On reduction of the wave-packet, decoherence, irreversibility and the second law of thermodynamics
aa r X i v : . [ m a t h - ph ] M a r On reduction of the wave-packet, decoherence,irreversibility and the second law ofthermodynamics
H. NarnhoferInstitut f¨ur Theoretische PhysikUniversit¨at WienBoltzmanngasse 5A1090 Vienna, Austria [email protected]
W. F. WreszinskiInstituto de FisicaUniversidade de S˜ao PauloRua do Mat˜ao, Travessa R 18705508-090 S˜ao Paulo, Brazil [email protected]
June 21, 2018
Abstract
We prove a quantum version of the second law of thermodynamics:the (quantum) Boltzmann entropy increases if the initial (zero time)density matrix decoheres, a condition generally satisfied in Nature. Itis illustrated by a model of wave-packet reduction, the Coleman-Heppmodel, along the framework introduced by Sewell (2005) in his ap-proach to the quantum measurement problem. Further models illus-trate the monotonic-versus-non-monotonic behavior of the quantum oltzmann entropy in time. As a last closely related topic, decoher-ence, which was shown by Narnhofer and Thirring (1999) to enforcemacroscopic purity in the case of quantum K systems, is analysedwithin a different class of quantum chaotic systems, viz. the quantumAnosov models as defined by Emch, Narnhofer, Sewell and Thirring(1994).A review of the concept of quantum Boltzmann entropy, as well asof some of the rigorous approaches to the quantum measurement prob-lem within the framework of Schr¨odinger dynamics, is given, togetherwith an overview of the C* algebra approach, which encompasses therelevant notions and definitions in a comprehensive way. Keywords
Quantum Boltzmann entropy, Wave packet reduction, De-coherence, Quantum measurement theory, Irreversibility, Relative entropy,Second law of thermodynamics, Coleman-Hepp model, Models of decoher-ence, Quantum chaotic systems, Quantum Anosov systems, Irreversibiltyversus collapse.
Contents
Reduction of the wave-packet: the approaches of Hepp andSewell 18 β = ∞ . . . . . . . . . . . . . . 435.3 The mixed case β < ∞ . . . . . . . . . . . . . . . . . . . . . . 445.4 The ”bad” microstates . . . . . . . . . . . . . . . . . . . . . . 455.5 Reduction of the wave packet according to definitions 2 and 3 465.6 Miscellaneous remarks: the role of the quantum Boltzmannentropy in the vanishing of cross-terms, and the notion of tem-perature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.7 Entropy reduction in events causing purification by state col-lapse and the role of quantum chaotic systems (quantum Ksystems) in the purification process . . . . . . . . . . . . . . . 495.8 Monotonic versus non-monotonic behavior of the quantumBoltzmann entropy as a function of time: some models . . . . 505.9 Decoherence for quantum Anosov systems . . . . . . . . . . . 535.10 Quantum Anosov systems: definitions, motivation and con-nection to classical mechanics . . . . . . . . . . . . . . . . . . 543.11 The upper quantum Lyapunov exponent . . . . . . . . . . . . 565.12 Quantum Anosov models of decoherence . . . . . . . . . . . . 585.13 Remarks on the connection with the ideas of Peres and Zurekand collaborators. The dependence of decoherence on specialparameter values in the purely quantum case: breakdown ofuniversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 The subject of quantum measurement theory and the related problems ofirreversibility and decoherence has a long history, beautifully reviewed andsummarized in [1],[2]. We refer to the classic work of von Neumann [3], aswell as to the ”big red book”, as Wightman calls it [4], as early references,and to the review articles of Zurek [5], [6], as well as the books [7], [8] [9],and the recent article [10] for comprehensive treatments of the subject andreferences.The recent review by Allahverdyan, Balian and Nieuwenhuizen [11] com-plements our treatment nicely, both from the conceptual point of view (thecentral concepts we use, the quantum Boltzmann entropy and quantumAnosov systems, are not contemplated in [11]) and the models which areanalysed. Indeed, the authors of [11] treat most extensively a mean fieldmodel, of interaction between Curie-Weiss spins and a phonon bath (see also[12]), while in our applications in chapter 5 we study models of local inter-action between a particle and a spin chain.On the other hand, rigorous treatments of the quantum theory of mea-surement within the framework of Schr¨odinger dynamics, which we adopt,have been extremely rare. We are only aware of the papers of Hepp [13],Sewell ([14], [15]), Whitten-Wolfe and Emch [16], see also [17] for a compre-hensive review. A recent paper on irreversible behavior and collapse of thewave-packets in a quantum system with point interactions is [18]: the modellacks, however, for the moment, a physical interpretation.4n section 3 we provide a review of the approaches of [13] and ([14], [15])to the quantum measurement problem. Since we hope to make the presentpaper accessible to theoreticians, not only mathematical physicists or math-ematicians, in section 2 an overview of what is needed from the C* algebraapproach of infinite quantum systems is given (for a more comprehensiveaccount, see [17] and [19]), as well as [20]).In section 4 we discuss the problem of irreversibility and its connectionto the quantum measurement problem. The quantum Boltzmann approachhas been discussed by C. Maes and coworkers in a similar spirit, albeit dif-ferent direction: see [21] for a comprehensive exposition and references. Inparticular, the model [22] is an example of a unitary quantum dynamicswith positive entropy production. The subject has also been recently illu-minated in a thorough and precise discussion of quantum probability by JFr¨ohlich and B. Schubnel [23]. In particular, the operator algebra viewpointis treated in greater depth there, and a unified discussion of concepts suchas decoherence, information loss and entanglement is provided.A central object in our approach is what we call the quantum Boltmannentropy. In order to explain its main features, it is convenient to considerclassical systems first, for which a comprehensive theory of the approachto equilibrium, characterized by the growth of the Boltzmann entropy to alimiting value, exists ([24], see also [25]).
We now follow [25] quite closely, for the reader’s convenience, because Lebowitz’sdiscussion contains all the essential points.The microscopic state of a classical system of N particles is represented bya point X ≡ ( ~r , ~p , · · · , ~r N , ~p N ) in its phase space Γ, ~r i and ~p i , i = 1 , · · · , N denoting the position and momentum of the ith particle. We suppose thedynamics to be Hamiltonian, associated to a flux T t which is phase-spacevolume preserving, by Liouville’s theorem, describing the macroscopic state.The latter will be denoted by M and consists of a system of N (of order ofAvogadro’s number) atoms in a box V , with volume | V | ≥ N l , where l isa typical atomic scale. Divide V into K ”cells”, where K is large but still K ≪ N , and specify number of particles, momentum and energy in each cell.All assertions in the following are to be taken with some tolerance, one thinksof the ”thickened” microcanonical ensemble, see, e.g., [26], pg. 1960. M is5hus determined by X - we write, always following [25], M ( X ), but there isa continuum of X corresponding to the same M . Let Γ M denote the regionin Γ consisting of all microstates X corresponding to a given macrostate M ,and | Γ M | = R Γ M Q Ni =1 d~r i d~p i N ! h N .Let, now, a gas of N atoms with energy E be initially confined by apartition to the left half of the box V , and suppose that this constraint isremoved at time t . The phase space volume available to the system fortimes t > t is, then, an enormous factor 2 N of the initial one. The space ofmacrostates consists in this case of all couples of pairs { ( N L , E L ) , ( N R , E R ) } satisfying both constraints N L + N R = N and E L + E R = E . Equivalently, wemay label a macrostate of this gas M = ( N L N , E L E ), the fraction of the particlesand energy in the left half of V ; the macrostate M at time t is M = (1 , X t , which at time t were in M , forms a region T t Γ M , whose volume is equal to | Γ M | , but whose shape will increasinglybe contained in regions Γ M corresponding to macrostates with larger andlarger phase space volumes, until almost all the phase points initially inΓ M are contained in Γ M eq , where M eq = (1 / , /
2) (up to fluctuations, see[25]). Boltzmann assumed that S B ( M eq ) ≡ k log | Γ M eq | is proportional to thethermodynamic entropy of Clausius, and generalised it to define an entropyalso for macroscopic systems not in equilibrium, by associating with eachmicroscopic state X the number S B ( M ( X )) ≡ k log | Γ M ( X ) | (1 . M , S B ( X t ) = k log | Γ M ( X t ) | typically increases in a way which qual-itatively describes the evolution towards equilibrium of macroscopic systems.”Typically” means that there will be ”bad” intitial microstates X ∈ Γ M -e.g., those which have all velocities directed away from the barrier lifted at t in the gas example - but the phase space volume of these is an utterlynegligible fraction of | Γ M | .We are here speaking of closed systems - systems completely isolatedfrom all external influences. Traditional thermodynamics takes this idealiza-tion as the starting point - see Clausius’ sweeping formulations of the firstand second laws, ”Die Energie der Welt ist konstant” and ”Die Entropie derWelt strebt einem Maximum zu” ([28]: ”The energy of the world is constant”6nd ”The entropy of the world tends to a maximum”). Accordingly, we takeclosed systems as our fundamental objects, hoping, as in [29] that, once ir-reversibility is understood for closed systems, ”it is possible to make someestimate of the severity of the approximations involved”. Although there hasbeen great progress in the theory of irreversibility for open quantum systems(see [30],[31] and references given there) examples such as the evolution ofthe Universe or the adiabatic irreversible expansion of a gas, in which theisolation of the system may be achieved to any degree of accuracy - a proto-type of events of daily experience - demonstrate the necessity of consideringboth open and closed systems. The procedure in thermodynamics [32] is alsoin conformance to our approach; compare also the discussion in ([33], firstparagraph page 311). For quantum systems, a macrostate M of a macroscopic quantum system isdescribed by von Neumann [3] (see the discussion in [25], section 8, whichwe follow below) by specifying the values of a set of commuting macroscopicobservables ˆ M , representing particle number, energy, etc., in each of thecells into which the box containing the system is divided. Labelling the setof eigenvalues of ˆ M by M α , with α = 1 , · · · , n , we have, corresponding to thecollection { M α } an orthogonal decomposition of the system’s Hilbert space H into linear subspaces ˆΓ α in which the observables assume the values M α .Let P α denote the projection onto ˆΓ α , ˆ µ the density matrix describing thesystem (ˆ µ = | Ψ)(Ψ | for a system in a pure state associated to a wave functionΨ), p α = T r ( P α ˆ µ ) (1 . µ = X ( p α | ˆΓ α | ) P α (1 . S mac (ˆ µ ) is then defined by S mac (ˆ µ ) ≡ − kT r (˜ µ log ˜ µ ) (1 . { M α } denote the set of possible macrostates of a classical system. Ithappens, however, that, whereas in the classical case the actual microstate7f the system is described by X ∈ Γ M α , for some α , so that the system isdefinitely in one of the macrostates M α , this is not so for a quantum systemspecified by ˆ µ or Ψ, i.e., the analogue of (1.1) is not valid. Indeed, even whenthe system is in a microstate Ψ or ˆ µ , corresponding to a definite macrostateat time t , only a classical system will be in a unique macrostate for all times t , because the quantum system may evolve to a superposition of differentmacrostates (”Schr¨odinger cats”), an example of which appears in section5.1. In addition, and related to this fact, is that, in contrast to the classicalcase, the state of a subsystem (”cell”) depends on possible correlations toother systems - i.e., if one extends the algebra of observables, different (e.g.”entangled”) states may be obtained. As Landsman aptly remarks [17], ”therestrictions of a generic state to all subsystems of a system does not, ingeneral, uniquely determine the state on the total system”. This means,grosso modo, that it is not possible to assign a wave-function to a subsystem- the analogue of the previous ”cells” in classical statistical mechanics.Both phenomena discussed above - Schr¨odinger cats and entanglement(in this connection see [34])- arise in quantum measurement theory. It seemstherefore of special interest to study irreversibility in the quantum theory ofmeasurement. This is attempted in sections 4 and 5.In order to do so, one really needs a formulation and proof of a form of thesecond law of thermodynamics for closed quantum systems, which includesboth the measurement and the thermal cases. This is done in section 4. Following Lindblad [35], we define a quantum Boltzmann entropy S QB (re-lated but not identical to S mac ) and show that for automorphic (unitary)time evolutions, with a initial state ˆ µ = ˆ µ (0) which is decoherent at timezero (definition 4), S QB (0) ≤ S QB ( t ) (1 . t ≥ We give a preliminary overview of the C ∗ algebra approach to infinite quan-tum systems as they are used in Hepp’s approach. According to the Haag-Kastler theory of local algebras of observables [42], one assumes that foreach bounded region Λ ⊂ R (Λ ⊂ Z for a lattice system) there is an as-9ociated algebra A (Λ), and Λ ⊂ Λ implies A (Λ ) ⊂ A (Λ ). By taking theunion over all Λ the local algebra A L ≡ ∪ Λ A (Λ) is obtained. Haag andKastler assumed that A (Λ) are C* algebras, which have a norm || . || and arecomplete in the topology defined by the norm, i.e., if { A n } ∞ n =1 ∈ A L andlim n →∞ || A − A n || = 0, then A ∈ A ≡ A L , where the bar denotes the abovedescribed norm closure: A is called the quasi-local algebra . This is a wayto arrive at an infinite system: although the limit points of A do not belongto the A (Λ) of any bounded region Λ, they may be viewed as ”essentiallylocal”.A C* algebra A is, thus, a vector space over the complex numbers, with anassociative multiplication operation compatible with the vector space struc-ture, an antilinear involution ∗ , satifying A ⋆⋆ = A and ( AB ) ⋆ = B ⋆ A ⋆ for all A, B ∈ A , a norm satisfying the triangle inequality, as well as the relation || AB || ≤ || A |||| B || , in such a way that A is complete in the topology in-duced by the norm, and, finally, the equality || A ⋆ A || = || A || . This definitionmay be motivated by the fact that it is satisfied by norm-closed subalge-bras of B ( H ), the set of bounded operators on a separable Hilbert space H ,with || A || = sup Φ (Φ , A ⋆ A Φ), with the supremum taken over all unit vectorsΦ ∈ H , where, above, ⋆ is the usual adjoint operation, and ( ., . ) denotes theinner product on H , antilinear in the first argument. One simple example ofthe latter structure is, on H = C , given by the Pauli matrices σ α , α = 1 , , σ σ = iσ (2 . . A Λ , for Λ ⊂ Z d ,for instance ( d ≥
1; in the example of section 3.1, d=1), becomes the concretespin algebra on H (Λ) = ⊗ i ∈ Λ C , generated by taking linear combinations,(ordinary and tensor) products of σ ji and i with i ∈ Λ and j = , , . Thelocal algebra is A L = ∪A (Λ) (2 . . A = A L (2 . . A spin .For particles, e.g. in one dimension, one considers H = L ( R ) on whichthe concrete algebra W of finite linear combinations of (bounded) Weyl op-erators W ( β, γ ) = exp[ i ( βx + γp )] (2 . . W ( β, γ ) † = W ( − β, − γ ) (2 . . W ( β, γ ) W ( β ′ , γ ′ ) = exp[ − / βγ ′ − β ′ γ )] W ( β + β ′ , γ + γ ′ ) (2 . . || W ( β, γ ) || = 1 || W ( β, γ ) − W ( β ′ , γ ′ ) || = 2 , β = β ′ ; γ = γ ′ (2 . . state on a C* algebra A is defined to be a positive linear form ω on A ,that is normalized to one on the unit element of A : ω ( αA + βB ) = αω ( A ) + βω ( B ) ω ( A ⋆ A ) ≥ ω ( ) = We shall see examples very shortly. A ⋆ -representation of A on a Hilbertspace H is a mapping Π of A into B ( H ), such thatΠ( αA + βB ) = α Π( A ) + β Π( B )Π( AB ) = Π( A )Π( B )Π( A ⋆ ) = [Π( A )] † The representation is cyclic, with cyclic vector Ψ, if the set of vectors { Π( A )Ψ; A ∈A} is dense in H . By the Gelfand-Naimark-Segal (GNS) construction ([43],see also [44], 1.8.1), if ω is a state on A , there exists a cyclic representationΠ ω ( A ) of A on a Hilbert space H ω , with cyclic vector Ψ ω , such that ω ( A ) = (Ψ ω , Π ω ( A )Ψ ω ) (2 . and Π of a given C* algebra A on Hilbert spaces H and H are equivalent iff there is a unitary map U : H → H such thatΠ ( A ) = U Π ( A ) U † for all A ∈ A . Related concepts of quasiequivalence anddisjointness are reviewed in [17], with references, but it should be at leastmentioned that two representations Π and Π are disjoint iff no subrepre-sentation of Π is unitarily equivalent to any subrepresentation of Π . Two11tates which induce disjoint representations are said to be disjoint: if theyare not disjoint, they are called coherent .Remark: For finite dimensional matrix algebras (with trivial center) allrepresentations are coherent.The concept of pure state is fundamental: a state is pure iff it cannot bedecomposed as a convex combination of other states, i.e., cannot be writtenas ω = λω +(1 − λ ) ω for some real 0 < λ < ω i = ω, i = 1 ,
2; otherwise,it is mixed . On B ( H ) the states ω ( A ) = tr H ( P Ψ A ) = (Ψ , A Ψ) with P Ψ = | Ψ)(Ψ | is pure ( || Ψ || = 1); the state ω ( A ) = tr H ( ρA ) where ρ = ρ † is apositive operator of trace one is mixed whenever ρ = ρ . A related conceptis that of a primary or factor representation Π: it is one which cannot bedecomposed as direct sum of two nontrivial disjoint subrepresentations; if Π ω is a primary representation, where Π ω is related to ω by (2.5), ω is said tobe a primary or factor state. Corresponding to the previous remark for finitedimensional matrix algebras all representations are factor representations.The set of all pure states on A falls into equivalence classes, whereby twostates ω and ω are defined to be in the same class if ω = ω B for some B ∈ A , where ω B ( A ) = ω ( B ⋆ AB ) ω ( B ⋆ B )and ω ( B ⋆ B ) = 0; these equivalence classes are called folia (see also [19] and[17]): we denote them by [ ω ] following [17]. Each folium [ ω ] of the set of purestates on A defines a superselection sector of the theory described by thealgebra of observables A . The fact that the abstract characterization of a C*algebra captures the concrete version exactly means that the representationsof the quasilocal algebra A occurring in different superselection sectors canbe isomorphic as C* algebras, even though they are not unitarily equivalent.This feature is a partial explanation of the naturalness of the C* algebraapproach to systems with infinite number of degrees of freedom ([19], [45]),and an explicit example of it will be given shortly.Concerning the GNS state ω , the following structure also arises nat-urally: a subalgebra of B ( H ω ) is called a von Neumann algebra if itis closed in the so-called weak topology: a sequence of operators { A i =Π ω ( A i ) , i = 1 , , , · · ·} converges weakly to A iff lim i →∞ (Φ , A i Ψ) = (Φ , A
Ψ)for all Φ , Ψ ∈ H ω . Then Π ω ( A ) ′ - the commutant of Π ω ( A ) - the setof all bounded operators on H ω which commute with every operator in12 ω ( A ), defines the algebra generated by the superselection rules. The center Z (Π ω ( A )) ≡ Π ω ( A ) ∩ Π ω ( A ) ′ . If ω is a factor state, the center Z = { λ } ,i.e., the operators in the center are dispersion free. This is the crucial prop-erty leading to the identification of factor states with pure thermodynamicphases (see [19]).The following construction is very important in section 3. Let D be a setof bounded regions Λ ⊂ R d such that ∪ D Λ = R d . For Λ ∈ D , let ˜ A Λ be theC* algebra generated by all A Λ ′ with Λ ′ ∈ D and Λ ∩ Λ ′ = ∅ and Π be arepresentation of A . The following algebra corresponds to operations whichmay be performed outside of any bounded set: L Π = ∩ D Π( A Λ ) (2 . algebra of observables at infinity in the representationΠ [46]. The double commutant of Π( A Λ ) in (2.6) is, by von Neumann’sbicommutant theorem ([43][44],(2.25)) equal to the weak closure of Π( ˜ A Λ ). L Π lies in the center of Π( A ) ′ and is, thus, abelian, a necessary prerequisitefor a set of classical observables.The macroscopic observables are special elements of L Π : for any sequenceΛ n ∈ D converging to infinity (i.e., almost all Λ n lie outside of any boundedregion), let A n ∈ A Λ n with || A n || ≤ b uniformly in n , and Π be a representa-tion of A . If w − lim N →∞ P Nn =1 Π( A n ) N = A (2 . A ∈ L Π . Similarly, we have: Lemma 6 of [13] If ω and ω are factor states on A , andlim N →∞ P Nn =1 ω i ( A n ) N = a i for i = 1 , . a = a , then ω and ω are disjoint.Indeed, let Π ω i denote the representations associated to ω i by (2.5) ( i =1 , ω i are factor states, (2.8) implieslim N →∞ P Nn =1 Π ω i ( A n ) N = a i for i = 1 , . .2 Examples of non-unitarily equivalent representa-tions and superselection sectors We now give an explicit example, following [38], which illustrates why (2.8)implies disjointness, and how this properties typically arises in infinite sys-tems. For each direction ~n i , ~n i = 1, there is a vector in the Hilbert space C such that ~σ i points in this direction:( ~σ i , ~n i ) | ~n i ) i = | ~n i ) i (2 . N spins the 2 N - dimensional Hilbert space is H N = ⊗ Ni =1 C and in thelimit N → ∞ the space becomes nonseparable, but separable subspaces -the so-called IDPS, isomorphic to Fock space, see Wehrl’s contribution to[47] - by letting A act on a reference vector | Ψ + ): H + = A| Ψ + ) where, for | Ψ + ) one may choose a polarized state in which all spins point in the samedirection ~n : | Ψ + ) = ⊗ ∞ i = −∞ | ~n ) i (2 . . H + one obtains an irreducible representation Π + of A (see, again, [47]).Weak limits such as the mean magnetization ~M + = w − lim N →∞ P Ni = − N Π + ( ~σ i )2 N + 1 = ~n (2 . . H + depend on the representation, and, on another reference state | Ψ − ) = ⊗ ∞ i = −∞ | ~m ) i (2 . . ~M − = w − lim N →∞ P Ni = − N Π − ( ~σ i )2 N + 1 = ~m (2 . . U − Π + ( ~σ ) U =Π − ( ~σ i ) would imply U − ( ~n ) U = ~ m1 , which is impossible because U cannotchange the unity . The | Ψ ± ) define states ω ± ( . ) = (Ψ ± , . Ψ ± ) (2 . . λω + + (1 − λ ) ω − (2 . . + and Π − | Ω S ) = q ( λ ) | Ψ + ) ⊕ q (1 − λ ) | Ψ − ) ∈ H + ⊕ H − ≡ H S Π S = Π + ⊕ Π − (2 . . S , Π S ( ~σ i )Ω S ) = λ~n + (1 − λ ) ~m (2 . . ~M = lim N →∞ P Ni = − N Π S ( ~σ i )2 N + 1= λ~n + (1 − λ ) ~m (2 . + ( A ) and Π − ( A ) areisomorphic as C* algebras, with the product operation (2.1), although notunitarily equivalent as announced.(2.14.1) generalizes to a finite number n of projectors { Π α } nα =1 , with H § = ⊕ nα =1 H α , Π S = ⊕ nα =1 Π α , and a state of the form ω S = (Ω , . Ω) = n X α =1 λ α ω α (2 . . ≤ λ α ≤ P nα =1 λ α = 1. Definition 1
The state ω S is a decoherent mixture over the macroscopic(or classical ) observables iff, for 1 ≤ α ≤ n , the states ω α in (2.16.1) satisfythe generalization of (2.12.1,2), i.e., ~M α = w − lim N →∞ P Ni = − N Π α ( ~σ i )2 N + 1= ~m α . . ~m α = ~m α ′ for α = α ′ (2 . . ω α are said to be macroscopically pure . In fact in thethermodynamic limit the definition of ”decoherent mixture ” is equivalent tothe fact, that the center of the representation is not trivial, whereas ”macro-scopically pure states” are those with trivial center. The advantage of thedefinition lies in the fact that it can be easily generalized to finite systems: Definition 1.1
The state ω over a finite-dimensional algebra A is calledcoherent with respect to an abelian subalgebra { P α } , P α P α = 1, if thereexist operators A ∈ A and α = α ′ such that ω ( P α AP α ′ ) = 0It is called macroscopically pure if there exists α such that ω ( P α ′ AP α ′ ) = 0 ∀ α ′ = α ∀ A ∈ A Otherwise it is called decoherent.It should be remarked that it is the abelian subalgebra of definition 1.1which actually defines the macroscopic observables.As a common example leading to macroscopically pure states (see alsoremark 6), let (2.16.1) be the decomposition into extremal invariant (er-godic) states ([43], 4.2.1). The elements (2.16.2) belong to the center Z ω α =Π ω α ( A ) ∩ Π ω α ( A ) ′ . By, e.g., [20], Corollary 4.1.5, if ω S is assumed to be anequilibrium (KMS) state at inverse temperature 0 < β < ∞ ([48],Chapter5.3), so are the ω α and, being extremal invariant, are primary or factor states,and thus Z ω α = { λ } , and therefore (2.16.2) holds. By (2.16.3) and the pre-viously mentioned lemma 6 of [13], the ω α are mutually disjoint, and thestructure of (2.16.1) is complete.There are, of course, obvious generalizations to particle systems, with theobservables ~M α replaced by the particle, energy or momentum density.A basic dynamical concept is that of automorphism : an automorphism τ ∈ Aut ( A ) is a one to one mapping of A onto A which preserves thealgebraic structure. The following lemma is important (for an alternativeproof to Hepp’s, see [43], Prop. 2.4.27):16 emma 2 of [13] If ω and ω are disjoint states on A , and τ ∈ Aut ( A ),then ω ◦ τ and ω ◦ τ are disjoint.Above, the circle denotes composition, i.e., ( ω ◦ τ )( A ) = ω ( τ ( A )). By thislemma, coherence cannot be destroyed during the measurement process, aslong as the time evolution is automorphic. It can only be shifted.What are the main properties of the group of time-translation automor-phisms t ∈ R → τ t ∈ Aut ( A )? For a large class of quantum spin systems -those with potentials of finite range - the local Hamiltonian H (Λ) generatesan automorphism of A (Λ) by A → A t = exp( itH (Λ)) A exp( − itH (Λ)) (2 . τ t ( A ) = lim Λ →∞ exp( itH (Λ)) A exp( − itH (Λ)) = lim Λ →∞ A Λ t (2 . → ∞ couldbe taken to mean along a sequence of hypercubes whose sides tend to infinity,but many other choices leading to the same element of A are possible. Thelimit is uniform for t in some circle in the complex plane around zero. Onethen extends τ t to the whole quasi-local algebra A by the density property(2.2.2), and then, by the group property of τ t on A L , to all t ∈ R : see([49],[50]). A Λ t in (2.18) is constructed as A Λ t = A + it [ H (Λ) , A ] + ( it ) / H (Λ) , [ H (Λ) , A ]] + · · · (2 . A ∈ A (Λ ) roughly toan A ∈ A (Λ + rd ), where d is the range of the interaction potential (see also[51] and [52] for localization properties of τ t ( A )) ; when r → ∞ , A becomesinfinitely extended, and, for that reason τ t becomes an automorphism only ofthe quasilocal algebra A given by (2.2.2) and not of any local algebra A (Λ)- this happens as long as there is any interaction between the spins, i.e., forany nonzero range d . This is a basic property of the quantum evolution,which will play an important role in section 3.1.17 Reduction of the wave-packet: the approachesof Hepp and Sewell
In this section we review the approaches of Hepp [13] and Sewell ([14][15])to the quantum measurement problem.
Hepp’s lemma 2 ([13] and section 2) shows that coherence cannot be de-stroyed by an automorphic time evolution during the process of measure-ment, but his lemma 6 ([13] and section 2) and the example in section 2suggest that it might be possible to find sequences ω ,n and ω ,n of coherentstates which converge weakly (denoted w − lim ω i,n = ω i ) towards disjointstates ω , ω , and one has the important Lemma 3 of [13]
Let Π n be representations of A and Ψ i,n ∈ H Π n with ω i,n = ω (Ψ i,n ) ◦ Π n , i = 1 ,
2, i.e., ω i,n ( A ) = (Ψ i,n , Π n ( A )Ψ i,n ), i = 1 , w − lim ω i,n = ω i for i = 1 , . . ω , ω disjoint. Then, for all A ∈ A ,lim n →∞ (Ψ ,n , Π n ( A )Ψ ,n ) = 0 (3 . . Proof If ω ( A ⋆ A ) = 0, | (Ψ ,n , Π n ( A )Ψ ,n ) | ≤ ω ,n ( A ⋆ A ) →
0; otherwise, ω ,n ( A ⋆ A ) = 0 for n sufficiently large and ω A ,n ≡ (Ψ ,n , Π n ( A ) † ( . )Π n ( A )) / (Ψ ,n , Π n ( A ⋆ A )Ψ ,n )is such that w − lim ω A ,n = ω ,A ω ,A ∈ [ ω ], the folium of ω . Since ω and ω are disjoint, ω and ω ,A are disjoint, and thus || ω − ω ,A || = 2 (3 . . || ω || = sup || A || =1 | ω ( A ) | , by the theorem of Glimm and Kadison ([53],see also [44], theorem 2.6.6 for a simple proof). By ([54], theorem 2, pg. 47), || ω ,n − ω A ,n || = [ τ ( T )] (3 . . τ ( T ) denotes the trace norm of the operator T ≡ Ψ ,n ⊗ Ψ ,n − Ψ A ,n ⊗ Ψ A ,n , where Ψ A ,n ≡ Π n ( A )Ψ ,n || Π n ( A )Ψ ,n || , and( x ⊗ x ) f ≡ ( f, x ) x for all f ∈ H Π n Note that || Π n ( A )Ψ ,n || 6 = 0 for sufficiently large n . The above trace normmay be evaluated by converting (Ψ ,n , Ψ A ,n ) to an orthonormal system by theGram-Schmidt procedure and equals τ ( T ) = 2 q − | (Ψ ,n , Ψ ,n ) | (3 . . τ t ∈ Aut ( A ) suchthat, for i = 1 , w lim t →∞ ω i ◦ τ t = ¯ ω i and ω , ω are coherent, but ¯ ω and ¯ ω are disjoint. Disjointness is brought about by the existence of a macroscopicobservable with different expectation values in ω , ω (lemma 6 of [13] andsection 2). We now come to an example, which was called by Bell [55] the Coleman-Heppmodel, a terminology we also adopt. We initially follow Bell [55] for clarityof exposition in the description of the model.The apparatus is a semi-infinite linear array of spin one-half particles,fixed at positions n = 1 , , · · · L + 1; the system is a moving spin one-halfparticle with position coordinate x and spin operators ~σ ≡ ( σ , σ , σ )- the19hird component σ is to be ”measured”. The combined system is describedby a wave function Ψ( t, x, σ , σ , · · · ) (3 . σ n are diagonal, and σ i = ± i =0 , , , · · · . The Hamiltonian is H = − i ∂∂x + ∞ X n =1 V ( x − n ) σ n (1 / − / σ ) (3 . V ( x ) = 0 for | x | > r (3 . . J = R ∞−∞ dxV ( x ) π = 1 / . . i ∂∂t Ψ = H Ψ is solved byΨ( t, x, σ , σ , · · · ) = ∞ Y n =1 exp[ − iF ( x − n ) σ n (1 / − / σ )]Φ( x − t, σ , σ , · · · ) (3 . . F ( x ) ≡ Z x −∞ dyV ( y ) (3 . . F has the properties F ( x ) = ( x < − r ,π/ x > r , (3 . + ( t, x, · · · ) = χ ( x − t )Ψ + ( σ ) ∞ Y n =1 Ψ + ( σ n )Ψ − ( t, x, · · · ) = χ ( x − t )Ψ − ( σ ) ∞ Y n =1 Ψ ′ + ( σ n , x − n )203 . ± ( σ ) = δ σ ∓ Ψ ′ + ( σ n , x − n ) = exp[ − iF ( x − n ) σ n ]Ψ + ( σ n ) (3 . ′ + ( σ n , x − n ) = ( Ψ + ( σ n ) for x − n < − r , − i Ψ − ( σ n ) for x − n > r , (3 . χ has compact support: χ ( x ) = 0 for | x | > w (3 . ′ + ( σ n , x − n ) = Ψ + ( σ n ) (3 . . n > x + r ≥ t − w + r (3 . . ′ + ( σ n , x − n ) = − i Ψ − ( σ n ) (3 . . n < x − r ≤ t + w − r (3 . . x plays only an intermediate role.Under conditions (3.10), the evolution does not depend on x , except by thewave function χ in (3.9). Due to the linearity of (3.3) in the momentumoperator, the evolution of χ is only a translation.Let the quasilocal algebra ˜ A be chosen as the tensor product˜ A = B ( C ) ⊗ A (3 . A is the quasilocal spin algebra (2.2.1), (2.2.2). We assume that thecoordinate x is not ”measured”, i.e., the corresponding part of the algebra isthe identity. Correspondingly, we omit the factor χ ( x − t ) in the forecomingformulas, because ( χ, χ ) = 1, which is not affected by a translation by t .Let Ψ + and Ψ − be given by (2.11.1), (2.11.2), where ~n ≡ (0 , , ~m ≡ (0 , , − i = 1 , , · · · in definitions (2.11.1), (2.11.2).The precise mathematical meaning of (3.6) and (3.10) is that Ψ( t, x, · · · )equals Ψ + and that Ψ( t, x, · · · ) equals Ψ − , the latter if the conditions in(3.10) are verified. More precisely, if Π n is the representation of A on theHilbert space H n = C ⊗ · · · ⊗ C , the states ω ± ,n ◦ Π n (3 . . ± ,n = Ψ , ± ⊗ ni =1 Ψ i, ± (3 . . σ Ψ , ± = ± Ψ , ± , tend weakly, as n → ∞ , on the algebra ˜ A , to states ω ± given by (2.13.1); by (2.11.1,2) (with index sum running from zero to n instead of − N to N ). By lemma 6 of [13] (see also section 2), they aredisjoint. Consider now the initial vectorΨ ,n ≡ Ψ ⊗ ni =1 Ψ i, + (3 . . ≡ c + Ψ , + + c − Ψ , − (3 . . ,n ( t ) = c + Ψ , + ⊗ ni =1 Ψ i, + + c − Ψ , − ⊗ ni =1 Ψ i, − (3 . . ω Ψ ,n ( t ) tends weakly,as n → ∞ , with t satisfying (3.10.2), which implies t → ∞ ), to the mixedstate ω = | c + | ω + + | c − | ω − (3 . . λ = | c + | and 1 − λ = | c − | .The Coleman-Hepp model has two main shortcomings from a physicalstandpoint: the linear, instead of quadratic, momentum term, i.e., absenceof dispersion, and the lack of interaction between the spins in (3.3). Weshall now dwell on the consequences of this last defect, but remark that,22n spite of that, it is quite ingenious, allowing for detailed estimates on thetime evolution, an extremely rare feature. For other examples, this time withinteraction, see [56], [57], as well as the model of Curie-Weiss spins weaklyinteracting with a phonon bath introduced in [12] (see also [11]).Consider, now, instead of the quasilocal algebra, the local subalgebra˜ A M = B ( C ) ⊗ A M (3 . . A M = A (Λ M ) (3 . . A (Λ) is the local spin algebra of section 2, withΛ M ≡ { , , · · · M } (3 . . d is zero see(2.19) and remarks thereafter. By (3.10),if t ≥ M + 1 − w + r (3 . ± ,n ( t ) , Π n ( A )Ψ ∓ ,n ( t )) = 0 if n > M and t satisfies (3.15) and A ∈ ˜ A M (3 . . A in B ( H n ) is given by A → A ⊗ ⊗ · · · . Since(3.16.1) occurs for finite times t (satisfying (3.15)), neither irreversibilitynor disjoint states - now of the form (3.12.1), (3.12.2), with n > M and t satisfying (3.15) - occur in this case!The estimate for the critical ”decoherence time” t resulting from (3.15), t = M + 1 − w + r (3 . . M , e.g., for M = 10 , corresponding to alength of the chain equal to M a ≈ × − ≈ cm , taking for the spacing23etween the atoms in the chain a ≈ − cm , t ≈ M ma ¯ h yields, takingfor m the electron mass, , t est ≈ − seconds, roughly comparable with spindecoherence times in quantum wells. Indeed, some of the largest decoherencetimes are found in electron spin coherence in GaAs quantum dots [58], whichis mostly due to hyperfine interactions with nuclear spins, and range from 0.1-100 microseconds, i.e, ranging from (1 − ) t est . Notice that this decoherencetime is nothing but the time taken by the electron to traverse the chain, asobserved by Sewell [14], [15], because it equals the distance M a divided bythe ”electron velocity” ¯ hk/m , where the wave vector k ≈ /a . For larger M this estimate quickly becomes highly unrealistic, suggesting us to adopta modified definition, to which we now turn. Definition 2
We say that a sequence of states ω Ψ ,n ( t ) ◦ Π n given by (3.13.3)exhibits reduction of the wave packet or decoherence with respect toa subalgebra A M ⊂ A stable under the time evolution ifflim n →∞ ,t>t n (Ψ ,n ( t ) , Π n ( A )Ψ ,n ( t )) = 0 (3 . A ∈ A M , where t n is a ”decoherence time”.We have included in the definition the possibility to consider the deco-herence time only with respect to a subalgebra. This opens the possibility toconsider a net of increasing subalgebras with finite though increasing decoher-ence time. The physically most interesting case in which t n is independentof n is, of course, included.The above definition is, in fact, very natural: it states that there is no”experiment” (represented by a quasilocal operator) connecting the ”pointerpositions” represented by the vectors Ψ ,n ( t ) and Ψ ,n ( t ) for sufficiently largesystems, and times greater than a critical value. Lemma 3 of [13] provesthat disjointness of the corresponding states w lim n,t →∞ ω t,n , ≡ ω Ψ , ,n ( t ) ◦ Π n is a way of achieving this aim, but only with t n → ∞ as n → ∞ . Thelatter happens in the Coleman-Hepp model, but definition 2 leaves open thepossibility of a fixed t independent of n of a realistic order of magnitude. Anapplication of definition 2 to the Coleman-Hepp model is made in proposition1 of the forthcoming section. There are, however, no examples yet with t n which display reduction of the wave-packet in accordancewith definition 2. We conclude our review of Hepp’s approach with some remarks on Bell’scriticism [55] of Hepp’s work. From (3.10.2) and (3.10.4) (we take w = r for simplicity: this does not affect the argument), the spin flips occur justafter the times t n = n . Again for simplicity, we consider only discrete times t n = n . Define the operator Z n = σ n Y k =1 σ k (3 . . λ n ≡ (Ψ ,n ( t = n ) , Z n Ψ ,n ( t = n )) = c > . . c is a constant independent of n (it ”undoes” the measurement, see [59]and [17]). Although Z n = Π n ( A n ) for some A n ∈ A given by (2.2.1),(2.2.2)(in fact, for an explicit element A n ∈ A L ), for each finite n , there is no element A ∈ A such that the infinite sequence { λ n } , given by (3.17.2),satisfies lim n →∞ (Ψ ,n ( t = n ) , Π n ( A )Ψ ,n ( t = n )) = lim n →∞ λ n = c (3 . . A ∈ A , given any ǫ >
0, there exists A L ∈ A L such that || A − A L || < ǫ ; writing(Ψ ,n ( t = n ) , Π n ( A )Ψ ,n ( t = n )) = (Ψ ,n ( t = n ) , Π n ( A − A L )Ψ ,n ( t = n )) ++(Ψ ,n ( t = n ) , Π n ( A L )Ψ ,n ( t = n ))and using || Π n ( A − A L ) || ≤ || A − A L || together with the fact that(Ψ ,n ( t = n ) , Π n ( A L )Ψ ,n ( t = n )) = 025or n > diameter of A L , because of the orthogonality of Ψ ,n ( t = n ) andΨ ,n ( t = n ) for each finite n , we obtain a contradiction with (3.17.3).One may ask why not restrict oneself to fixed, finite n . This is indeedpossible if one adopts the forthcoming definition 3 of the reduction, whichrefers only to the microsystem and not to the measuring instrument. If, how-ever, one wishes to probe into the observables of both system and apparatus,this is a bad choice, as we now show.Consider, now, that the dynamics, instead of being defined by the Hamil-tonian H defined in (3.3), is described by the slightly (for α small) perturbedHamiltonian H α,N ≡ H N + α N X m =1 σ m σ m +1 (3 . α real. Above, H N is the finite chain version of the Coleman-Heppmodel (the sum in (3.3) running from 1 to N , and the limit, as N → ∞ , isunderstood in the automorphism sense (see (2.19) et seq.). The effect of theinteraction in (3.18) is to render the observable algebra necessarily infinitelyextended, as described in (2.19) et. seq., so that no strictly local algebra ispreserved by the dynamics: we are forced to consider the algebras ˜ A M with M → ∞ as discussed after (3.14.1). Although we are not able to solve thequantum evolution for any α = 0, the possibility of such small interactionsbetween the spins should be allowed for in any model, and the choice of alocal algebra is not stable under these.Definition 2 probes a priori the macroscopic nature of the pointers: thevanishing of the cross terms (3.16) in the limit of infinite degrees of freedomis natural for ”essentially local”- supposedly physical - observables, which areblind to changes in the global structure. One may ask what is so importantabout requiring (3.17.3), i.e., A quasilocal. This is, of course, a philosophicalquestion related to the proposal by Haag and Kastler [42] of what is anadequate choice of observables. But, in our opinion, it is precisely in thepresent context that their argument is most convincing: (3.16) probes, as n → ∞ , a global change, but this limit does not exist mathematically: itis the would-be operator which intertwines two disjoint representations. Itseems also reasonable to suppose that the associated sequence of observablesis devoid of any physical meaning.It is possible to study reduction of the wave-packet using a definition (theforthcoming definition 3) introduced by von Neumann [3] and used by Sewell26n [14],[15], which involves only observables of the system, not the apparatus.This definition, by its own scope, does not by itself require macroscopicpointers (although this may be, and usually is, included in the definitionas extra requirement). It is, therefore, more general and possesses a higherdegree of flexibility than definition 2. This is demonstrated by the case β = ∞ in proposition 1. Following this idea, we use a modification of definition 2(definition 5 in section 5.12), which is actually implicitly widely used in thetheoretical physics literature, to study a class of models of decoherence. We now turn to Sewell’s approach ([14][15]), whose objective was, as Hepp’s,to reconsider the quantum measurement problem within the framework ofSchr¨odinger dynamics, but utilizing the language of quantum probability [3].He considers a composite system S c , consisting of a microsystem S coupledto a measuring instrument I , S c = S + I , symbolically, where I is a large,but finite, N - particle system. Also essential to his approach was to takeinto explicit account the macroscopic nature of the observables M , whichare taken to comprise a set of coarse-grained intercommuting extensive vari-ables, whose simultaneous eigenspaces correspond to the pointer positions of I . The approach leans on von Neumann’s picture of the measurement pro-cess discussed in the introduction, but more specifically according to whichthe coupling of S to I leads to the following two effects: (I) It converts apure state of S , as given by a linear combination P nr =1 c r u r of its orthonormalenergy eigenstates u r into a statistical mixture of these states for which | c r | is the probability of finding this system in the state u r ;(II) It sends a certain set of classical, i.e., intercommuting macroscopic vari-ables M of I to values, indicated by pointers, that specify which of the states u r is realized. Sewell assumes that the algebras A of bounded operators ofthe microsystem, B of the instrument I , and their composite S c = ( S + I ) arethose of the bounded operators on separable Hilbert spaces H , K and H ⊗ K ,respectively. Correspondingly, the states of these systems are represented bythe density matrices in the respective spaces. The density matrices for thepure states of any of these systems are then the projection operators P ( f ) oftheir normalized vectors f . For simplicity, it was assumed that H is of finite27imensionality n .The macroscopic description of I pertinent to the measuring process wasthen based on an abelian subalgebra M of B , which is generated by theaforementioned coarse-grained macroscopic observables, typically extensivevariables of parts or the whole of I . The choice of M yields a partition of K into the simultaneous eigenspaces K α of its elements. These subspaces of K were termed ”phase cells”, being the canonical analogues of classical phase”pointer positions” (compare (1c) et seq.) of this instrument.An important element of [14], [15] is an amplification property of the S −I - coupling, whereby different microstates of S give rise to macroscopicallydifferent states of I . Since I is designed so that the pointer readings arein one-to-one correspondence with the eigenstates u , · · · u n of S , we assumethat the index α of its macrostates also runs from 1 to n . hence, denotingthe projection operator for K α by Π α , it follows from the above specificationsthat: Π α Π β = Π α δ αβ (3 . . n X α =1 Π α = K (3 . . M of M takes the form M = n X α =1 M α Π α (3 . . M α are scalars.The above description corresponds to von Neumann’s description of amacrostate of a macroscopic quantum system [3], as sketched in the intro-duction - ”rounding off” and ”tolerance” effects may be added without prob-lems, but we shall omit them from our discussion. It will be assumed that S c is a conservative system described by a Hamiltonian H c = H ⊗ K + H ⊗ R + V (3 . V induces no transitions between theeigenstates u , · · · u n of the Hamiltonian H of S - I is, thus, an ”instrumentof the first kind” [60]. The systems S and I are coupled at t = 0 followingindependent preparation of S in a pure state and I in a (in general) mixed28ne, as represented by a normalized vector Ψ and a density matrix Ω, re-spectively ; the initial state of the composite S c is thus given by the densitymatrix ρ (0) = P (Ψ ) ⊗ Ω (3 . . t > U † c ( t ) ρ (0) U c ( t ) ≡ ρ ( t ) (3 . . U c ( t ) ≡ exp( iH c t ) (3 . . is a normalized vector, it is a linear combination of thebasis vectors Ψ = n X r =1 c r u r (3 . . n X r =1 | c r | = 1 (3 . . ρ ( t ) are defined, asusual, by E ( A ⊗ M ) = T r ( ρ ( t ) A ⊗ M ) for all A ∈ A and for all M ∈ M (3 . . T r is the trace on
H ⊗ K ; the conditional expectation value E ( A |K α )of A given the macrostate K α of I may then be defined by E ( A |K α ) ≡ E ( A ⊗ Π α ) /w α (3 . . A ∈ A , w α = 0, where w α = E ( H ⊗ Π α ) (3 . It follows from the above that the conditions for the realization of conditions(I) and (II) may be summarized in the following29 efinition 3
The system displays reduction of the wave packet or deco-herence iff E ( A ) = n X r =1 | c r | ( u r , Au r ) for all A ∈ A (3 . . φ of the set 1 , , · · · n such that E ( A |K α ) = ( u φ ( α ) , Au φ ( α ) ) for all A ∈ A (3 . . t such that t > t ,n , where t ,n is a critical time (”decoherencetime”).In other words, the pointer reading α signifies that the resulting state of S is u φ ( α ) .It was proved in [14][15] that, under suitable conditions on H c , given by(3.20), and Ω (given by (3.21.1)), (3.24) holds, and, furthermore, that theseconditions are fulfilled by the finite version of the Coleman-Hepp model (3.3),where the infinite sum is replaced by a finite one running from 1 to N , i.e., N is the number of points in the chain, with a suitable choice of ”phase cells”,to which we come back in 5.1. Sewell also showed that that the conditionsare satisfied by the previously mentioned model [12], see also [11]. Actually,(3.24) was proven up to exponentially small corrections in N . The estimatefor the critical time t agrees with (3.15), with the correspondence N = M .Sewell’s method entails several important aspects which are absent inHepp’s discussion: explicit use of the ”cell’ decomposition (3.19) (which willplay a central role in our illustration of the second law in section 4), and aproperty of stability against local perturbations of the state Ω. This is anessential property of any measuring apparatus, which leads to considerationof mixed states, in addition to the ground state, a fact which will also play avery important role in sections 5.3 and 5.4. Finally, the method used to con-trol the above mentioned exponentially small corrections allows a discussionof the ”bad microstates”, see remark 3 in section 5.4. Although the applica-tion of this method to the Coleman-Hepp model is elementary, [15] indicatesa more general result, which shows that these corrections are governed bythe large deviation principle (see, e.g., [61]).30 Irreversibility and a quantum version of thesecond law of thermodynamics
In [13], pg. 247, Hepp remarks: ”The solution of the problem of measurementis closely connected with the yet unknown correct description of irreversibilityin quantum mechanics”. And, further, at the end: ”...while the automorphicevolution between finite times is reversible, it is precisely the irreversibilityin the limit of infinite times which reduces the wave packets”.To the extent that irreversibility (in closed systems) is observed in Natureat definite (finite) times, ”here and now”, so to speak, and not just as anapproximation, we believe that the weak limit t → ∞ in Hepp’s article isnot adequate to relate the problem of measurement to irreversibility (thisconcern was, however, beyond the proposed scope of Hepp in [13]). On theother hand, we shall see that Sewell’s approach is natural for that purpose.The problem of irreversibility has been previously discussed in the litera-ture by the introduction of various types of ”coarse graining” : for beautifuldiscussions, see the articles by Lebowitz [25] and Griffiths [29]. There arestrong arguments (see [29] and section 5.3) that the ”arrow of time” is pro-vided by the direction of increase of the the quantum version of Boltzmann’sentropy, the von Neumann macroscopic entropy (see [25], sec.8 and the intro-duction). Of course, if one wishes to relate the quantum Boltzmann approachto irreversibility to the theory of measurement, it is essential to include themeasuring apparatus as part of the closed quantum mechanical system (seealso the remarks in [29], pg.154).The following version of the second law seems to encompass both obser-vational (i.e. measurement) and thermal phenomena (theorem 1). Althoughseveral mathematical tools have already been developed by Lindblad in aseries of beautiful papers ([35], [62], [63]), which had precisely measurementtheory in mind, we use them in a different direction. In section 5a we illus-trate theorem 1 by the Coleman-Hepp model.In a letter to Niels Bohr (28-1-1947), Wolfgang Pauli remarks: ”Thediscussions I had here with Stern (he left Z¨urich a few days ago) concerned thequantitative side of the connection of the concepts of entropy and observation,a connection which, as we all agree, is of a very fundamental character”.In this section, we revisit this topic, with a view to try to illuminate theaforementioned connection. 31 .1 Definition of the quantum Boltzmann entropy We adopt the setting of section 3.5, and define the projectors occurring in(3.19): P α = ⊗ Π α for α = , · · · n (4 . A ∈ A is measured on the system S , initially in a state of thecomposite system described by a density matrix ρ , the value φ ( α ) ∈ [1 , n ]is obtained with probability w α = T r ( ρP α ), after which the state of thecomposite system is described by the density matrix ρ ′ α ≡ P α ρP α w α (4 . . ρ ′ = X w α ρ ′ α = X α P α ρP α (4 . . A has a definite value. We definethe quantum Boltzmann entropy S QB ( ρ ) of a state ρ by S QB ( ρ ) ≡ − kT r ( ρ ′ log ρ ′ ) (4 . ρ ′ is given by (4.2.2), provided ρ ′ log ρ ′ is of trace class.Definition 4.3 is adopted by several authors in the field, see, e.g., Zurekin [5]. The transformation ρ → ρ ′ = X α P α ρP α may be viewed as a loss of information contained in the non-diagonal terms P α ρP α ′ with α = α ′ in ρ = X α,α ′ P α ρP α ′ .2 Histories and decoherence A generalization of (4.2.2), viz. when a sequence of measurements is carriedout, and letting a time evolution intervene between measurements, leads tothe assignment to a sequence of ”events” P α ( t ) P α ( t ) · · · P α n ( t n )( a ”history”, briefly written α for the index set of the corresponding vector)a probability distribution W , W ( α ) = T rP α n ( t n ) · · · P α ( t ) ρP α ( t ) · · · P α n ( t n ) (4 . . ρ = ρ (0), over the set of histories, where the P satisfy the relations(3.19.1),(3.19.2) (written for Π α ↔ P α ). This framework was proposed inde-pendently in [64], [65], [66]. Let D ( α ′ , α ) == T rP α ′ ( t ) · · · P α ′ n ( t n ) ρP α n ( t n ) · · · P α ( t ) (4 . . Definition 4
A history is said to decohere iff D ( α ′ , α ) == δ α ′ ,α ρ α (4 . P α iff P α ′ ρ (0) P α = 0 for all α = α ′ (4 . T rP α ′ ρP α A = 0 ∀ α = α ′ and this is equivalent to [ P α , ρ ] = 0 ∀ α In contrast to infinite systems (see definition 1) where there is no need torefer to a choice of projections, decoherent mixed states (over the macroscopic33bservables) can be described by relations between the density matrix a ρ m and the projectors. They are of the form ρ m = | Ψ)(Ψ | (4 . ≡ X λ α P α Φ α with X α | λ α | = 1 andΦ α ∈ H for α ∈ [1 , n ] (4 . X α<α ′ ( P α ′ ρ m P α + P α ρ m P α ′ ) = 0 (4 . L Π of section (2), which defines the ”pointer po-sitions” in Hepp’s formulation [13], may also be viewed as a main feature -besides the inevitable indeterminacies in the values of macroscopic quantities- which lies behind the definition (4.3) of the quantum Boltzmann entropy.This is, of course, not new, and basic to von Neumann’s approach [3]. Theelements of L Π have to be considered as the only ones accessible to experi-ment, and, indeed, all information on microscopic quantities, such as the spinof one electron in the Stern-Gerlach experiment, are, in principle, derivablefrom them, provided we can manipulate the time evolution in the system inan appropriate way. Therefore the reduced description of (4.3) is also naturalfrom a purely conceptual point of view.As Lindblad remarks ([35], pg. 314)), (4.3) may be viewed as an averagingover the relative phases between the subspaces P α ( H ⊗ K ). We shall need the concept of relative entropy S ( ρ | ρ ) (called by [35] con-ditional entropy, but actually corresponding to the finite-dimensional versionof the concept introduced subsequently by Araki [68] in the general context34f states on von Neumann algebras) between two states (positive operatorsof trace one), which is defined by S ( ρ | ρ ) = kT r ( ρ log ρ − ρ log ρ ) (4 . Lemma 1 S ( ρ | ρ ) ≥ . . S ( ρ | ρ ) = 0 iff ρ = ρ (4 . . λρ ≤ ρ for some λ ∈ (0 , S ( ρ | ρ ) ≤ − k log λ (4 . . S ( λρ + (1 − λ ) ρ | λσ + (1 − λ ) σ ) ≤ λS ( ρ | σ ) + (1 − λ ) S ( ρ | σ )(4 . . S ( ρ .γ | ρ .γ ) ≤ S ( ρ | ρ ) (4 . . γ is a completely positive map, e.g. an imbedding. The last twoinequalities are known aa joint concavity and monotonicity of the relativeentropy, both follow from Lieb’s concavity theorem [72]. We shall denotevon Neumann’s entropy S ( ρ ) = − kT r ( ρ log ρ ) simply by S and call it ”theentropy”. 35 .4 A quantum version of the second law Theorem 1 A quantum version of the second law
Let the (initial) density matrix be assumed to be decoherent at zero time(4.6) with respect to P α and to have finite entropy, i.e., ρ (0) = X α P α ρ (0) P α S QB ( ρ (0)) = S ( ρ (0)) = − kT rρ (0) log ρ (0) < ∞ (4 . ρ ( t ), t >
0, beany subsequent state of the system, possibly an equilibrium state. Then, foran automorphic (unitary) time-evolution of the system between 0 ≤ t ≤ t , S QB (0) ≤ S QB ( t ) (4 . S QB (0) = S QB ( t ) iff X α<α ′ P α ρ ( t ) P α ′ + P α ′ ρ ( t ) P α = 0 (4 . Proof
With ρ ′ ( t ) = P α P α ρ ( t ) P α = ρ ( t ) .γ where we indicate that ρ ′ is obtained from ρ by a completely positive map γ , we estimate S ( ρ ′ ( t ) | ρ ′ (0)) = − S ( ρ ′ ( t )) − X α kT r ( ρ ( t ) P α log ρ (0) P α ) = − S QB ( t ) − kT r ( ρ ( t ) log ρ (0)) = ≤ S ( ρ ( t ) | ρ (0)) = − S ( ρ (0)) − T r ( ρ ( t ) log ρ (0)) (4 . ρ ′ , in the second equality we used the decoherence of ρ (0),the next inequality is a consequence of (4.11.5). In the last equality we usethat the evolution is unitary and therefore preserves the entropy. Togetherthis implies S QB ( t ) ≥ S QB (0) . . The equality condition in (4.14) follows from (4.11.2). q.e.d.36 .5 Miscellaneous remarks: monotonicity in time, ther-mal and observational systems, Griffiths’ ideas andthe ”arrow of time” Remark 1
The entropy growth in theorem 1 is not necessarily monotonicin the time variable. For this reason, we refer to fixed initial and final statesin that theorem. For thermal systems, e.g., the adiabatic expansion of agas discussed in the introduction, a natural choice of the final state is theequilibrium state of the gas. In the case of observational systems, it may bean incoherent (macroscopically) mixed state such as the forthcoming (section5a) density matrix of the Coleman-Hepp model for t ≥ t , where t is the”decoherence time”. Remark 2
This remark contains some (at this point) entirely speculativeobservations related to theorem 1 and remark 1. Following an intuition dueto Griffiths [29] for thermal systems (eg. the irreversible expansion of a gasdiscussed in the introduction), the entropy growth is expected to be relatedto the property that, at time t > P α ′ ρ ( t ) P α = 0 for some P α ′ > P α ;and, successively, P α ′′ ρ ( t ′ ) P α ′ = 0 with P α ′′ > P α ′ and t ′ > t , that is, themacrostate will gradually be associated, with high probability, to subspacesof increasing dimension (in the simplified case that the latter are finite, alsotheir number has to decrease), whereas S QB is still calculated with respectto the initial P α . In this process, superpositions between states in different H α must be occurring all the time, leading to a final equilibrium entropy inagreement with theorem 1.The above conjectures by Griffiths were formulated in an informal man-ner, with no claim to precision whatever. In the following we try to conveythe logic of his formulation in a slightly more concrete manner, but muchwork will be necessary in order to find precise conditions under which thislogic is correct, together with models which might serve as testing grounds.The basic idea stems from an analogy with classical statistical mechanics,more specifically the fact that, as mentioned in (1.2), the totality of phasespace points which at time t were in a certain macrostate M forms a re-gion with fixed phase-space volume | Γ M | by Liouville’s theorem, but whoseshape will, for mixing systems, ”spread” over regions Γ M corresponding tomacrostates M with increasingly large phase space volumes. Therefore, theprobability that the macrostate is found in successively larger phase space37olumes grows with time, and, with it, the classical Boltzmann entropy.Following [29], consider now a quantum mechanical system which at time t is away from equilibrium, e.g., two interacting metal blocks at differenttemperatures which can exchange energy through a thin wire. Let a setof projections Π α satisfying (3.19.1),(3.19.2) be defined, which represent a”coarse graining” in the sense that each projector Π α specifies, within a rea-sonable tolerance, the energy of each of the metal blocks. Choose then aninitial normalized state | Ψ ) ”at random” in the appropriate subspace of theHilbert space corresponding to the specified initial energies. The correspond-ing D = | Ψ )(Ψ | serves as initial condition upon which the histories withevents at later times drawn from the set { Π α } are conditioned. Due to theunitarity of the dynamics (the analogue of Liouville’s theorem in the classicalargument), T rD ( t ) = || Ψ ( t ) || = 1 = X j ∈ S t | c j ( t ) | (4 . c j ( t ) are coefficients associated to the expansion of Ψ ( t ) in a fixedbasis. A ”quantum ergodicity” or ”quantum weak mixing” (see later) impliesthat the cardinality of the set S t (supposed finite as in [29]) increases with t ,i.e, Ψ ( t ) ”spreads” (the analogue of the ”spreading ” of the initial volumeof phase space in the classical case, keeping the phase space volume fixed).Therefore, the chances are that most of the probability in (4.16) (in the senseof (4.4.1)) will be associated to sequences Π α with increasing dimensionality,implying the increase of the quantum Boltzmann entropy with time.It should be remarked that even in the simplest cases, e.g., Laplacians orSchr¨odinger operators on compact (or finite volume) Riemannian manifolds,with or without boundary, i.e., with ”chaotic” geodesic flow, it does not seemeasy to prove a conjecture of the type of the one formulated in the previousparagraph. If Ψ is an eigenfunction of the Laplacian on this manifold, andthe c j ( t ) in (4.16) are the coefficients of Ψ ( t ) in the basis of the eigenfunctionsof the same operator, the ”spreading ” has been proved in a certain precise,but semiclassical, sense - see the results on ”ergodic eigenfunctions” and”quantum weak mixing” in [73] and references given there.Concerning the possible non-monotonicity of the quantum Boltzmann en-tropy in time mentioned in remark 1, Griffiths remarks ([29], pg. 152) thatit is crucial in the probability Ansatz he uses (for details, see [29]) that itrefers to the beginning t of the period of time in which the system’s entropyincreases. Choosing a microstate at random within the phase space cell cor-38esponding to the microstate reached by the original system at a certainintermediate time t > t and integrating the classical equations of motionbackwards in time, the result would - with overwhelming probability - not agree with the typical behavior of the entropy with the analogous specifica-tion at time t . ”To put it another way, statistical mechanics provides an ex-planation of an irreversible phenomenon by a probabilistic hypothesis whichitself singles out a (thermodynamic) direction of time”. The nonmonotonicfeature may be related to van Kampen’s suggestion [74] that an additionalassumption, e. g. of repeated random phases analogous to Boltzmann’smolecular chaos, is indispensable for a theory of irreversibility. Indeed, the-orem 1 suggests the possibility of a behavior of ”average growth”, similar tothe analogous conjectured behavior for the Boltzmann H function (see, e.g.,Fig. 1.4 in the elementary but very clear introduction in chapter 1 of [75]).Finally, what determines the ”arrow of time”? Griffiths [29] remarks that,since two bodies at unequal temperatures but in thermal contact are knownto exchange energy in such a way that the temperatures approach each othereven in the presence of a magnetic field, which breaks time-reversal invari-ance, the latter is certainly not the key for understanding macroscopic irre-versibility (see also the pioneer paper of Aharonov, Bergmann and Lebowitz[76], reprinted in [4], as well as the discussion in [77] in the framework ofclassical statistical mechanics). Like Griffiths [29], we believe that it is thegrowth of the Boltzmann entropy that determines the arrow of time: inquantum theory, it points from an event away from equilibrium towards asituation of higher quantum Boltzmann entropy (compare also the discussionin ([33], second paragraph of page 312). For the role of the arrow of time inrelativistic quantum field theory, see [78].Some model results on the monotonic- versus - non-monotonic behaviorof the entropy as a function of time are given in section 5.8. In spite of the existence of an enormous literature on decoherence and thetransition from quantum to classical (see the books [8] [7], the review articles[5] [6], [2], there have been few papers dedicated to an exact purely quantumapproach to the problem: notable are the aforementioned [38], [67], see alsoprevious work by Narnhofer [79] and Narnhofer and Robinson [80], the re-39ent three dimensional model of decoherence induced by scattering [81], andthe exact quantum Brownian model of Unruh and Zurek [82] and Joos andZeh [83]. The reason is, of course, the difficulty in obtaining some detailedinformation on the quantum evolution.In sections 5.1 to 5.7 we illustrate theorem 1 by a specific model, a finiteversion treated by Sewell ([14], [15]) of the Coleman-Hepp model introducedin section 3.2, which we extend to infinite space and times. In section 5.8we look into models of the behavior of the quantum Boltzmann entropy forintermediate times,, introducing interactions between the chain spins. Insections 5.9 et seq. we consider a class of models of quantum chaotic systems(quantum Anosov systems) as models of decoherence.
We consider the finite Coleman-Hepp model , whose infinite version wasintroduced in section 3.2, following closely the approach of Sewell and thecorresponding notation described in 3.5. According to (3.20) and (3.2), wetake the Hilbert space of S to be H = L [ − r, r ] ⊗ C (5 . H = H ⊗ p (5 . p is the self-adjoint momentum operator on L [ − r, r ], with someboundary condition Φ( x + 2 r ) = exp( iθ )Φ( x ), θ ∈ R , and V = ( P − ) ⊗ L +1 X n =1 v ( x − n ) ⊗ σ n (5 . N = 2 L + 1 of spins and n = 2 ; the index1 , ± corresponding to the eigenvalues of σ . We define asin [14],[15] the macroscopic phase cells K α of section 3.5, now denoted K ± , asthe subspaces of K = ⊗ C spanned by the eigenvectors of the ”polarizationoperator” Σ L ≡ L +1 X n =1 σ n (5 . ± denote the associate pro-jection operators. Of course, the restriction to these phase cells correspondsto an enormous reduction of information. we can consider the time evolutionfor steps. It becomes τ n P (Φ) P (Ψ ) ⊗ L +1 l =1 P ( c l + | +) + c l − |− )) == c P (Φ( x − n )) ⊗ | +) ⊗ L +1 l =1 P ( c l + | +) + c l − |− ))++ c − P (Φ( x − n )) ⊗ |− ) ⊗ nl =1 P ( c l + |− ) + c l − | +)) ⊗ L +1 l = n +1 P ( c l, + | +) + c l − |− )) τ kx P (Φ) = P (Φ( x − k )) τ L +1+ k = τ kx × τ L +1 (5 . t ∞ = t = 2 L + 1 − w + r , see (3.15)):we interpret this as a statement that at this point and time the interactionwith the chain ceases. Since the free Hamiltonian is a shift, no problem withunitarity arises. As remarked after (3.15), this yields a physically reasonablevalue of t for a suitable choice of L , and from this point of view is thuspreferable to Hepp’s choice of limits. In the rest of the paper, when we speakof the Coleman-Hepp model, we shall be referring to this version.The correspondence between the phase cells and the eigenstates of S in(3.24 2) is given by the mapping r → a ( r ) with a ( ± ) = ± ; i.e., the phase cells K ± are the indicators for the vector states u ± = |± ) , with σ |± ) = ±|± ) .The M α in (3.19.3) are the eigenvalues of Σ L in K ± .We consider two situations, characterized by two distinct types of initialstate (3.21.1): one which is pure (over the microscopic states), originallyexplicitly considered in [13]; another which is not pure over the microscopicstates, but is decoherent at zero time, complying with the hypothesis oftheorem 1, which was treated in [14], [15]. We recall definitions (4.1) and(4.3), where, now, α = ± . We shall take ρ (0) = P (Φ) ⊗ P (Ψ ) ⊗ Ω (5 . P (Φ) = | Φ)(Φ | (5 . . P (Ψ ) = | Ψ )(Ψ | (5 . . = c + | +) + c − |− ) (5 . . ± ,L ( β = ∞ ) = 2 − (2 L +1) ⊗ L +1=1 ( n ± σ ) (5 . . + ,L ( β < ∞ ) = 2 − (2 L +1) ⊗ ( n + m ( β ) σ )= exp( βB Σ L ) Z (5 . . − ,L ( β < ∞ ) = 2 − (2 L +1) ⊗ L +1 n =1 ( n − m ( β ) σ )= exp( − βB Σ L ) Z (5 . . m ( β ) = tanh( βB ) (5 . . L is given by (5.4), Z = T r K exp( βB Σ L ) = T r K exp( − βB Σ L ) (5 . . ± m ( β ) = T r K (Ω ± ,L ( β < ∞ )Σ L )2 L + 1 (5 . . ± ,L ( β = ∞ ) istransformed into Ω ± ,L ( β < ∞ ) by small perturbations of the global polar-ization m , it is natural to consider both, and, thus, only the ”apparatus”corresponding to finite temperature is stable under such small perturbations- a natural requirement on any measuring apparatus.In [14] , [15] values different from the value one-half for the quantity(3.4.2) were considered, but, except for some remarks in connection withtypicality, we shall take J = 1 / .2 The microscopically pure case β = ∞ In this case ρ (0) is pure (over the microscopic states) given by (5.6) and(5.8.1). Hypothesis (4.12) of theorem 1 is thus true a fortiori. By (4.1) and(4.3), we have the general form: S QB ( ρ ) = − kT r ( P + ρP + log P + ρP + ) − kT r ( P − ρP − log P − ρP − ) (5 . H ⊗ K . We shall denote any t satisfying (3.10.4) for n = 2 L + 1 by t = ∞ . By (5.7), ρ (0) = | Φ)(Φ | ⊗ | Ψ )(Ψ | ⊗ L +1 i =1 | +) i (+ | i (5 . ρ ± ≡ P ± ρP ± (5 . ρ + (0) = ρ (0) (5 . . ρ − (0) = 0 (5 . . S β = ∞ QB ( ρ (0)) = 0 (5 . t = ∞ (i.e. 2 L + 1 < t << r ). Define˜Ψ t ≡ | Φ t ) ⊗ ( c + | +) ⊗ (2 L +1) i =1 | +) i + c − |− ) ⊗ (2 L +1) i =1 |− ) i ) (5 . . t ( x ) = Φ( x − t ) (5 . . P + ˜Ψ t = | Φ t ) ⊗ ( c + | +) ⊗ (2 L +1) i =1 | +) i ) (5 . . P − ˜Ψ t = | Φ t ) ⊗ ( c − |− ) ⊗ (2 L +1) i =1 |− ) i ) (5 . . ρ β = ∞ ( ∞ ) = | Φ t )(Φ t | ⊗ ( | ˜Ψ t )( ˜Ψ t | . . P ± = C ⊗ Π ± (5 . . P + ρ β = ∞ ( ∞ ) P + + P − ρ β = ∞ ( ∞ ) P − = | Φ t )(Φ t | ⊗ ( | c + | | +) (+ | ⊗ (2 L +1) i =1 | +) i (+ | i ++ | c − | |− ) ( −| ⊗ (2 L +1) i =1 |− ) i ( −| i ) (5 . S β = ∞ QB ( ρ ( ∞ )) = − k | c + | log | c + | − k | c − | log | c − | (5 . Q ≡ c + ¯ c − | +) ( −| ⊗ (2 L +1) i =1 | +) i ( −| i ++ c − ¯ c + |− ) (+ | ⊗ (2 L +1) i =1 |− ) i (+ | i (5 . P + Q P − = 0. The effect is stable under perturbationsof the initial state: β < ∞ In the case β < ∞ and taking ω = Ω + ,L ( β < ∞ ) in (5.6), we see that ρ (0)is mixed state, but (5.8.2) shows explicitly that ρ is decoherent at zero time,because [ P ± , ρ ] = 0 . The density matrix at t = 0 is given by ρ β< ∞ L ( t = 0) = P (Φ) ⊗ P (Ψ ) ⊗ Ω + ,L (5 . E i denote the (degenerate) eigenvalues of Σ L on the basis {| i ) } (2 L +1) i =1 ofeigenstates of ⊗ L +1 i =1 σ i . These eigenstates can be characterized by partitionsof the set { l = 1 , .... = 2 L + 1 } = I S J so that44 ± ,I = ⊗ l ∈ I |± ) l ⊗ k ∈ J |∓ ) k (5 . t ∞ into | c | +) + c − |− ) ⊗ Ψ ± ,I → c | +) ⊗ Ψ ± ,I + c − |− ) ⊗ Ψ ∓ ,I (5 . P (Ψ ± ,I ) so that the density opera-tor remains decoherent, but now with different weights for the contribution,namely ρ = P + ρP + + P − ρP − == X I ( c α I + c − α I c ) | Ψ + I hi Ψ + I | ++ X I ( c − α I + c α I c ) | Ψ − I hi Ψ − I | As for the ground state the coherence of the system that is measured turnedinto a decoherence of the apparatus so that S QB ( ρ ( t ∞ )) = S ( c ρ β + c − ρ − β ) (5 . Remark 3
In the classical theory discussed in the introduction there arise”bad microstates”, e.g., those microstates consisting of gas molecules whosevelocity vectors are directed away from the barrier lifted at time t . The anal-ogous states in the Coleman-Hepp model are just those leading to nonzero”nondiagonal” terms Π − Ω + ,L and Π + Ω − ,L , which hamper the approach ofthe coherent mixture (5.21) to the decohered or incoherent mixture | c + | P T r (Ω + ,L . ) + | c − | P − T r (Ω − ,L . )45he fact that their contribution is ”small” is measured by m + ≡ T r (Π − Ω + ,L ( β < ∞ )) = O (exp[ − c (2 L + 1)]) (5 . . m − ≡ T r (Π + Ω − ,L ( β < ∞ )) = O (exp[ − c (2 L + 1)]) (5 . . c = log cosh( βB ) (5 . . From the point of view of the reduction of the wave-packet, we have:
Proposition 1
Reduction of the wave packet in the sense of definition2 occurs for both β = ∞ and β < ∞ . Reduction of the wave packet in thesense of definition 3 occurs for β = ∞ for any L ≥ β < ∞ , buthere only in the limit L → ∞ for all A or for all L but [ A, P + ] = 0 . Proof
We have h Ψ + ,I | Ψ − ,I i = 0For β = ∞ only I = { ... L + 1 } contributes. For β < ∞ we have to takeinto account the contribution of h Ψ + ,I | A | Ψ − ,I i This contribution vanishes for all I that are larger than the localization of A , therefore for almost all I that contribute to the expectation value with ρ ( t ∞ ) if L tends to ∞ . More precisely h Ψ + ,I | A | Ψ − ,I i < O ( exp [ − c (2 L + 1)]) (5 . . Remark 4
The case β = ∞ in proposition 1 exemplifies the fact thatdefinition 3 - in contrast to definition 2 - does not rely a priori on the appa-ratus I having a large number of degrees of freedom - see also the commentsafter definition 2. The case L = 0, i.e., only one degree of freedom, is takenover in section 5c (see definition 5 there), in a model of decoherence, and46eems to agree well with the ideas of Zurek and collaborators, expounded inthe introduction and in that chapter, that only one degree of freedom in achaotic environment may be more effective for decoherence that the quantumBrownian motion models with reservoirs with a large number of degrees offreedom.On the other hand, the case β = ∞ treated by Hepp is very special, andSewell showed that natural stability requirements imposed on the apparatuslead to consider the mixed case β < ∞ , for which the number of degreesof freedom being large is crucial , being related to the exponentially smallcorrections mentioned in proposition 1.It is to be noted that ρ β< ∞ ( t = ∞ ) contains terms Q = c + ¯ c − | +) ( −| ⊗ | Ψ + ,I )(Ψ − ,J | + ¯ c + c − |− ) (+ | ⊗ | Ψ − ,J )(Ψ + ,I | (5 . . P + | Ψ + ,I ) = | Ψ + ,I ) , P − | Ψ − ,J ) = | Ψ − ,J )As a consequence Lemma 2
For L large but finite P + ρ β< ∞ ( t = ∞ ) P − + P − ρ β< ∞ P + = 0 (5 . . Proof
Let Θ ≡ | +) ⊗ | Ψ + ,L ), Θ ≡ |− ) ⊗ | Ψ − ,L ), and denote by D ± ≡| Ψ ± ,L )(Ψ ± ,L | the diagonal terms in ρ β< ∞ . We have that Θ = 0 and Θ = 0(they are vectors of norm one each), and the following hold, for some c > , Q Θ ) = c + c − + terms of order O (exp[ − c (2 L + 1)]) (5 . . | (Θ , P − ρ β< ∞ P + Θ ) | = O (exp[ − c (2 L + 1)]) (5 . . , D ± Θ ) = O (exp[ − c (2 L + 1)]) (5 . . ρ β< ∞ ( t = ∞ ) is (macroscopically) mixed . Theorem 1 is applicableto this more general case and yields the expected growth of the quantumBoltzmann entropy, here nontrivial because a direct treatment is hinderedby the cross terms Q (5.24), which do not disappear as in the case β = ∞ by forming P + Q P + + P − Q P − . However, again as in remark 3, | T r ( P + Q P + + P − Q P − ) | = O (exp[ − c (2 L + 1)]) (5 . . Proposition 2
For any initial state (5.7.3) such that c + c − = 0, we have:if β = ∞ , ∆ S β = ∞ QB ≡ S β = ∞ QB ( ρ ( ∞ )) − S β = ∞ QB ( ρ (0)) == − k ( | c + | log | c + | + | c − | log | c − | ) > . . < β < ∞ ,∆ S β< ∞ QB ≡ S β< ∞ QB ( ρ ( ∞ )) − S β< ∞ QB ( ρ (0)) > . . Proof (5.26.1) follows from (5.13) and (5.17), (5.26.2) from (4.14) oftheorem 1 and Lemma 2. q.e.d.
Remark 5
It is interesting that, by definition (4.3) of the quantum Boltz-mann entropy, the cross-terms Q and Q are either vanishing or small: there-fore, the density matrix behaves, in a sense, as a coherent mixture (with(5.26.1) as expected result), and the reduction of information in the defini-tion of the quantum Boltzmann entropy eliminates the cross-terms alreadyfor finite systems and times.For β < ∞ the positive term on the r.h.s. of (5.26.2) may be extensive ,but this extensive term tends to zero, as β → ∞ . Remark 6
It should be emphasized that β < ∞ and β = ∞ are sugges-tive of the type of state we considered, but has nothing to do with ”inversetemperature of the system”, a notion reserved for equilibrium states. If theinfinite-volume version of the state described in theorem 1 is a stationarystate (e.g. for t = ∞ in Hepp’s formulation of the Coleman-Hepp model)with the property of passivity [84], it follows [84] that it is either a groundstate or a temperature (KMS) state, and that the temperature parameterdefined in the latter case may indeed be identified with the absolute temper-ature of thermodynamics. 48 .7 Entropy reduction in events causing purificationby state collapse and the role of quantum chaoticsystems (quantum K systems) in the purificationprocess Remark 7
The model with( β < ∞ ) provides an example of a density matrixat zero time which is a mixed state but decoheres. By proposition 1, itsweak limit, as L → ∞ , (represented by state collapse defined by a familyof projectors, i.e. the final state is macroscopically pure according to thegiven probability) is long enough, these events purify the mixed state on theclassical quantities. In the case n = 2 in (2.16.1), we have ω = W ω ( α ) ω ( α ) The authors of [38] prove that for almost all histories (i.e. α ) either lim α ω α = ω (5 . . or lim α ω α = ω (5 . . ω or ω dominates, but no mixture of them. One cannotdominate over all histories because X α W ω ( α ) = X α W ω ( α ) = 1 (5 . . undetermined indi-vidual outcome, and it happens under the assumption of collapse of the stateat each individual measurement.Quantum K systems are closest to classical chaotic systems in the sensethat the time correlations factorize when the differences in the time argu-ments become very long. to the extent that most Hamiltonians are quantiza-tions of classical chaotic Hamiltonians in a generic classical sense, the resultof [38] is a partial explanation of the macroscopic purity found in Nature.On the other hand, there are problems with ”what happens if thingsare not measured”: Griffiths [29] remarks that a colleague once asked him,half-seriously, at a conference, ”Is Jupiter really there when no one has his49elescope pointed in that direction?”. Concerning this last question, one mayquote the following from Narnhofer and Thirring [38], which relates to theobservables assuming definite values in e.g. (2.12.1). They observe that, e.g.below the phase transition, domains of a magnet will be magnetized in adefinite direction - for instance, the value of the mean magnetization (2.15)for some fixed ~n is obtained. ”Even if nobody looks at them, there will beenough ’events’-i.e., interactions with the environment - to purify the stateover the classical part”.It is important to stress that the relation between the above considera-tions and the main text concerns only the preliminaries, that is to say, asindications that the assumption in theorem 1 that ρ decoheres at zero timeis a natural one. In fact, the events referred to above act to decrease ouruncertainty regarding the state, and may be expected, in general, to lead toa decrease of the quantum Boltzmann entropy: Lemma 3
The state collapse of a decoherent state produces ,in theaverage, a reduction of the quantum Boltzmann entropy.
Proof
In a state collapse the state corresponding to the density matrix ρ becomes one of the states corresponding to ρ i = ( T rρP i ) − P i ρP i withprobabilities T rρP i . Therefore in the average the entropy will be X i ( T rρP i ) S ( ρ i ) = X i ( T rρP i ) S (( T rρP i ) − ρ / P i ρ / ) > t ).Another possibility is an interaction inside of the apparatus, such thatthe interaction between S and I produces again a coherent state that evolvesonly in I to a decoherent state with respect to macroscopic observables. Thismodel is inspired by an apparatus where a photon sets an electron free thatitself will produce an avalanche of other photons. We take for S a spinsystemthat interacts with a tensor product of four spin systems that in addition canmove. We start with the initial stateΨ = ( c + | +) + c − |− )) ⊗ | , , , S and I changes the state toΨ = ( c + | +) ⊗ | , , , c | − ) ⊗ | − , , , I move and interact such that the states evolveΨ = ( c + | +) ⊗ | , , , c − |− ) ⊗ | − , , , c + | +) ⊗ | , , , c − |− ) ⊗ | − , − , , c + | +) ⊗ | , , , c − |− ) ⊗ | − , − , − , c + | +) ⊗ | , , , c − |− ) ⊗ | − , , − , − )Ψ = ( c + | +) ⊗ | , , , c − |− ) ⊗ | , , − , − )Ψ = ( c + | +) ⊗ | , , , c − |− ) ⊗ | , , − , c + | +) ⊗ | , , , c − |− ) ⊗ | , , − , c + | +) ⊗ | , , , c − |− ) ⊗ | , , − , − ) (5 . . c − . Thedimension of the Hilbert space is now d = 16 , whereasdim [ l V l | Ψ − i = 6 , dim P ( − [ l V l | Ψ − i = 251im P (0) [ l v l | Ψ − i = 2 dim P (1) [ l V l | Ψ − i = 2As other example we took n = 6 . Now the dimension of the Hilbert space is d = 64 . Otherwisedim [ l V l | Ψ − i = 30 , dim P ( − [ l V l | Ψ − i = 3dim P ( − [ l v l | Ψ − i = 3 dim P (0) [ l V l | Ψ − i = 12dim P (2) [ l v l | Ψ − i = 6 dim P (4) [ l V l | Ψ − i = 16For other choice (e.g the permutations (254613) or (234516)) we obtaindim [ V l | Ψ − i = 26 , η l ( h Ψ − | V l σ z V − l | Ψ − i ) = − / , − / n of spins correspond-ingly that I is taken to be macroscopic and to a more random movement ofthe spins. More precisely we consider a perturbation of the points Π( j ) andinteraction between ordered pairs U Φ − = V Π( j ) ⊗ n/ k U k − , k Φ − mit U , | , | , U , | − , | − , − ) U , | , − ) = | , − ) U , | − , − ) = | − , . . c − will leadto an expectation value 0 for the total spin, the vector corresponding to c + remains unchanged and will give an expectation value 1 for the mean spin,which makes it possible to choose the projections on a macroscopic level. Ofcourse the time evolution of a finite system is periodic, therefore we can ex-pect monotonic increase of the quantum Boltzmann entropy and macroscopicdecoherence only for some time interval according to the considerations ofSewell.As for the Coleman-Hepp model we can make the apparatus infinite, i.e.take n = ∞ and replace the perturbation of the points by a movement whereevery particle moves to the next place, where the spin is not turned around,in the ordering in which they were hit. This corresponds to a shift in theHilbert space. Since the Hilbert space is infinite, we do not worry, whether itis unitary. Therefore the number of turned spins will increase exponentiallyin time and the state will converge in the weak limit but nevertheless fast toa decoherent state. Comparing the Coleman-Hepp model with the models in5.8 we realize that in both models we obtain immediately entanglement ofthe system with the apparatus and this entanglement is kept in the course oftime. In the Coleman-Hepp model the passage to macroscopic observation isdue to a macroscopic duration of the interaction between the system and theapparatus whereas in the other models the interaction between system andapparatus is instantaneous and the passage to macroscopic values happensonly in the apparatus. however now this asks differently frome the Coleman-Hepp model for a precise initial state, that is only metastable, thereforeexceptional for the otherwise random evolution of the apparatus. In remark 7, we have dwelt on the important role of quantum K systemsin promoting macroscopic purification. In this section we revisit a differentclass of quantum chaotic models - quantum Anosov systems - for which someexact results have been obtained, in particular an estimate for the time scaleof decoherence. 53 .10 Quantum Anosov systems: definitions, motiva-tion and connection to classical mechanics
In classical statistical mechanics, ergodicity and mixing play a crucial rolein the theory of approach to equilibrium ([26]). Classical Anosov systemsdisplay these properties due to the phenomenon of trajectory instability (see,again, [26] for a brief review and references), or sensitive dependence on initialconditions. The intuition behind the latter property is that next to any pointon any trajectory there is a point at distance s such that their distance growsexponentially in the course of time. Quantum Anosov systems have beenintroduced by Emch, Narnhofer, Sewell and Thirring [39], see also [85], [86],[87] and [88]. If τ t is the time translation automorphism (of the algebra A asin section 2), and σ s the space translation automorphism, τ t ◦ σ s = σ s exp( − λt ) ◦ τ t (5 . t, s ) ∈ Z × R + , Anosov group if ( t, s ) ∈ Z × R , or continuous Anosovgroup if ( t, s ) ∈ R × R [89]. In the case of abelian C* algebras, i.e., spaces ofcontinuous functions, we have the classical Anosov groups, of which a pro-totype is represented by repulsive harmonic forces, which displays sensitivedependence on initial conditions because the particle runs with exponentiallyincreasing velocity to infinity. The classical Hamiltonian H and the classicaldilation K are given by H = λ ( p − x )2 (5 . . K = p − x (5 . . τ t ( x, p ) = (cosh tx + sinh tp, cosh tp + sinh tx ) (5 . . σ s ( x, p ) = ( x + s, p + s ) (5 . . iKs exp( − λt )] exp( iHt ) = exp( iHt ) exp( iKs ) (5 . H and K are assumed to be essentially self-adjoint on C ∞ ( R ), andtherefore generate automorphisms of B ( L ( R ) by τ t ( A ) = exp( iHt ) A exp( − iHt ) (5 . . σ s ( A ) = exp( iKs ) A exp( − iKs ) (5 . . x the multiplication operator and p = − i ∂∂x , the momentum oper-ator on L ( R ). The algebra A is the Weyl algebra W generated by W ≡ W ( β, γ ) = exp[ i ( βx + γp )] (5 . . m = 1): H ( t ) = p / f ( t ) x / . . f is a periodic function of period T : f ( t + T ) = f ( t ) (5 . . K α ≡ α p p + α x x with α ≡ ( α p , α x ) ∈ R (5 . K α , W ( β, γ )] = ( α p β − α x γ ) W ( β, γ ) (5 . W , U † ( t, t ) AU ( t, t ) ∈ W for all A ∈ W and t, t ∈ R (5 . U ( t, t ) is the unitary family of propagators associated to (5.35.1).55 .11 The upper quantum Lyapunov exponent Under the above assumptions, one may define [87] the upper quantumLyapunov exponent ¯ λ as ¯ λ = sup α ∈ R ¯ λ α (5 . . λ α ( U, L α , A, t ) ≡ lim sup t →∞ log || [ L α , A ( t, t )] || t (5 . . A ( t, t ) ≡ U † ( t, t ) AU ( t, t ) (5 . . A ∈ W . The norm, as in section 2, is || A || = sup Ψ ∈H || A Ψ || / || Ψ || .Example (5.31) may be considerably extended to include the case ofan almost periodic quantum parametric oscillator [88], i.e., for which f in(5.35.1) is periodic (5.35.2), quasi-periodic or almost-periodic; accordingly, aso-called generalized Floquet Hamiltonian (originally defined by Jauslin andLebowitz in [90]), a self-adjoint operator K on an enlarged Hilbert space K = L ( M , µ ) ⊗ H , may be defined, where M is a compact metric spaceendowed with a probability measure µ ; it is a circle, a torus or the hull of analmost-periodic function in the three cases above. Defining U K ( t, t ) ≡ exp[ − i ( t − t ) K ] (5 . is exp( λ j ( t − t ) K α j )] U † ( t, t ) = U † ( t, t ) exp( isK α j ) (5 . H ( t ) = p / x T A ( t ) x ] / . x T ≡ ( x , · · · x n ), p T ≡ ( p , · · · p n ), T denoting the transpose of thecolumn vectors ( x and p ). A is a real symmetric matrix depending almost-periodically on time (see remark 4 of [88]). In (5.42), λ j are 2 n complexnumbers such that ℜ ( λ ) ≤ · · · ℜ ( λ n ) < < ℜ ( λ n +1 ) ≤ · · · ℜ ( λ n ) (5 . . λ n is the generalization of the upper quantum Lyapunov exponent de-fined as in (5.39) with R replaced by R ; analogously (5.36) becomes K α = α T x + α Tp p with α ∈ R (5 . . α denoting the column vector with components ( α x , α p ), and (5.34)becomes W ( α ) = exp[ i ( α Tx x + α Tp p )] (5 . . α ∈ R . Finally, (5.37) is replaced by[ L α , W ( α ′ )] = − σ ( α, α ′ ) W ( α ′ ) for all α, α ′ ∈ R (5 . . σ ( α, α ′ ) = α Tx α ′ p − α Tp α ′ x (5 . . α x , α p ) in the R - case be such that α p = 0 (5 . . α p n = 0 (5 . . t , see [88]. It is discussed in[89] and [87] why (5.32), (5.42) are a natural quantum version of the classicalconcept of sensitive dependence on initial conditions.We refer to (5.32) and (5.42) (with α = α n corresponding to λ n ) as( H, K ) - case (a)- (resp. (
K, K α n ) - case (b)-) Anosov systems.Since Hamiltonians such as (5.31.1) and (5.35.1) are not semibounded(in the latter case we should more properly speak of the generalized Floquetoperator), they are only an approximation to realistic systems, similarly tothe Stark Hamiltonian for an atom in an external electric field, in whichcase the approximation may be excellent, under suitable physical conditions;as an example, (5.35.1) describes well the behavior of ions in so-called Paultraps [91].We shall now introduce a class of models for decoherence, constructedfrom quantum Anosov systems as a starting point, in similar way as theColeman-Hepp model in section 3.2. 57 .12 Quantum Anosov models of decoherence We adopt the general framework of section 3.5, although we shall be describ-ing an interacting system, without necessarily identifying I to a measuringapparatus. In proposition 1 we remarked on the greater flexibility of defini-tion 3. We now take for the system S just one spin one-half, and I a particle(oscillator) described by a ( H, K ) or (
K, K α ) - Anosov system (the general n case is analogous). The Hamiltonian H c given by (3.20), with H a multipleof σ , R the first member of either a ( H, K ) (case (a)) or (
K, K α ) Anosovsystem (case (b)), and V = µKσ (5 . . V = µK α σ (5 . . H is a multiple of σ it will not play any role in thedynamics and will be omitted (the choice H = σ is nontrivial and may betreated by perturbation theory, but we shall not do it in this paper). TheHamiltonians are, thus: h = H + µKσ (5 . . h = K + µK α σ (5 . . S c aredescribed, respectively, by the wave vectorsΨ , (0) = Ψ ⊗ Ψ , (5 . . = c + | +) + c − |− ) (5 . . = φ (5 . . = ⊗ φ (5 . . φ ∈ C ∞ ( R ) (5 . . H ⊗ K , , with H = C and K = L ( R ), K = L ( M , dµ ) ⊗ L ( R ).By (5.46) and (5.47), the time evolutes of Ψ , (0) at time t ≥ i ( t ) = c + | +) ⊗ exp[ − it ( H + µK )]Ψ i + c − |− ) ⊗ exp[ − it ( H − µK )]Ψ i for i = 1 , . Definition 5
The systems S i , i = 1 ,
2, described by (5.46.1),(5.46.2) aresaid to display decoherence or reduction of the wave-packet if there exists a0 < t < ∞ (”critical” or ”decoherence” time) and real numbers d i ± , i = 1 , E i ( A ) ≡ (Ψ i ( t ) , A Ψ i ( t )) = d i + (+ | A | +) + d i − ( −| A |− ) (5 . A ∈ B ( C ) and for all t ≥ t , i = 1 , S of arbitrary finite dimension may be included, asin (3.24.1). This means, we are now interested in decoherence of the stateof the small system B ( C ) and interpret its behaviour as reaction on theinteraction with an environment.Definition 5 is implicitly adopted in most applications in theoretical physics([5],[6],[92],[93]), i.e., it takes over (3.24.1) without assuming the remainingstructure (which is, however, of crucial importance in the mixed case and ingeneral statistical mechanics, see remark 3). This definition is of relevancewhen the ”pointer” has few degrees of freedom: see, in particular, studies[93] which have shown that an environment with a single number of degreesof freedom but displaying chaotic dynamics can be far more effective at de-stroying quantum coherence than a heat bath with infinitely many degreesof freedom - e.g., ”quantum brownian motion” [82].The following two lemmas will play a major role. We state them for( H, K )- Anosov systems, but they are also applicable to (
K, K α n )-Anosovsystems using (5.42). Lemma 4
Let (
H, K ) define a quantum Anosov system. Then: { exp( itH/n ) exp( itµK/n ) } m == exp[ it (exp( − λt/n ) + exp( − λt/n ) · · · + exp( − mλt/n )) µK ] exp( imtH/n )(5 . . exp( − itH/n ) exp( itµK/n ) } m == exp( − itmH/n ) exp[ iµKt/n (exp( − λt/n )+exp( − λt/n )+ · · · +exp( − mλt/n )](5 . . m, n arbitrary positive integers. In addition, e it ( H + µK ) = e iµK ( − e − λtλ ) e iHt e − it ( H − µK ) = e − itH e − iµK ( − e − λtλ ) (5 . . Proof
We use induction on m . For m = 1, (5.50.1) reduces to (5.32).Assume, now, (5.50.1) valid for m = m , and consider the l.h.s. of (5.50.1)for m = m + 1. By the induction hypothesis it equalsexp( itH/n ) exp( itµK/n ) exp[ it (exp( − λt/n ) + · · · + exp( − m λt/n )) µK ] exp( im tH/n )= exp( itH/n ) exp[ itµK/n (1 + exp( − λt/n ) + · · · + exp( − m λt/n )] exp( im tH/n )(5 . s = 1+exp( − λt/n )+ · · · +exp( − m λt/n ),we obtain (5.50.1) for m = m + 1, concluding the proof. The exponentsin (5.50.1) and (5.50.2) are Riemann sums for the integral R t exp( λu ) du = − exp( − λt ) λ . Hence, by the strong continuity of the group α → exp( iαK ), andtaking m = n → ∞ in (5.50.1), (5.50.2), we obtain (5.50.3) by the Trotterproduct formula (see, e.g., [94], pp.295-296). q.e.d. Theorem 2
The system composed of a two-level system and a quantum(
H, K ) Anosov system in the initial state (5.47.1) decoheres exponentiallyfast for t → ∞ , λ <
0. If in addition Φ has compact support it decoheresin finite time in the sense of Definition 5, if Φ has compact support. Thedecoherence time depends on λ and the localization of Φ.If λ > t → ∞ only if the support of φ satisfies S < µ/λ. In this case the decoherence time is given by t = | log 2 | λ (5 . . a ≡ Sλ µ < . . H, K ) system, and t = t + | log 2 |ℜ λ (5 . . a ≡ S ℜ ( λ ) exp( t ℜ ( λ ))4 µα p < . . K, K α ) system. Above S denotes the support of the function Φ in(5.47.5), i.e, the smallest real number such that Φ( x ) = 0 if | x | > S .If however these conditions are not satisfied it does not decohere at all.Therefore the decoherence properties are time irreversible! Proof
In order to prove (5.49) for (
H, K ) Anosov systems, it suffices, by(5.48), to prove that
I ≡ (Φ , exp[ it ( H + µK )] exp[ − it ( H − µK )]Φ) = 0 (5 . t > t under assumption (5.52.2). Using (5.50.3) in (5.53) we obtain I = ( φ, exp[2 iµK − exp( − λt ) λ ] φ ) (5 . i ( αx + βp )] = exp( iαx ) exp( iβp ) exp( − iαβ/
2) and the fact that exp( iβp )acts as a translation by β to obtain (5.53) for t > t under assumption(5.52.2). The proof of assertion (5.53) for t > t under assumption (5.52.4)for ( K, K α ) Anosov systems is the same, using the Trotter product formulawith t = t as initial time (see (5.40) and the remark after (5.36.2)). q.e.d. Remark 8
Interactions such as (5.45.1,2) generalize the interaction linearin the x coordinate in quantum Brownian motion ([82], [83]), by replacing it61y the dilation operator K α . The latter, as (5.37) shows, may be interpretedas a derivation in the direction of phase space determined by ( α p , α x ), andis, thus, naturally suggested by classical mechanics, see also [87]. Remark 9
The phenomenon of decoherence, according to definition 5, may also becharacterized in specific models, e.g., a particle in a double well or a caricaturethereof as a two-level system (see, e.g., [95], [96] for examples), as instabilityof tunneling. We refer to [1], [2] for lucid reviews, in particular the relationto the problem of the inversion line, shape and chiral superselection sectorsassociated to pyramidal molecules. This tunneling instability is universal in the semiclassical regime (outside of which we refer to as ”purely quantumcase”), viz., it occurs for an arbitrarily small disturbance of the double wellpotential (and even localized far away from the minima of the potential): see[97]. This phenomenon, dubbed by Simon ”flea on the elephant” [98] plays amajor role in the approach of Landsman and Reuvers [10] to the importantissue of how to obtain just one outcome in a given measurement.Theorem 2 provides a concrete realization of an idea of Peres [7] andZurek and collaborators ([93], [92])- that two nearly identical systems, pre-pared in identical states, but obeying slightly different dynamics, i.e., subjectto slightly different Hamiltonians H and H + δH , will evolve into two dif-ferent states, whose inner product decays exponentially in time, when H isthe quantization of a chaotic Hamiltonian - in a purely quantum context.There is, however, a proviso: conditions (5.52.2) and (5.52.4) do not holdfor µ small, except if λ or ℜ ( λ ) is small - and, then, t and t will becorrespondingly large. This breakdown of the universality present in thesemiclassical domain, due to the dependence of the tunneling instability ordecoherence on special parameter values - was also observed in the purelyquantum proofs of tunneling instability (see, e.g., [95][96]), as well as in theAnderson transition (see, e.g., [99] and references given there). In thermodynamics irreversibility is expressed as monotonic increase of en-tropy. This thermodynamical concept of entropy is believed to be related tothe entropy of states defined both in classical and quantum theory, here espe-cially by von Neumann. Under an automorphic time evolution this entropy is62onserved. To explain an increase we can either include a suitable interactionwith a surrounding or a kind of coarse graining becomes necessary, togetherwith conditions on the initial state to explain the arrow of time. In quantumtheory this coarse graining can be considered as a reduction of the algebra,that can also be interpreted as ignoring some unobservable quantum corre-lations. This corresponds to decoherence effects as they are also importantin the theory of quantum measurements and formally appear in the sameway. Entropy is related to our knowledge of the system. Thus models thatappear in the theory of measurements can support also our explanations forthe origin of the second law of thermodynamics.However, decoherence in the theory of measurements should be inter-preted with more care. In classical theory optimal knowledge of the systemguarantees certainty in the outcome of a measurement. This is no longer truein quantum theory. Our ignorance on the outcome of the experiment is ex-pressed in the increase of entropy in theorem 1. If however the measurementis performed and we have observed the position of the pointer, the proba-bility of its position can only be evaluated if we repeat the measurement.In the individual observation it has a definite value. We may assume if e.g.we follow the interpretation of von Neumann, that the measured system isnow in a state corresponding to the position of the pointer. Reduction of thewave function turns into the collapse of the wave function. We have gainedknowledge though not necessarily in the individual case, but in the averagethe entropy is reduced (Lemma 3).If we believe in the validity of the second law and also in the fact, thatthe occurrence of state collapse as it happens in a measuring process is notrestricted to the observation by an observer who is not part of the physicalsystem (compare [33], VII.1-VII.3 , see also [100]) we run into two competingeffects: one is the increase of entropy corresponding to the fact that corre-lations become unobservable. The other is a decrease of entropy due to nonautomorphic events. Both effects yield irreversibility, but nevertheless theyare not cooperative but contrary to one another. The fact that the secondlaw holds up to any doubts tells us that these non automorphic events mustbe rare in comparison to the time scale that is relevant in thermodynamics.
Acknowledgements
We are deeply indebted to Geoffrey L. Sewell forseveral enlightening remarks on previous drafts of this paper, in particularfor his emphasizing the nonmonotonic character of the entropy growth intheorem 1 and referring us to the work of van Kampen, as well as for pointing63ut some physical inconsistencies in a previous version of the model in section5.12. We are also grateful to the referee for several helpful remarks.
References [1] A. S. Wightman. In M. Ioffredo F. Guerra and C. Marchioro, editors,
Proc. Int. Workshop on probability methods in mathematical physics .World Scientific Singapore, 1992.[2] A. S. Wightman. Superselection sectors: old and new.
Nuovo Cim. B ,110:751, 1995.[3] J. von Neumann.
Mathematical foundations of quantum mechanics .Princeton university press, 1955.[4] J. A. Wheeler and W. H. Zurek.
Quantum theory and measurement .Princeton University Press, 1983.[5] W. H. Zurek. In J. Perez-Mercader J. Halliwell and W. Zurek, editors,
Physical origins of time asymmetry . Cambridge University Press, 1994.[6] W. H. Zurek. Decoherence, einselection and the quantum origins of theclassical.
Rev. Mod. Phys. , 75:715, 2003.[7] A. Peres.
Quantum theory: concepts and methods . Kluwer academicpublishers, 1995.[8] C. Kiefer J. Kupsch I O. Stamatescu D. Giulini, E. Joos and H. D.Zeh.
Decoherence and the appearance of a classical world in quantumtheory . springer Berlin, 1996.[9] M. Schlossauer.
Decoherence and the quantum-to-classical transition .Springer Heidelberg Berlin, 2007.[10] N. P. Landsman and R. Reuvers. A flea on Schr¨odinger’s cat.
Found.Phys. , 43:373–407, 2013.[11] R. Balian A. E. Allahverdyan and T. M. Nieuwenhuizen. Understand-ing quantum measurement from the solution of dynamical models.
Phys. Rep. , 525:1–166, 2013. 6412] R. Balian A. E. Allahverdyan and T. M. Nieuwenhuizen. Curie-Weissmodel of the quantum measurement process.
Eur. Phys. lett. , 61:452,2003.[13] K. Hepp. Quantum theory of measurement and macroscopic observ-ables.
Helv. Phys. Acta , 45:237, 1972.[14] G. L. Sewell. On the mathematical structure of quantum measurementtheory.
Rep. Math. Phys. , 56:271, 2005.[15] G. L. Sewell. Can the quantum measurement problem be resolvedwithin the framework of Schr¨odinger dynamics?
Markov Proc. Rel.Fields , 13:425, 2007.[16] B. Whitten-Wolfe and G. G. Emch. A mechanical quantum measuringprocess.
Helv. Phys. Acta , 49:45, 1976.[17] N. P. Landsman. Algebraic theory of superselection sectors and themeasurement problem in quantum mechanics.
Int. Jour. Mod. Phys.A , 6:5349, 1991.[18] I. Guarneri. Irreversible behaviour and collapse of wave packets in aquantum system with point interactions.
J. Phys. A Math. Theor. ,44:485304, 2011.[19] G. L. Sewell.
Quantum theory of collective phenomena . ClarendonPress, Oxford, 1986.[20] N. M. Hugenholtz. In R. F. Streater, editor,
Mathematics of Contem-porary Physics . Academic Press, 1972.[21] C. Maes. On the origin and the use of fluctuation relations for theentropy.
Seminaire Poincar´e , 2:29–62, 2003.[22] W. deRoeeck, T. Jacobs, C. Maes, and K. Netocny. An extension ofthe Kac ring model.
Jour. Phys. A , 36:11547, 2003.[23] J. Fr¨ohlich and B. Schubnel. Quantum probability theory and thefoundations of quantum mechanics. arXiv 1303.1484.6524] S. Goldstein and J. L. Lebowitz. On the boltzmann entropy of non-equilibrium systems.
Physica D , 193:53–66, 2004.[25] J. L. Lebowitz. Time-asymmetric macroscopic behavior: an overview.In W. L. Reiter G. Gallavotti and J. Yngvason, editors,
BoltzmannsLegacy , pages 63–87. Eur. Math. Soc., 2008.[26] O. Penrose. Foundations of statistical mechanics.
Rep. Progr. Phys. ,42:1937, 1979.[27] O. Penrose.
Foundations of statistical mechanics . Oxford, PergamonPress, 1970.[28] D. ter Haar and H. Wergeland.
Elements of Thermodynamics . AddisonWesley Publ. Co., 1966.[29] R. B. Griffiths. In J. Perez-Mercader J. Haliwell and W. Zurek, editors,
Physical origins of time asymmetry . Cambridge University Press, 1994.[30] V. Jaksic and C. A. Pillet. Mathematical theory of nonequilibriumstatistical mechanics.
Jour. Stat. Phys. , 108:787, 2002.[31] V. Jaksic and C. A. Pillet. NESS in quantum statistical mechanics:where are we after ten years? IAMP News Bulletin, January 2011,available at the web.[32] E. H. Lieb and J. Yngvason. The physics and mathematics of thesecond law of thermodynamics.
Phys. Rep. , 310:1–96, 1999.[33] R. Haag.
Local quantum physics - Fields, particles, algebras . SpringerVerlag, 1996.[34] W. H. Zurek. Basis of classical apparatus: into what mixture does thewave packet collapse?
Phys. Rev. D , 24:1516, 1981.[35] G. Lindblad. Entropy, information and quantum measurements.
Comm. Math. Phys. , 33:305, 1973.[36] H. Narnhofer and W. Thirring. Algebraic K systems.
Lett. Math. Phys. ,20:231, 1990. 6637] G. G. Emch. Generalized K-flows.
Comm. Math. Phys. , 49:191–215,1976.[38] H. Narnhofer and W. Thirring. In A. Amann H. Atmanspacher andU. M¨uller-Herrold, editors,
On quanta, mind and matter . Kluwer aca-demic publishers, 1999.[39] G. G. Emch, H. Narnhofer, G. L. Sewell, and W. Thirring. Anosovactions on noncommutative algebras.
Jour. Math. Phys. , 35:5582, 1994.[40] E. H. Lieb and M. B. Ruskai. Proof of the strong subadditivity ofquantum mechanical entropy.
J. Math. Phys. , 14:1938, 1973.[41] H. Araki and G. L. Sewell. KMS conditions and local thermodynamicalstability of quantum lattice systems.
Comm. Math. Phys. , 52:103, 1977.[42] R. Haag and D. Kastler. An algebraic approach to quantum field theory.
Jour. Math. Phys. , 5:548, 1964.[43] O. Bratelli and D. W. Robinson.
Operator algebras and quantum sta-tistical mechanics I . Springer, 1987.[44] I. F. Wilde. Lecture notes on local quantum theory and operator alge-bras. available from Ivan F. Wilde homepage.ntl.world.com.[45] Domingos H. U. Marchetti and Walter F. Wreszinski.
Asymptotic TimeDecay in Quantum Physics . World Scientific, 2013.[46] O. E. Lanford and D. Ruelle. Observables at infinity and states withshort range correlations in statistical mechanics.
Comm. Math. Phys. ,13:194, 1969.[47] A. Wehrl M. Guenin and W. Thirring. Introduction to algebraic thech-niques. Lectures given at theoretical seminar series CERN 68-69.[48] O. Bratelli and D. W. Robinson.
Operator algebras and quantum sta-tistical mechanics II . Springer, 2nd edition, 1997.[49] R. F. Streater. The Heisenberg ferromagnet as a quantum field theory.
Comm. Math. Phys. , 6:233, 1967.6750] D. W. Robinson. The statistical mechanics of quantum spin systems.
Comm. Math. Phys. , 6:151, 1967.[51] E. H. Lieb and D. W. Robinson. The finite group velocity of quantumspin systems.
Comm. Math. Phys. , 28:251, 1972.[52] B. Nachtergaele and R. Sims. Lieb-Robinson bounds and the exponen-tial clustering theorem.
Comm. Math. Phys. , 265:119–130, 2006.[53] J. Glimm and R. V. Kadison. Unitary operators in C*-algebras.
Pac.J. Math. , 10:547, 1960.[54] R. Schatten.
Norm ideals of completely continuous operators . SpringerBerlin, 1960.[55] J. S. Bell. On wave packet reduction in the Coleman-Hepp model.
Helv.Phys. Acta , 48:93, 1975.[56] H. Narnhofer and W. Thirring. Galilei invariant quantum field theorieswith pair interaction - a review.
Int. Jour. Mod. Phys. A , 17:2937–2970,1991.[57] H. Narnhofer and W. Thirring. Quantum field theories with Galilei-invariant interactions.
Phys. Rev. Lett. , 64:1863, 1990.[58] D. Loss D. Burkard and P. di Vicenzo. Coupled quantum dots asquantum gates.
Phys. Rev. B , 59:2070, 1999.[59] C. M. Lockhart and B. Misra. Irreversibility and measurement in quan-tum mechanics.
Physica A , 136:47, 1986.[60] J. M. Jauch.
Foundations of quantum mechanics . Addison Wesley,1968.[61] R. S. Ellis.
Entropy, large deviations and statistical mechanics . SpringerBerlin, 1985.[62] G. Lindblad. Expectations and entropy inequalities for finite quantumsystems.
Comm. Math. Phys. , 39:111, 1974.6863] G. Lindblad. Completely positive maps and entropy inequalities.
Comm. math. Phys. , 40:147, 1975.[64] R. B. Griffiths. Consistent histories and the interpretation of quantummechanics.
J. Stat. Phys. , 36:219–272, 1984.[65] M. Gell-Mann and J. B. Hartle. In K. K. Phua and Y. Yamaguchi,editors,
Proceedings of the 25th international conference on high energyphysics . World Scientific Singapore, 1991.[66] R. Omn´es.
The interpretation of quantum mechanics . Princeton Uni-versity Press, 1994.[67] W. Thirring. The histories of chaotic quantum systems.
Helv. Phys.Acta , 69:706, 1996.[68] H. Araki. Relative entropy of states of von Neumann algebras.
Publ.RIMS Kyoto Univ. , 11:809, 1976.[69] M. Ohya and D. Petz.
Quantum entropy and its use . Springer Verlag,1993.[70] A. Wehrl. General properties of entropy.
Rev. Mod. Phys. , 50:221,1978.[71] H. Kosaki. Interpolation theory and the Wigner-Yanase-Dyson-Liebconcavity.
Comm. Math. Phys. , 87:315, 1982.[72] E. H. Lieb. Convex trace functions and the Wigner-Yanase-Dysonconjecture.
Adv. Math , 11:267, 1973.[73] S. Zelditch. In G. Naber J. P. Fran¸coise and T. S. Tsun, editors,
Quan-tum ergodicity and mixing of eigenfunctions in Encyclopedia Mathemat-ical Physics . Academic Press NY, 2006.[74] N. G. van Kampen. In E. G. D. Cohen, editor,
Fundamental problemsin statistical mechanics . North Holland Amsterdam, 1962.[75] C. J. Thompson.
Mathematical statistical mechanics . Macmillan, 1971.6976] P. G. Bergmann Y. Aharonov and J. L. Lebowitz. Time symmetry inthe quantum process of measurement.
Phys. Rev.B , 134:1410, 1964.[77] C. Maes. Fluctuation relations and positivity of the entropy productionin irreversible dynamical systems.
Nonlinearity , 17:1305, 2004.[78] D. Buchholz. New light on infrared problems: sectors, statistics, spec-trum and all that. arXiv 1301.2516 v.2 (14-1-2013).[79] H. Narnhofer. In R. Kotecky, editor,
Phase Transitions . World Scien-tific Singapore, 1993.[80] H. Narnhofer and D. W. Robinson. Dynamical stability and thermo-dynamic phases.
Comm. Math. Phys. , 41:89, 1975.[81] R. Carlone C. Cacciapuoti and R. Figari. Decoherence induced byscattering: a three-dimensional model.
J. Phys. A Math. Gen. , 38:4933,2005.[82] W. G. Unruh and W. H. Zurek. Reduction of the wave packet inquantum brownian motion.
Phys. Rev. D , 40:1071, 1989.[83] E. Joos and H. R. Zeh. The emergence of classical properties throughinteraction with the environment.
Zeitschr. Physik B , 59:223, 1985.[84] W. Pusz and S. Woronowicz. Passive states and KMS states for generalquantum systems.
Comm. Math. Phys. , 58:273, 1978.[85] W. A. Majewski and M. Kuna. On quantum characteristic exponents.
J. Math. Phys. , 34:5007, 1993.[86] R. Vilela Mendes. Lyapunov exponent in quantum mechanics. A phasespace approach.
Phys. Lett. A , 171:253, 1992.[87] H. R. Jauslin, O. Sapin, S. Gu´erin, and W. F. Wreszinski. Upperquantum Lyapunov exponent and parametric oscillators.
Jour. Math.Phys. , 45:4377, 2004.[88] O. Sapin, H. R. Jauslin, and S. Weigert. Upper quantum Lyapunov ex-ponent and Anosov relations for quantum systems driven by a classicalflow.
Jour. Stat. Phys. , 127:699, 2007.7089] W. Thirring. What are the quantum mechanical Lyapunov expo-nents. In P. Urban, editor,
Proceedings of the 34 Internationale Univer-sit¨atswoche f¨ur Kern und Teilchenphysik Schladming , pages 223–237.Springer, 1996.[90] H. R. Jauslin and J. L. Lebowitz. Spectral and stability aspects ofquantum chaos.
Chaos , 1:114, 1991.[91] J. Wu M. Feng and K. Wang. A study of the characteristics of thewave packets of a Paul trapped ion.
Commun. Theor. Phys. , 29:497,1998.[92] C. Jarinski Z. P. Karkusevski and W. H. Zurek. Quantum chaoticenvironments, the butterfly effect, and decoherence.
Phys. Rev. Lett. ,89:170405, 2002.[93] R. Blume-Kohout and W. H. Zurek. Decoherence from a chaotic en-vironment: An upside-down ”oscillator” as a model.
Phys. Rev. A ,68:032104, 2003.[94] M. Reed and B. Simon.
Methods in modern mathematical physics - v.1,Functional Analysis . Academic Press, 1st edition, 1972.[95] V. Grecchi and A. Sacchetti. Critical metastability and destruction ofthe splitting in non-autonomous systems.
Jour. Stat. Phys. , 103:339,2001.[96] W. F. Wreszinski and S. Casmeridis. Models of two-level atoms inquasiperiodic external fields.
Jour. Stat. Phys. , 90:1061, 1998.[97] G. Jona Lasinio, F. Martinelli, and E. Scoppola. New approach to thesemiclassical limit of quantum mechanics. I. Multiple tunnelings in onedimension.
Comm. Math. Phys. , 80:223, 1981.[98] B. Simon. Semiclassical analysis of low-lying eigenvalues: the flea onthe elephant.
J. Func. Anal. , 63:123, 1985.[99] Walter F. Wreszinski. Progress in the mathematical theory of quantumdisordered systems.