The semigroup L_nX₁is semitopological when endowed with the pointwise weak^∗topology, that is, the multiplication is separately continuous. This means that the maps T → T ◦S and S → T ◦S are both continuous with respect to the pointwise weak^∗ topology. Finally we remark that it is important to note that L_nX₁ is not closed in LX₁ with respect to the pointwise weak^∗ topology. Moreover, recall that an operator T ∈ LX is normal if and only if there exists a unique predual operatorT_∗ fromX_∗ into X_∗, defined by Tx, ϕ x, T_∗ϕ,x ∈ X, ϕ∈X∗.

In this section, our goal is to study one-parameter semigroups on dual Banach spaces.

A contractive one-parameter semigroup9,10is a family{Tt}_t≥0 of linear and contractive operators onX, such thatTs◦Tt Tstfor alls, t ≥ 0 andT0 id_X. The semigroup is called weak^∗ continuous provided eachTt is a normal operator and0,∞ t → Ttxis weak^∗ continuous for any x ∈ X. For a weak^∗ continuous semigroup there exists the following concept of a weak^∗generatorZ:

Zxlim

t↓0

Ttx−x

t in the weak^∗ topology, 2.2

domZ{x∈X: the limit in2.2exists}. 2.3 The predual semigroup{Tt,∗}_t≥0of a weak^∗continuous semigroup{Tt}_t≥0 is strongly contin- uous, and the adjoint of its generatorZ∗is equal to the weak^∗generatorZ.

Suppose now that{Tt}_t≥0is a weak^∗continuous contractive semigroup onX, and write S0 {Tt :t ≥ 0} ⊆ LnX₁. In the following, we assume that the closure of S0 in LX₁with respect to the pointwise weak^∗topology consists of normal operators, that is, we assume that S S₀ ⊆ L_nX₁, where the bar denotes closure in the pointwise weak^∗ topology. Then S is a compact commutative semitopological subsemigroup of LnX₁. We now use the fact that every compact commutative semitopological semigroup has a unique minimal ideal G⊆ S, the so-called Sushkevich kernel9, which is given by

G

R∈S

R◦S, 2.4

andQ∈G will denote the unit of G. We then have GQ◦S. By compactness of G, it follows that G is, in fact, a commutative topological group. In the following, we will simplify our notation by writingT1T2instead ofT1◦T2.

We are now able to prove a weak^∗version of the Jacobs-de Leeuw-Glicksberg splitting theorem, originally going back to Jacobs11and de Leeuw and Glicksberg12,13, see also 14. The present proof mimics the one given in9for weakly almost periodic one-parameter semigroups.

Theorem 2.1. Let S0 {Tt}_t≥0 be a weak^∗ continuous contractive one-parameter semigroup with generatorZ. Assume that SS0consists of normal operators. Then there exist weak^∗closed subspaces Xs,XrofXinvariant under all operatorsTt,t≥0, such thatXXs⊕Xr, and

Xs

x∈X: 0∈ {Ttx:t≥0}^w∗

, 2.5

Xr lin{x∈domZ:∃α∈Rsuch thatZxiαx}^w∗ lin{x∈X :∃α∈Rsuch thatTtx e^iαtx∀t≥0}^w∗.

2.6

Proof. SinceQ² Q, the unitQis a normal projection such thatQ, Tt 0 for allt≥0. The theorem will be established once we prove thatXs kerQandXrranQ.

Letx ∈ kerQ. SinceQ ∈ S, there is a net {Ti}_i∈I ⊆ S0 such thatTi → Qrelative to the pointwise weak^∗topology; henceTix → Qx 0, so 0∈ {Ttx:t≥0}^w∗. Conversely, assume 0 ∈ {Ttx:t≥0}^w∗for somex. Then there is a net{Ti}_i∈I ⊆S0 such thatTix → 0 relative to the weak^∗ topology. By compactness of S, there is a subnet{Tj}_j∈J ⊆ {Ti}_i∈I with Tj → Rrelative to the pointwise weak^∗topology for someR∈S, and it follows thatRx0.

HenceRQRx 0 for allR ∈ S. ChoosingRto be the inverse ofQRin G, we getQx 0, hencex∈kerQ. We have thus proved thatXs kerQ.

LetG be the character group of G. For eachχ∈G define the operator

Xx−→Pχx

G

χSSx dμS, 2.7

whereμis the normalized Haar measure of G. The integral is to be understood as a weak^∗ integral, thusPχis a well-defined bounded operator in LXwithPχ ≤1. Then for allR∈G we get

RPχx

G

χSRSxdμS χR

G

χRSRSxdμS χR

G

χSSx dμS χRPχx,

2.8

in particularQPχ Pχ; therefore,TtPχ TtQPχ χTtQPχfor allt≥0. Sincet→χTtQis continuous and satisfies the functional equation

χTtQ·χTsQ χTtsQ∈ {z∈C:|z|1} 2.9

for allt, s≥0, we haveχTtQ e^iαtfor someα∈R. ThusTtPχ e^iαtPχ, hencePχX⊆domZ andZPχiαPχfor allχ∈G. We next define the subspace

Mlin

⎧⎨

⎩

χ∈G

PχX

⎫⎬

⎭

w^∗

⊆Xr. 2.10

We prove that ranQ⊆M⊆Xr. Letϕ∈M^⊥{ϕ∈X_∗:x, ϕ0∀x∈M}. ThenPχx, ϕ0 for allx∈X,χ∈G, that is,

G

χS Sx, ϕ

dμS 0 2.11

for allx∈X,χ∈G. Since the character groupG is total inL²G, μby the Stone-Weierstraß theorem and sinceS → Sx, ϕis continuous it follows thatSx, ϕ 0 for allS ∈ G and x ∈ X. Take S Q, then we obtain ϕ ∈ ranQ^⊥ and thus M^⊥ ⊆ ranQ^⊥. By the bipolar theorem we obtain ranQ⊆coMM, since ranQis a weak^∗closed subspace. Conversely, let x ∈ domZ with Zx iαx for someα ∈ R. It follows that Ttx e^iαtxfor all t ≥ 0 and consequentlyRx e^iαxforR ∈S. Thus there existsβ ∈Rsuch thatQx e^iβx Q²x.

Consequently, we must haveβ0 which impliesQxx∈ranQ, henceXr⊆ranQ, and the proof is finished.

Corollary 2.2. Under the hypothesis ofTheorem 2.1, there exists a weak^∗continuous one-parameter group{αt}_t∈Rof isometries onXrsuch thatαtTtXr fort≥0.

Proof. LetT ∈S, thenQT ∈G, and letRbe the inverse ofQT in G, that is,RQT Q. Then for allx∈Xr, we haveRTxRQTxQxx. Now writeαtQTtfort≥0 and letα_−tbe the inverse ofαtin G. The foregoing calculation shows that{αt}_t∈Ris a one-parameter group on Xr. Moreover, it is clear that it is weak^∗continuous and contractive. Now assume that there is x∈ Xrandt≥ 0 such thatαtx <x. Then it follows thatα_−t> 1, contradiction; thus {αt}_t∈Ris isometric.

The subspaceXris called the reversible subspace andXsis called the stable subspace; its elements are sometimes called flight vectors.

In applications it is sometimes desirable to have a stronger characterization of the subspace Xs, namely, we are interested in a stronger stability property of the elements in Xs. In particular, this is relevant in the applications to decoherence we discuss inSection 4.

The next result provides a sufficient condition for weak^∗stability to hold onXs based on the boundary spectrum specZ∩iRof the generatorZ.

Proposition 2.3. Assume that the hypothesis of Theorem 2.1 is satisfied and additionally that specZ ∩iRis at most countable. Then the stable subspace2.5is given by

Xs

x∈X: lim

t→ ∞Ttx 0 relative to the weak^∗topology

. 2.12

Moreover, the convergence in2.12is uniform forxinXs∩X1.

Proof. Consider the predual semigroup{Tt,∗}_t≥0with generatorZ∗; as already remarked,Z∗^∗ isZ. The predualQ_∗ofQis a projection and induces a splittingX_∗X_r,∗⊕X_s,∗by way ofX_r,∗

ranQ∗andXs,∗ kerQ∗. Let{T_t,∗^s}_t≥0be the restriction of{Tt,∗}_t≥0 toXs,∗. SinceXs,∗is closed;

the generatorZ^s_∗of{T_t,∗^s }_t≥0is given by the restrictionZ^s_∗Z∗Xs,∗, domZ_∗^sdom Z_∗∩Xs,∗. A similar construction applies to the reversible subspaceXr. We check that specZ_∗^s ⊆specZ∗. Letλ ∈ ρZ_∗, that is, the mapλ1−Z_∗ : domZ_∗ → X_∗ is bijective. Then clearly the map λ1−Z_∗^s λ1−Z∗Xs,∗: domZ∗∩Xs,∗ → Xs,∗is injective. It is also surjective: letϕ∈Xs,∗, then there isψψs⊕ψr∈domZ∗such thatλ1−Z∗ψ ϕ. Now

λ1−Z∗ψ λ1−Z∗ ψs⊕ψr

ϕ⊕0, 2.13

so λ1−Z∗ψr 0 andψr 0 by injectivity. Thus λ1−Z_∗^sis bijective andλ ∈ ρZ_∗^s. In particular, using specZspecZ∗we find that

specZ_∗^s∩iR is countable. 2.14

We now see that

spec_pZs∩iR∅, 2.15

for if iλ ∈ spec_pZs,λ ∈ R, then the corresponding eigenvector x ∈ domZs ⊆ Xs satisfying Zxiλxmust lie inXrby2.6, hencex0, contradiction.

From2.14and 2.15, it follows by the Arendt-Batty-Lyubich-Vutheorem15,16, see also9, that the semigroup{T_t,∗^s }_t≥0is strongly stable, that is, for all x∗ ∈ Xs,∗we have lim_{t→ ∞}T_t,∗^s x_∗0. Thus ifx∈Xsandx_∗∈X_∗it follows that

|Ttx, x_∗|

Q^⊥Ttx, x_∗ x, T_t,∗

Q^⊥_∗x_∗

≤ x · T_t,∗^s

Q^⊥_∗x∗ −→0

2.16

ast → ∞uniformly forx∈Xs∩X1, whereQ^⊥id_X−Qdenotes the projection ontoXs.

3. Semigroups on von Neumann Algebras

The results of the previous section apply to the case of Neumann algebras. LetHbe a Hilbert space. A von Neumann algebra is a^∗-subalgebra of the Banach-^∗-algebra LHof all bounded linear operators acting on H, which is additionally closed in the weak or equivalently strongoperator topology. The identity operator will be denoted by1, and we will always assume that1∈M. The ultraweak topology onMis defined by the seminormspρx |trρx|, where ρ runs through the trace class operators on H, it agrees with the weak operator topology on bounded portions ofM. The set of all ultraweakly continuous linear functionals onMforms a Banach space, and this Banach space is the uniqueup to isomorphismpredual space ofM, for this reason we denote it byM∗. The ultraweak topology onMcan be shown to be equivalent to theσM,M∗ i.e., weak^∗topology. Hence the setup of the previous section applies to this case. The set of all positive operators ofMwill be denoted byM. A functional

ϕinM∗ which is positivei.e.,ϕx ≥ 0 provided x ∈ Mand normalizedi.e.,ϕ 1, equivalentlyϕ1 1will be called a normal state. A state is called faithful ifx ∈ Mand ωx 0 impliesx0. For proofs of these results, we refer to10,17.

Let T ∈ LM. Then T is called positive if TM ⊆ M. A positive operator is normali.e., weak^∗ continuousif and only if for every uniformly bounded increasing net {xi}_i∈I ⊆ M we have sup_iTxi Tsup_ixi. Furthermore, T is called strongly positive whenever it satisfies Kadison’s inequality, that is,T1Tx^∗x≥Tx^∗Txfor anyx∈M.

Clearly strong positivity implies positivity. An even stronger notion of positivity is complete positivity:T∈LMis called completely positive whenever_n

i,j1y_i^∗Tx^∗_ixjyj ≥0 for alln∈N and allx1, . . . , xn and y1, . . . , yn from M. The mapT is called unital ifT1 1; a positive unital map is automatically contractive, that is,Tx ≤ xfor allx∈M.

The following result has been established in18.

Proposition 3.1. Suppose thatS⊆L_nM₁is a subset of normal contractive linear operators. Then the following assertions are equivalent.

1The set{T_∗ϕ:T ∈S} ⊆M_∗is relatively weakly compact for everyϕ∈M_∗.

2The set S is equicontinuous when M is endowed with the Mackey topology (i.e., the τM,M∗topology).

3The pointwise weak^∗closure ofSconsists of normal operators:S⊆L_nM₁.

Moreover, these conditions are satisfied whenever there is a faithful normal stateωonMsuch that

ωTx^∗Tx≤ωx^∗x for anyT ∈S, x∈M. 3.1

In particular, if each element in S is strongly positive we conclude that 3.1 can be rewritten as ωTx≤ωxfor allx∈M,T ∈S, or brieflyω◦T ≤ω, for allT∈S.

IfS⊆L_nM₁is a subset of normal contractive linear operators andωa normal state, we call ω an invariant state under S providedωTx ωx for allx ∈ Mand T ∈ S.

We now apply the results ofSection 2 to weak^∗ continuous semigroups on von Neumann algebras. This gives us the following result.

Corollary 3.2. Suppose that{Tt}_t≥0is a weak^∗continuous contractive strongly positive one-param- eter semigroup on a von Neumann algebra Mwith ultraweak generatorZ, and suppose that there exists a faithful normal invariant stateω. Then there exist weak^∗closed andTt-invariant subspaces MrandMsofM, given by2.5and2.6, such thatMMr⊕Ms.

Proof. By Kadison’s inequality, 3.1 holds; thusProposition 3.1 implies that the pointwise weak^∗ closure S S₀, with S₀ {Tt}_t≥0, consists of normal operators. Hence Theorem 2.1 applies.

It is worth pointing out that a similar result was recently established in19for general semigroups acting on a W^∗-algebra and possessing a faithful family of subinvariant states.

We now prove thatMris actually a von Neumann subalgebra. Recall that a conditional expectation Q from a C^∗-algebraA onto a C^∗-subalgebra B ⊆ A is a completely positive contraction withQx xforx∈BandQxyx xQyxforx∈B,y∈A.

Proposition 3.3. Let{Tt}_t≥0be a weak^∗ continuous semigroup of strongly positive unital operators and suppose there exists a faithful normal invariant stateω. ThenMris a von Neumann subalgebra ofMand there exists a group of^∗-automorphisms{αt}_t∈RonMrsuch thatTtMrαtfor allt≥0.

Moreover, there exists a normal conditional expectationQfromMontoMr such thatω◦Q ω.

Finally,Msis^∗-invariant.

Proof. Since each Tt is a contraction; Corollary 3.2 applies. Let M0 {x ∈ M : ∃α ∈ R such that Ttx e^iαtx∀t ≥ 0}, that is, we have linM0

w^∗

Mr. As in20we define the sesquilinear mapD:M×M → MbyDx, y Ttx^∗y−Ttx^∗Ttyfor some fixedt≥0.

By Kadison’s inequality, the sesquilinear formϕ◦Dis positive-definite for anyϕ ∈M_∗, so by the Cauchy-Schwarz inequality,Dx, x 0 if and only ifDx, y 0 for ally∈M. Now letx ∈ M0, thenTtx^∗x ≥ Ttx^∗Ttx e^−iαte^iαtx^∗x x^∗x. Thus 0 ≤ ωTtx^∗x−x^∗x ≤ ωx^∗x−x^∗x 0, and by faithfulnessTtx^∗x x^∗x, henceDx, x 0 for allx ∈ M0. So Dx, y 0 for allx, y ∈ M0, that is,Ttx^∗y Ttx^∗Tty e^iα2−α1tx^∗y, and we conclude thatxy∈M0wheneverx, y ∈M0. It follows that linM0is a^∗-subalgebra ofMcontaining 1 and consequently Mr is a von Neumann subalgebra, and Ttx^∗y Ttx^∗Tty for all x, y ∈ Mr. By Corollary 2.2, the restriction of Tt to Mr extends to a one-parameter group {αt}_t∈Rof isometries and the above argument shows thatαtmust be a^∗-homomorphism. Let Qbe the Sushkevich kernel of the semigroupS⊆LnM₁. SinceQis a projection andQ1 it follows from Tomiyama’s theorem21thatQis a conditional expectation; sinceQ∈S; it is also clear thatω◦Qω. The last assertion is clear as well.

In the following, we will be interested in the stronger characterization of Ms by a stability property as in2.12. We start by quoting the following result.

Lemma 3.4. Suppose that{Tt}_t≥0is a weak^∗continuous one-parameter semigroup of strongly positive unital operators on the von Neumann algebraMwith a faithful normal invariant stateω. Introduce the subsets

M

x∈M:Ttx^∗x Ttx^∗Ttx∀t≥0 , M^∗

x∈M:Ttxx^∗ TtxTtx^∗ ∀t≥0 , M1M∩M^∗.

3.2

ThenM1is aTt-invariant von Neumann subalgebra ofM, and there exists a group of^∗-automorphisms {αt}_t∈RonM1such thatTtM1αtfort≥0. Moreover,M1is a maximal (in the sense of not being properly contained in a larger von Neumann subalgebra) von Neumann subalgebra on which the restriction of{Tt}_t≥0is given by a group of^∗-automorphisms.

A proof can be found in22 see the proof of Proposition 2. It is easy to see that we always haveMr⊆M1.

Lemma 3.5. Under the assumptions ofLemma 3.4, for everyx∈Mthe weak^∗limit points of the net {Ttx}_t∈Rlie inM1.

A proof of this statement is contained in the proof of Theorem 3.1 of23.

We can now establish the following result.

Proposition 3.6. Let{Tt}_t≥0be a weak^∗ continuous semigroup of strongly positive unital operators on the von Neumann algebraMwith a faithful normal invariant stateω. IfMrM1, it follows that

Ms

x∈M: lim

t→ ∞Ttx 0 in the weak^∗ topology

. 3.3

Proof. Letx∈Msand assume without loss of generality thatx ≤1. By Alaoglu’s theorem the net{Ttx}_t∈Rcontained in the unit ball ofMhas a limit pointx0fort → ∞. Then using Lemma 3.5, we find thatx0 ∈M1 Mr. But sincex∈Ms, it follows that alsox0 ∈Ms, that is,x0 ∈Ms ∩ Mr {0}; hencex0 0. This proves that any limit point of the net{Ttx}_t∈R

is equal to 0; therefore we conclude that lim_{t→ ∞}Ttx 0 in the weak^∗ topology for allx ∈ Ms.

Moreover, let us remark the following: suppose that{Tt}_t≥0is a weak^∗continuous semigroup of strongly positive unital operators with generator Z having a faithful normal invariant stateω, and assume that the peripheral spectrum specZ∩iRis at most countable. Then by usingProposition 2.3the conclusion ofProposition 3.6holds. These results will be used in the next section when we discuss the notion of decoherence for uniformly continuous quantum dynamical semigroups.

4. Applications to Decoherence

4.1. The Notion of Decoherence in the Algebraic Framework

Consider a closed quantum system whose algebra of observables is a von Neumann algebra N, and its reversible time evolution is given by a weak^∗ continuous group of ^∗- automorphisms{βt}_t∈RonN. A subsystem can be described by a von Neumann subalgebra M ⊆ N containing the observables belonging to the subsystem. We will assume that there exists a normal conditional expectationEfromNontoM. In this situation, we can define the reduced dynamics as follows:

Ttx E◦βtx, x∈M, t≥0. 4.1 This is the Heisenberg picture time evolution an observer whose experimental capabilities are limited to the system described byMwould witness. Since it is the time evolution of an open system it is, in general, irreversible. From4.1we can isolate some mathematical properties of the reduced dynamics.

1{Tt}_t≥0is a family of completely positive and normal linear operators onM.

2Tt1 1for allt≥0, in particular; eachTtis contractive.

3t→Ttxis weak^∗continuous for anyx∈M.

In general the reduced dynamics{Tt}_t≥0 is not Markovian, that is, memory-free, and hence the operators{Tt}_t≥0do not form a one-parameter semigroup. However, in many physically relevant situations it is a good approximation to describe the reduced dynamics by a semigroup satisfying the above properties1–3, that is, a weak^∗ continuous semigroup of completely positive unital maps on the von Neumann algebra M. Such a semigroup is called a quantum dynamical semigroup. We remark that in many physically relevant models

we have the following structure: N M⊗M0, acting on a tensor productH ⊗ H0 of two Hilbert spaces, whereM0 describes the environment of the systeme.g., a heat bath. The time evolution of the system and environment is Hamiltonian, that is,βtx e^itHxe^−itHwith HH1⊗11⊗H2Hint, whereH1andH2are the Hamiltonians belonging to the system and its environment, andHintis an interaction term. Moreover, the conditional expectationEωis given with respect to a reference stateωof the environment, that is,ϕ ⊗ωx ϕEωxfor allx ∈Nandϕ ∈ M∗. In this situation, the predual time evolution is given by the familiar formula

T_t,∗

ϕ tr₂

e^−itH ϕ⊗ω

e^itH

, 4.2

whereϕis a normal state onMand tr₂denotes the partial trace with respect to the degrees of freedom of the environment.

Since the reduced dynamics is in general not reversible, new phenomena like the approach to equilibrium can appear. In this paper, we are particularly interested in an effect called decoherence. The following general and mathematically rigorous characterization of decoherence in the algebraic framework was introduced by Blanchard and Olkiewicz3,4.

Its present form is taken from24; the relation of this form and that given in3is discussed in25.

Definition 4.1. We say that the reduced dynamics{Tt}_t≥0displays decoherence if the following assertions are satisfied: there exists aTt-invariant von Neumann subalgebraM1ofMand an weak^∗continuous group{αt}_t∈Rof^∗-automorphisms onM1 such thatTtM1αtfort≥0, and aTt-invariant and^∗-invariant weak^∗closed subspaceM2ofMsuch that

MM1⊕M2, 4.3

t→ ∞limTtx 0 in the weak^∗ topology for anyx∈M2. 4.4

Moreover, we require thatM1 is a maximal von Neumann subalgebra ofMin the sense of not being properly contained in any larger von Neumann subalgebraon which{Tt}_t≥0 extends to a group of^∗-automorphisms. We callM1the algebra of effective observables.

The physical interpretation of this definition is rather clear: if decoherence takes place there is amaximalvon Neumann subalgebra on which the reduced dynamics is reversible, that is, given by an automorphism group, and a complementary subspace on which the expectation values with respect to any normal state of all its elements tend to zero in time.

Thus any observable x ∈ Mhas a decomposition x x1 x2, where xi ∈ Mi, such that ϕTtx2 → 0 ast → ∞for any normal stateϕonM. Hence after a sufficiently long time the system behaves effectively like a closed system described byM1 and{αt}_t∈R. By analyzing the structure of the algebra of effective observablesM1 and the reversible dynamics{αt}_t∈R various physically relevant and well-known phenomena of decoherence can be identified, like the appearance of pointer states, environment-induced superselection rules, and classical dynamical systems. In this way it is possible to obtain an exhaustive classification of possible decoherence scenarios, see3for a thorough discussion of this point. Particularly interesting is the case when M1 is a factor, that is, after decoherence we still have a system of pure

quantum character. This is of interest in the context of quantum computation since in this way one may obtain a system which retains its quantum character despite decoherence.

According toDefinition 4.1, if{Tt}_t≥0is a group of automorphisms, decoherence takes place and the splitting 4.3 is trivial withM2 {0}. However, we will keep this slightly unfortunate terminology since it simplifies the statements of theorems, keeping in mind that physically decoherence corresponds to the case whenM2/{0}. This can only happen if{Tt}_t≥0is irreversible.

We remark that the algebraM1has been studied in26and explicit representations of M1are obtained for quantum dynamical semigroups with unbounded generators. Moreover, in27a different notion of decoherence for quantum dynamical semigroups is introduced in a mathematically rigorous way, and its connection to the Blanchard-Olkiewicz notion is briefly discussed in28. In29an asymptotic property similar to4.3and4.4is discussed under the designation “limited relaxation.”

In connection withDefinition 4.1, the following question arises: if there exists a maxi- mal von Neumann subalgebra ofMon which{Tt}_t≥0extends to a group of automorphisms, is this subalgebra necessarily unique? The following theorem answers this question.

Theorem 4.2. Let{Tt}_t≥0be a quantum dynamical semigroup and suppose it has a faithful normal invariant stateω. Then there exists a unique maximal von Neumann subalgebra on which{Tt}_t≥0 extends to a group of automorphisms.

Proof. LetMi, wherei∈Iis some index set, be a collection of von Neumann subalgebras on each of which{Tt}_t≥0extends to a group of automorphisms. Then we put

B

i∈I

Milin{xi1· · ·xik :i1, . . . , ik∈I, k∈N, xi ∈Mi}, 4.5

where the closure is taken in the ultraweak topology. We proceed as in the proof of Proposition 3.3and introduceDx, y Ttx^∗y−Ttx^∗Ttyfor a fixedt≥0 andx, y ∈M.

ThenDx, x 0 if and only ifDx, y 0 for ally∈M. Now ifx∈Mi, we getDx, x 0, thusDx, y 0 fory∈M, wherei∈I, that is,Ttxy TtxTty. Proceeding inductively we have

Ttxi1· · ·xik Ttxi1· · ·Ttxik, 4.6

for an arbitrary monomial xi1· · ·xik, where xi ∈ Mi, i1, . . . , ik ∈ I. Therefore, if B0 lin{xi1· · ·xik : i1, . . . , ik ∈ I, k ∈ N}, thenTt : B0 → B0 is a^∗-homomorphism of the^∗-sub- algebraB0, andTt:B → Bis a^∗-homomorphism as well.

Next we note thatTt B is injective. Namely, ifx ∈ kerTt B, we get ωx^∗x ωTtx^∗x ωTtx^∗Ttx 0, thusx0 by faithfulness ofωand so kerTtB {0}.

We now establish that Tt : B → B is surjective. Since Tt is an injective ^∗-homo- morphism on the C^∗-algebra B, it follows that Ttx x for all x ∈ B. Moreover, notice thatTt:B0 → B0is invertible since each restrictionTtMiis invertible. This implies T_t⁻¹xTtT_t⁻¹xxfor anyx∈B0. Now for the proof of surjectivity choosey∈B.

By the Kaplansky density theorem, there exists a net{yj}_j∈J ⊆ B0 such thatyj ≤ Cfor all j ∈ J, and such that limyj y. Put xj T_t⁻¹yj. Thenxj yj ≤ C, and by Alaoglu’s

theorem the net{xj}_j∈Jhas an ultraweak limit pointx0. Thus there is a subnet{xj} ⊆ {xj}_j∈J such that limxjx0, hence

yj Tt

xj

−→yTtx0. 4.7

We conclude by injectivity that any ultraweak limit point of the net {xj}_j∈J is equal tox0, hence this net is convergent tox0, andTtx0 y, establishing surjectivity. We have thus proved thatTtBis a^∗-automorphism.

To finish the proof, we have to choose the collection{Mi :i∈I}in4.5to consist of all von Neumann subalgebras on which{Tt}_t≥0extends to a group of automorphisms.

We next prove that in certain cases the splitting4.3is always given by a conditional expectation.

Proposition 4.3. Suppose that the (not necessarily Markovian) reduced dynamics{Tt}_t≥0on the von Neumann algebraMdisplays decoherence. IfTtM1 id_M1, then there exists a normal conditional expectationEfromMontoM1.

Proof. Letx ∈ Mand write x x1x2 with xi ∈ Mi,i 1,2, and defineE : M → M1

byEx x1. ThenEM M1 andE² E. SinceM2 is^∗-invariant,Ex^∗ Ex^∗₁x^∗₂ x^∗₁ Ex^∗, henceEM^sa ⊆M^sa. Now letx∈M, consider the decompositionx x1x2, xi∈Mi,i1,2, and suppose thatx1∈M^sais not positive. Then there existsϕ∈M_∗ such that ϕx1<0. This implies

0≤ϕTtx ϕTtx1 ϕTtx2 ϕx1 ϕTtx2, 4.8 and lettingt → ∞ yields ϕTtx2 → 0, a contradiction. Thus EM ⊆ M, and since E1 1it follows thatE ≤1. FromE² E, we getE1, soEis a projection of norm 1 and hence by Tomiyama’s theorem a conditional expectation. Since kerE{x∈M:Ex x1 0} M2 is ultraweakly closed, we obtain by a theorem of Tomiyama 30 thatE is normal.

Given a reduced dynamics, the question arises of under what conditions decoherence will occur. In the Markovian case, sufficient conditions for the appearance of decoherence in the sense of Definition 4.1have been formulated in 31. In fact, we have the following theorem.

Theorem 4.4. Let{Tt}_t≥0 be a weak^∗ continuous one-parameter semigroup with a faithful normal stateω. Assume that the following conditions are satisfied.

1EachTt,t≥0, is strongly positive and unital.

2Let{σ_t^ω}_t∈Rdenote the modular group (see, e.g., [10]) corresponding to the stateω. Assume thatTt, σ_s^ω 0 for alls∈Randt≥0.

Then{Tt}_t≥0 displays decoherence and there exists a normal conditional expectationEfromMonto M1such thatTt, E 0 for allt≥0 andω◦Eω.

The splittingMM1⊕M2of{Tt}_t≥0provided by this theorem is called the isometric- sweeping splitting. In31the theorem was proved under the more general hypothesis that

ω is only a faithful semifinite normal weight. Then some additional technical assumptions about{Tt}_t≥0are necessary. A simpler proof for the theorem as stated above has been given in24.

The Jacobs-de Leeuw-Glicksberg splitting for weak^∗ continuous semigroups on von Neumann algebras established inCorollary 3.2can now be applied to establish decoherence.

Corollary 4.5. Let{Tt}_t≥0be a weak^∗continuous one-parameter semigroup of strongly positive unital operators with a faithful normal stateω. Assume thatMrM1. Then{Tt}_t≥0displays decoherence.

Proof. According to Corollary 3.2, the Jacobs-de Leeuw-Glicksberg splitting exists, and by Proposition 3.6we conclude that the requirements ofDefinition 4.1are satisfied.

Remark 4.6. Whenever a conditional expectationEfrom a von Neumann algebraMonto a von Neumann subalgebraM1satisfiesω◦Eωfor a faithful normal stateω, it is uniquely determined by these conditions17, Corollary II.6.10. Since in case of the isometric-sweeping splitting as given byTheorem 4.4we haveM1 EMandM2 1−EM, and in the Jacobs- de Leeuw-Glicksberg splitting as given byCorollary 3.2we haveMr QMandMs 1− QM, it follows that the isometric-sweeping and Jacobs-de Leeuw-Glicksberg splittings agree wheneverMrM1.

4.2. Uniformly Continuous Semigroups

The purpose of this section is to show how the Jacobs-de Leeuw-Glicksberg splitting can be applied to establish decoherence of quantum dynamical semigroups in the sense of Definition 4.1. To avoid complications arising from unbounded generators, we will concentrate on the case of uniformly continuous quantum dynamical semigroups. We will arrive at a result which avoids assumption2inTheorem 4.4.

LetMbe a von Neumann algebra acting on a separable Hilbert spaceH, and let{Tt}_t≥0 be a quantum dynamical semigroup such thatt→Ttis continuous in the uniform topology.

Then by32the generatorZof{Tt}_t≥0, which is a bounded operator onM, is given by

ZxG^∗xxG Φx, 4.9

whereG ∈LHandΦ: M → LHis a normal completely positive map. Since we have Tt1 1for allt≥0, it follows thatZ10 which forcesG^∗−G−Φ1. Upon introducing the operatorH iG 1/2iΦ1, it is seen thatHis a bounded selfadjoint operator onH andZmay be written as

ZxiH, x−1

2{Φ1, x} Φx, 4.10

where{·,·}denotes the anticommutator. Let us suppose now thatΦhas a Kraus decomposi- tion

Φx ^∞

n1

A^∗_nxAn, 4.11

where{An}_n∈Nis a sequence of bounded linear operators onHand the series converges in the weak^∗topology. This is always the case ifMis injective or equivalently, by the Connes

theorem33and separability ofH, thatMis hyperfinitethis includes the caseMLH.

The preadjoint operator ofZonM_∗then has the familiar Lindblad form34

Z_∗ρ−i

H, ρ −1 2

∞ n1

ρA^∗_nAnA^∗_nAnρ ^∞

n1

AnρA^∗_n, 4.12

whereρ∈M∗. We are now able to prove the following theorem.

Theorem 4.7. Let{Tt}_t≥0be as above. Assume that there is a faithful normal stateωsuch thatω ◦ Ttωfor allt≥0 and thatHin4.10has pure point spectrum. Then{Tt}_t≥0displays decoherence, and for the effective subalgebraM1we have

M1⊆ {An, A^∗_n:n∈N}∩M, 4.13 where the prime denotes the commutant. Moreover, there exists a normal conditional expectationQ fromMontoM1such thatω◦Qω. If the derivationx→iH, xleaves the subalgebra{An, A^∗_n: n∈N}∩Minvariant, equality holds in4.13.

Proof. First note that the assumptions ofProposition 3.3 are satisfied, that is, Mr is a von Neumann subalgebra. Consider the subalgebra M1 defined in Lemma 3.4, then {Tt}_t≥0 restricted toM1extends to a group of automorphisms. We start by proving4.13. By a simple calculation as in20, one obtains

Zx^∗x−Zx^∗x−x^∗Zx x^∗Φ1x Φx^∗x−Φx^∗x−x^∗Φx ^∞

n1

An, x^∗An, x. 4.14

The generatorZ, when restricted toM1, is a^∗-derivation; thus ifx∈M1, then

0Zx^∗x−x^∗Zx−Zx^∗x^∞

n1

An, x^∗An, x, 4.15

that is,An, x 0 for alln ∈ N, and, moreover, A^∗_n, x 0 for alln ∈ NsinceM1 is a^∗- subalgebra. This proves thatM1⊆ {An, A^∗_n :n∈N}∩M. Conversely, under the assumption that iH,·leaves the right-hand side of 4.13invariant, we haveZx iH, xor Ttx e^itHxe^−itHon{An, A^∗_n:n∈N}∩M, which implies equality in4.13.

Now letx∈Ms∩M1,x /0. Then there exist eigenvectorsξ, η ∈ HofHwith corres- ponding eigenvaluesEξandEηsuch thatξ, xη/0, thus

ξ, Ttxη

e^−itHξ, xe^−itHη

e^itEξ−Eη ξ, xη

4.16

is bounded away from 0, so 0 is not a weak^∗limit point of{Ttx:t≥0}. Sincex∈Ms, this is a contradiction in view of2.5, henceMs∩M1{0}. Now letx∈M1and writexxsxr∈ Ms⊕Mr. SinceMr ⊆M1, we havexr ∈M1, andxs x−xr ∈Ms∩M1 {0}, thusx∈Mr.

This provesM1Mrand it follows byProposition 3.6thatMshas the property3.3. So we conclude that{Tt}_t≥0 displays decoherence. The last assertion is clear fromProposition 3.3.

We remark that in29, equation34, a class of generators has been given for which equality in4.13always holds. As a corollary, we obtain the following result which is similar to the one proved in35and is also contained in36.

Corollary 4.8. Let{Tt}_t≥0 be a uniformly continuous semigroup on Mconsisting of normal com- pletely positive and contractive operators, and suppose it has a faithful normal invariant stateω. If {An, A^∗_n:n∈N}C1, then

t→ ∞limTtx ωx1 in the weak^∗topology 4.17 for anyx∈M.

Thus ifM1is trivial the semigroup describes the approach to equilibrium.

We remark that the last theorem can be generalized to certain cases when the semigroup{Tt}_t≥0is not uniformly continuous but has an unbounded generator of the form 4.10.

The existence of a faithful normal invariant state of a quantum dynamical semigroup as required by Theorems 4.7and4.4has been discussed in the literature. It is particularly simple in the case of a finite-dimensional von Neumann algebra, that is, a matrix algebra.

Suppose M MCd is the d × d-matrix algebra and consider a quantum dynamical semigroup {Tt}_t≥0 onM, then its generator is given by4.10 and its preadjoint by4.12, thus if theAn are normal,ρ0 1/d1 is a faithful normal invariant state for{Tt}_t≥0. Such generators arise, for example, in the singular coupling limit ofN-level systems, see8.

Acknowledgments

Thanks are due to Professor Ph. Blanchard Bielefeld for various discussions and to the anonymous referees for a number of remarks which helped to improve the paper and for pointing out an error in it.

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