aa r X i v : . [ qu a n t - ph ] D ec Quantum State Reduction
Dorje C. Brody , and Lane P. Hughston Department of Mathematics, Brunel University London, Uxbridge UB8 3PH, UK Department of Optical Physics and Modern Natural Science,St Petersburg National Research University of Information Technologies,Mechanics and Optics, 49 Kronverksky Avenue, St Petersburg 197101, Russia (Dated: July 12, 2018)We propose an energy-driven stochastic master equation for the density matrix as adynamical model for quantum state reduction. In contrast, most previous studies ofstate reduction have considered stochastic extensions of the Schr ¨odinger equation,and have introduced the density matrix as the expectation of the random pureprojection operator associated with the evolving state vector. After working outproperties of the reduction process we construct a general solution to the energy-driven stochastic master equation. The solution is obtained by the use of nonlinearfiltering theory and takes the form of a completely positive stochastic map.
Keywords: quantum mechanics, collapse of wave function, measurement problem, density matrix,master equation, stochastic analysis, nonlinear filtering
I. INTRODUCTION
Many physicists have expressed the view that quantum mechanics needs to be modified toprovide a mechanism for “collapse of the wave function” (Pearle 1976, Penrose 1986, Bell1987, Diosi 1989, Ghirardi 2000, Adler 2003a, Weinberg 2012). Among the ways forwardthat have been proposed, perhaps the most fully developed, at least from a mathematicalpoint of view, are the so-called stochastic models for state reduction, in connection withwhich there is now a substantial body of literature. In such models, the quantum systemis usually taken to be in a pure state, represented by a vector in Hilbert space, evolvingas a stochastic process. The state of the system evolves randomly in such a way that iteventually approaches an eigenstate of a preferred observable, such as position or energy.In the situation where the reduction is to a state of definite energy, which is the casethat will concern us here, the setup is as follows. The Hilbert space H is taken to be offinite dimension N , and the state-vector process {| ψ t i} t ≥ is assumed to satisfy an Ito-typestochastic di ff erential equation of the formd | ψ t i = − i ~ − ˆ H | ψ t i d t − σ ( ˆ H − H t ) | ψ t i d t + σ ( ˆ H − H t ) | ψ t i d W t . (1)Here { W t } t ≥ is a standard Brownian motion, and | ψ t i ∈ H is the state vector at time t . Theinitial state vector | ψ i is an input of the model. We write H t = h ψ t | ˆ H | ψ t ih ψ t | ψ t i (2)for the expectation value of the Hamiltonian operator ˆ H in the state | ψ t i . The reductionparameter σ , which has dimensions such that σ ≈ [energy] − [time] − , (3)determines the characteristic timescale τ R associated with the reduction of the state, whichis of the order τ R ≈ / ( σ ∆ H ) , where ∆ H is the initial uncertainty of the energy. Thus astate with high initial energy uncertainty has a shorter characteristic reduction timescalethan a state with low energy uncertainty. After a few multiples of τ R , the system will benearly in an eigenstate of energy. The determination of σ is an empirical matter. Oneintriguing possibility suggested by a number of authors (Karolyhazy 1966, Karolyhazyet al 1986, Penrose 1986, 1996, Diosi 1989, Percival 1994, Hughston 1996) is that statereduction is determined in some way by gravitational phenomena. In that case we mightsuppose that σ is given by a relation of the form σ ≈ E P − T P − , where E P is the Planckenergy and T P is the Planck time, and hence of the order σ ≈ √ G ~ − c − . (4)A surprising feature of this expression is that the large numbers associated with thevarious physical constants cancel out, and we are left with a reduction timescale that is inprinciple observable in the laboratory, given by τ R ≈ (cid:18) . ∆ H (cid:19) s . (5)Going forward, we shall not make any specific assumptions regarding the magnitude ofthe reduction parameter. Nevertheless, to get a feeling for the numbers involved, we notethat the binding energies per nucleon of low mass nuclei are of the order of 1.1 MeV for thedeuteron, 2.6 MeV for He , and 7.1 MeV for He . Since the fusion reactions leading to theproduction of such nuclei are essential in normal stellar evolution, it is not unreasonableto suppose that some form of observer-free “objective” state reduction is involved in theprocess, and that gravitational e ff ects play a role as well.No attempt will be made to review the extensive literature of dynamical collapsemodels, of which the energy-driven model described above is an example, or to discussin any detail the relative merits of the various models that have been proposed. See Bassi& Ghirardi (2003), Bassi (2007), Pearle (2007, 2009), Bassi et al (2013), Ghirardi (2016) forsurveys. For aspects of the energy-driven models, we refer the reader to Gisin (1989),Ghirardi et al (1990), Percival (1994, 1998), Hughston (1996), Pearle (1999, 2004), Adler& Horwitz (2000), Adler & Mitra (2000), Adler et al (2001), Brody & Hughston (2002a,b,2005, 2006), Adler (2003a,b, 2004), Brody et al (2003, 2006), Gao (2013), Meng ¨ut ¨urk (2016).Adler (2002), in an empirical study of energy-driven models, concludes thus:Our analysis supports the suggestion that a measurement takes place whenthe di ff erent outcomes are characterized by su ffi ciently distinct environmentalinteractions for the reduction process to be rapidly driven to completion.Although other collapse models have been considered at length in the literature, in-cluding, for example, the GRW model (Ghirardi, Rimini & Weber 1986) and so-calledcontinuous spontaneous localization (CSL) models (Diosi 1989, Pearle 1989, Ghirardi,Pearle & Rimini 1990), the energy-driven reduction models stand out, in our view, onaccount of (a) their parsimonious mathematical structure, and (b) the fact that they areuniversal. By “universal”, we mean applicable to any quantum system. We point out thatenergy-driven models maintain the conservation of energy in a well-defined probabilisticsense, as an extension of the Ehrenfest theorem, whereas models driven by observablesthat do not commute with the energy, such as position, do not conserve energy (Pearle2000, Bassi, Ippoliti & Vacchini 2005). Furthermore, energy-driven models give the Bornrule and the L ¨uders projection postulate as exact results (Adler & Horwitz 2000, Adleret al 2001, Adler 2003b), whereas other models do not. For these reasons we emphasizehere the role of energy-driven models. This is not to say that energy-driven models arethe only ones to be taken seriously. But if one wishes to propose a stochastic reductionmodel that it is applicable to any nonrelativistic system, without qualification, includingfinite dimensional systems, then it must be an energy-driven model.In that case, is the dynamics necessarily of the form (1) given above? Clearly not, since,for a start, one could consider the possibility that other forms of noise than Brownianmotion act as a basis for the stochastic dynamics of the state, and indeed there is a sizableliterature dealing with dynamical reduction models based on other types of noise. To keepthe discussion focussed, we stick here with models based on Brownian motion, though inthe final section of the paper we comment briefly on a generalization to models based onL´evy noise. One might also introduce time-dependent coupling (Brody & Hughston 2005,2006, Brody et al 2006, Meng ¨ut ¨urk 2016), which o ff ers an approach to the “tail problem”(Shimony 1990, Pearle 2009). Again, we pass over such considerations for the present.There is, however, an important aspect of the dynamical equation (1) that seems tobuild in what might be viewed as an unnecessary assumption, even if one accepts theprinciple that reduction must be energy driven, and even if one narrows the scope tomodels based on a Brownian filtration. This concerns the issue of what constitutes a“state” in quantum mechanics. The physics community seems to be divided on thematter. It is worth recalling that in von Neumann’s highly influential 1932 book, the term“state” is reserved for pure states, and the statistical operator is introduced to describemixtures. He introduces the notion of a statistical ensemble, corresponding to a countablecollection of quantum systems, each of which is in a pure state, and he distinguishes twocases. In the first case, the individual systems of the ensemble can be in di ff erent states,and the statistical operator is determined by their relative frequencies. In the second case,which he calls a homogeneous ensemble, the various individual systems are in the samestate. The statistical operator for a homogeneous ensemble is identical to the state of anyone of its elements, and takes the form of a pure projection operator.In his consideration of statistical ensembles von Neumann (1932) was motivated in partby the frequentist theories of von Mises (1919, 1928). In particular, von Neumann identifieshis concept of ensemble with von Mises’s idea of a “Kollektiv” (random sequence):Such ensembles, called collectives, are in general necessary for establishingprobability theory as the theory of frequencies. They were introduced byRichard von Mises, who discovered their meaning for probability theory, andwho built up a complete theory on this foundation.According to von Mises, “Erst das Kollektiv, dann die Wahrscheinlichkeit”. At about thesame time that these developments were under way, Kolmogorov (1933) revolutionizedclassical probability theory by giving it a set-theoretic foundation and providing it with asubtle measure-theoretic definition of conditional expectation that allows one to handle ina satisfactory way the logical issues associated with conditioning on events of probabilityzero. The mathematics community took on board Kolmogorov’s innovations, and successfollowed success, with the introduction of many further new ideas, including, amongothers, martingales, stochastic calculus, and nonlinear filtering. Von Mises’s theory,despite its attractive features, was eventually dropped by mathematicians, even thoughthe ensemble concept (and elements of the frequentist thinking underpinning it) has beenkept alive by physicists, and is still taught to students (Isham 1995, for instance, gives agood treatment of the relevant material). See van Lambalgen (1999) for a rather detaileddiscussion of where von Mises’s ideas stand today. It appears that the more general use ofthe term “state” (to include mixed as well as pure states) was introduced by Segal (1947),in his postulates for general quantum mechanics. Segal’s point of view was adopted byHaag & Kastler (1964), and also by Davies (1976), who says:The states are defined as the non-negative trace class operators of trace one,elsewhere called mixed states or density matrices.If the matter were purely one of terminology, there would be no point in worrying aboutit very much. The problem is that in the language physicists use there can be assumptionsthat are implicit in the choice of words, and these in turn can guide the direction of thesubject as it moves forward. The issue of what exactly constitutes a “state” is such a case.The point that concerns us here is that most of the models that have been developedin detail in the collapse literature treat the quantum system as a randomly evolving pure state. This point of view is represented, for example, in Ghirardi, Pearle & Rimini (1990) inthe context of their development of the CSL model, where we find the following succinctaccount of their stance on the matter:The theory discussed here allows one to describe naturally quantum measure-ment processes by dynamical equations valid for all physical systems. It isworthwhile repeating, that, in this theoretical scheme, any member of the sta-tistical ensemble has at all times a definite wave function. As a consequence,the wave function itself can be interpreted as a real property of a single closedphysical system.The emphasis placed on the role of pure states reflects a view held by many physiciststhat pure states should be treated as being fundamental. See, for example, Penrose (2016),who argues persuasively concerning the preferred status of pure states. According to thisview, which, as we have indicated, is generally in line with that of von Neumann (1932),individual systems are represented by pure states. Physicists are likewise divided on theissue of the status of statistical ensembles. Are they essential to the theory? Mielnik (1974)o ff ers the following:It is an old question whether the formalism of quantum theory is adequate todescribe the properties of single systems. What is verified directly in the mostgeneral quantum experiment are rather the properties of statistical ensembles.Although our brief remarks cannot do justice to the deep insights of the authors men-tioned above, one will be impressed by the diversity of opinion held by physicists onthe nature of quantum states and the role of statistical ensembles. It should be empha-sized, nevertheless, that, as far as we can see, there is no empirical basis for assumingthat individual quantum systems are necessarily in pure states. Nor is there any evi-dence showing that density matrices necessarily have to be interpreted as representingensembles. In fact, it seems to be accepted in the quantum information community thatthe state of an individual system should be represented, in certain circumstances, by ahigher-rank density matrix. This can happen, for example, if the system is entangledwith another system and the state of the composite system is pure, in which case the stateof the first system is obtained by taking the reduced density matrix of the system as awhole, where we trace out the degrees of freedom associated with the second system. Itthus seems reasonable to take matters a step further and drop altogether the assumptionthat individual systems are necessarily in pure states. It also seems reasonable to dropthe assumption that statistical ensembles play a fundamental role in the theory. In ourapproach, therefore, we make no use of frequentist thinking, and we avoid referenceto observers, measurements, and ensembles. We regard state reduction as an entirelyobjective phenomenon, and even in the case of an individual system we model the stateas a randomly evolving density matrix. We denote the density matrix process by { ˆ ρ t } t ≥ ,and we require that ˆ ρ t should be nonnegative definite for all t and such that tr ˆ ρ t =
1. Thedynamical equation generalizing (1) then takes the following form:
Definition 1
We say that the state { ˆ ρ t } t ≥ of an isolated quantum system with Hamiltonian ˆ Hsatisfies an energy-driven stochastic master equation with parameter σ if d ˆ ρ t = − i ~ − [ ˆ H , ˆ ρ t ]d t + σ (cid:16) H ˆ ρ t ˆ H − ˆ H ˆ ρ t − ˆ ρ t ˆ H (cid:17) d t + σ (cid:16) ( ˆ H − H t ) ˆ ρ t + ˆ ρ t ( ˆ H − H t ) (cid:17) d W t , (6) where H t = tr ˆ ρ t ˆH . We take a moment to spell out some of the mathematical ideas implicit in the dy-namics. In accordance with the well-established Kolmogorovian outlook, we introducea probability space ( Ω , F , P ) as the basis of the theory. We do not necessarily say indetail what the structure of this space is, but we assume that it is endowed with su ffi cientrichness to support the various structures that we wish to consider. Thus Ω is a set onwhich we introduce a σ -algebra F (no relation to the σ above) and a probability measure P . By an algebra we mean a collection of subsets of Ω such that Ω ∈ F , A ∈ F implies Ω \ A ∈ F , A ∈ F and B ∈ F implies A ∪ B ∈ F . If for any countable collection ofelements A i ∈ F , i ∈ N , it holds that ∪ i ∈ N A i ∈ F , then we say that F is a σ -algebra. Thepair ( Ω , F ) is called a measurable space. By a probability measure on ( Ω , F ) we mean afunction P : F → [0 ,
1] satisfying P [ Ω ] =
1, and P [ ∪ i ∈ N A i ] = P i ∈ N P [ A i ] for any countablecollection of elements A i ∈ F , i ∈ N , such that A i ∩ A j = ∅ if i , j . A measurable spaceendowed with a probability measure defines a probability space. A function X : Ω → R is said to be F -measurable, or measurable on ( Ω , F ), if for all A ∈ B R , where B R is theBorel σ -algebra on R , it holds that { ω : X ( ω ) ∈ A } ∈ F . Thus for each A ∈ B R we require X − ( A ) ∈ F . If X is a measurable function on a probability space ( Ω , F , P ), we say that X is a random variable, and the associated distribution function is defined for x ∈ R by F X ( x ) = P [ X < x ], where P [ X < x ] denotes the measure of the subset { ω ∈ Ω : X ( ω ) < x } .By a random process on ( Ω , F , P ) we mean a family of random variables { X t } t ≥ parametrized by time. To formulate a theory of random processes some additional struc-ture is required. First we need the idea of a complete probability space. A σ -algebra F P issaid to be an augmentation of the σ -algebra F with respect to P if F P contains all subsets B ⊂ Ω for which there exist elements A , C ∈ F satisfying A ⊆ B ⊆ C and P [ C \ A ] = F P = F , we say that ( Ω , F , P ) is complete. Next we need the idea of a filtration on( Ω , F , P ), by which we mean a nondecreasing family F = {F t } t ≥ of sub- σ -algebras of F .We say that a filtration F is right continuous if for all t ≥ F t = F t + where F t + = ∩ u > t F u . If additionally we assume, as we do, that for any A ∈ F such that P [ A ] = A ∈ F , then we say that the filtered probability space ( Ω , F , P , F ) satisfies theusualconditions. A random process { X t } is said to be adapted to F if the random variable X t is F t -measureable for all t ≥
0. We say that { X t } is right continuous if the sample paths { X t ( ω ) } t ≥ are right continuous for almost all ω ∈ Ω . By a standard Brownian motion orWienerprocesson a filtered probability space ( Ω , F , P , F ) we mean a continuous, adaptedprocess { W t } t ≥ such that (a) W = W t − W s is normally distributed withmean 0 and variance t − s for t > s ≥
0, and (c) W t − W s is independent of F s for t > s . Thefiltration F may be strictly larger than that generated by the Brownian motion itself. Theexistence of processes satisfying these conditions is guaranteed by the following (Hida1980, Karatzas & Shreve 1986). Let Ω = C [0 , ∞ ) be the space of continuous functionsfrom R + to R . Each point ω ∈ Ω corresponds to a continuous function { W t ( ω ) } t ≥ , and wewrite F = σ [ { W t } t ≥ ] for the σ -algebra generated by { W t } t ≥ . The σ -algebra generated by acollection C of functions X : Ω → R is defined to be the smallest σ -algebra Ξ on Ω suchthat each function X ∈ C is Ξ -measurable. Then there exists a unique measure P on the( Ω , F ), called Wienermeasure, such that properties (a), (b) and (c) hold, and we take F tobe the filtration {F t } t ≥ generated by { W t } t ≥ , defined by F t = σ [ { W s } ≤ s ≤ t ] for each t ≥ Ω , F , P , F ) satisfies the usual conditions. Equalitiesand inequalities for random variables are understood to hold P -almost-surely. One checksby the use of Ito calculus that if ˆ ρ t takes the form of a pure projection operatorˆ ρ t = | ψ t ih ψ t |h ψ t | ψ t i , (7)then the stochastic Schr ¨odinger equation (1) for the state-vector implies that the puredensity matrix (7) satisfies the stochastic master equation (6). The relevant calculation isshown, for example, in sections 6.1-6.2 of Adler (2004). Since (6) is a nonlinear stochasticdi ff erential equation, it does not immediately follow that (6) should be applicable to gen-eral states rather than merely to pure states. Nevertheless, this is what we propose, and,as we shall see, the theory that follows from Definition 1 has many desirable properties,both physical and mathematical. For some purposes it is useful if we write equation (6)in integral form, incorporating the initial condition explicitly. In that case we haveˆ ρ t = ˆ ρ − i ~ − Z t [ ˆ H , ˆ ρ s ]d s + σ Z t (cid:16) H ˆ ρ s ˆ H − ˆ H ˆ ρ s − ˆ ρ s ˆ H (cid:17) d s + σ Z t (cid:16) ( ˆ H − H s ) ˆ ρ s + ˆ ρ s ( ˆ H − H s ) (cid:17) d W s . (8)Then it follows, by taking the expectation of each side, which eliminates the term involvingthe stochastic integral, that the mean state of the system satisfies h ˆ ρ t i = ˆ ρ − i ~ − Z t [ ˆ H , h ˆ ρ s i ]d s + σ Z t (cid:16) H h ˆ ρ s i ˆ H − ˆ H h ˆ ρ s i − h ˆ ρ s i ˆ H (cid:17) d s . (9)Here h ˆ ρ t i = E [ ˆ ρ t ], where E [ · ] denotes expectation under P . One recognizes (9) as theintegral form of a master equation of the type derived by Lindblad (1976), Gorini et al(1976), and, in a di ff erent context, Banks et al (1984), and we have the following: Proposition 1
If the state of a quantum system satisfies the energy-driven stochastic masterequation, then the mean state of the system satisfies a linear master equation of the form d h ˆ ρ t i d t = − i ~ − [ ˆ H , h ˆ ρ t i ] + σ (cid:16) H h ˆ ρ t i ˆ H − ˆ H h ˆ ρ t i − h ˆ ρ t i ˆ H (cid:17) . (10)In the pure case, it is well known (see, for example, Gisin 1989) that if | ψ t i satisfies (1) thenthe expectation of the corresponding pure density matrix, given by h ˆ ρ t i = E " | ψ t ih ψ t |h ψ t | ψ t i , (11)satisfies the autonomous stochastic di ff erential equation (10). This is not so obvious if oneworks directly with the dynamics of a state vector, but if one takes the stochastic masterequation as the starting point then the linearity of the dynamics of h ˆ ρ t i is immediate.Proposition 1 shows that in the generic situation where the density matrix is of rankgreater than unity and follows the general nonlinear stochastic dynamics given by (6), theassociated mean density matrix h ˆ ρ t i still satisfies (10).We are thus led to postulate that the energy-driven stochastic master equation pre-sented in Definition 1, with a prescribed initial state ˆ ρ , characterizes the stochastic evo-lution of the state of a quantum system as reduction proceeds. In saying that we takethe initial state as prescribed, we avoid for the moment entering into a discussion abouthow that can be achieved. Likewise, we avoid asking how one can determine what theinitial state of the system is. It is meaningful to ask such questions, but we separate theproblem of working out the consequences of the evolution of the state from the problemof working out what the state of the system is in the first place, or how to create a systemin a given state. In Sections II, III and IV below, we work out properties of the energy-driven stochastic master equation. A number of the results obtained are generalizationsof corresponding results known to hold in the case when the state is pure. In Proposition2 we show that the expectation of the variance of the energy goes to zero in the limit as t grows large. In Proposition 3 we show that there exists a random variable H ∞ = lim t →∞ H t taking values in the spectrum of the Hamiltonian such that we have E [ H ∞ ] = tr ˆ ρ ˆ H andVar [ H ∞ ] = tr ˆ ρ ˆ H − (tr ˆ ρ ˆ H ) . The proofs of Propositions 2 and 3 generalize argumentsappearing in Hughston (1996). In Section V we present a derivation of the Born rule forgeneral states, summarized In Proposition 4, extending arguments of Ghiradi et al (1990),Adler & Horwitz (2000), and Adler et al (2001). In the case of a degenerate Hamiltonian,the reduction leads for a given outcome to the associated L ¨uders state. Then in SectionsVI and VII we proceed to construct a general solution of the energy-driven stochasticmaster equation using techniques of nonlinear filtering theory. Here we extend resultsknown for the dynamics of pure states (Brody & Hughston 2002). The solution, whichtakes the form of a completely positive stochastic map, is obtained by the introduction ofa so-called informationprocess { ξ t } t ≥ defined by ξ t = σ tH + B t where the random variable H takes values in the spectrum of the Hamiltonian operator, and { B t } t ≥ is an independentBrownian motion. We show that it is possible to construct the processes { ˆ ρ t } t ≥ and { W t } t ≥ in terms of { ξ t } t ≥ in such a way that { ˆ ρ t } t ≥ satisfies the energy-driven stochastic masterequation and { W t } is a standard Brownian motion on ( Ω , F , P , F ), where F is the filtrationgenerated by { ξ t } . The results are summarized in Propositions 5 and 6. Then we introducethe notion of a potential and in Propositions 7 and 8 we show that the decoherence of thedensity matrix can characterized in a rather natural way by the fact that its o ff -diagonalterms are potentials. Section VIII concludes. II. DYNAMIC PROPERTIES OF THE ENERGY VARIANCE
We proceed to show that many of the important properties of the pure state dynamics (1)carry forward to the general state dynamics (6). First, one can check that the trace of ˆ ρ t ispreserved under (6). Thus, if tr ˆ ρ = ρ t = t > { H t } t ≥ defined by H t = tr ˆ ρ t ˆ H isa martingale. In fact, even in the pure case the result can be obtained rather more directlyby use of (6) than (1), for if we transvect each side of (6) with ˆ H and take the trace we areimmediately led to the following dynamical equation for the energy:d H t = σ V t d W t . (12)Here we have written V t = tr ˆ ρ t ( ˆ H − H t ) for the variance of the energy. Thus we have H t = H + σ Z t V s d W s . (13)Since the expectation and the variance of the energy are bounded random variables, itfollows from (13) that { H t } t ≥ is a martingale. Letting E t [ · ] = E [ · | F t ] denote conditionalexpectation with respect to F t , we have E s [ H t ] = H s for 0 ≤ s ≤ t . The martingale propertyrepresents conservation of energy in a conditional sense. This property is known to besatisfied by the energy expectation process in the case of a pure state, and we see that themartingale property holds more generally in the case of a mixed state governed by theenergy-driven stochastic master equation. A further calculation shows thatd V t = − σ V t d t + σβ t d W t , (14)where { β t } t ≥ denotes the so-called energy skewness process, defined by β t = tr ˆ ρ t ( ˆ H − H t ) . (15)The dynamical equation (14) can be obtained as follows. Write the variance in the form V t = tr ˆ ρ t ˆ H − H t . (16)The dynamics of the term tr ˆ ρ t ˆ H can be worked out by transvecting each side of equation(6) with ˆ H . The dynamics of the second term can be deduced by applying Ito’s lemmato H t and using (12). The two results combined give (14).The stochastic equation satisfied by the variance of the Hamiltonian in the case of ageneral state has the same form that it has in the pure case. In the pure case (14) impliesthat the variance tends to zero asymptotically, and thus that the state evolves to an energyeigenstate. We shall show that the argument carries through to the case of a general initialstate. That is to say, for any initial state the result of the evolution given by (6) is an energyeigenstate. By an energy eigenstate with energy E we mean a state ˆ ρ such that ˆ H ˆ ρ = E ˆ ρ .If the Hamiltonian is nondegenerate, then the energy eigenstates are pure states. In thecase of a degenerate Hamiltonian, the situation is more complicated. If the outcome ofthe collapse is an eigenstate with energy E r , then it can be shown that the state that resultsis the so-called L ¨uders state given by outcome of the L ¨uders (1951) projection postulateassociated with that energy and the given initial state (Adler et al 2001). Definition 2
Let ˆ P r denotes the projection operator onto the Hilbert subspace H r consisting ofstate vectors that are eigenstates of ˆ H with eigenvalue E r . Then for any initial state ˆ ρ the associatedL ¨uders state ˆ L r is defined by ˆ L r = ˆ P r ˆ ρ ˆ P r tr ˆ ρ ˆ P r . (17)If the Hamiltonian is degenerate, and if the initial state is pure, then the final state willbe pure. On the other hand, if the initial state is impure, then the final state need not bepure, and in general will be impure.To show that collapse to an energy eigenstate occurs as a consequence of (14) for ageneral initial state, we establish the following, which is known to hold for pure states: Proposition 2
Let { ρ t } satisfy the energy-driven stochastic master equation. Then the expectationof the variance of the Hamiltonian vanishes asymptotically: lim t →∞ E [ V t ] = . (18) Proof
We integrate (14) to obtain V t = V − σ Z t V s d s + σ Z t β s d W s . (19)The integrals are defined since the variance and the skewness are bounded. Since thedrift in (14) is negative, we see that E s [ V t ] ≤ V s for 0 ≤ s ≤ t and hence that { V t } t ≥ is asupermartingale. Taking the unconditional expectation on each side of (19), we have E [ V t ] = V − σ E "Z t V s d s , (20)which shows that E [ V t ] decreases as t increases, and hence that lim t →∞ E [ V t ] exists. Wesay that an R -valued random process { X t } t ≥ on a probability space ( Ω , F , P ) is measurable if for all A ∈ B R , where B R is the Borel σ -algebra on R , it holds that { ( ω, t ) : X t ( ω ) ∈ A } ∈ F × B R + , (21)where B R + denotes the Borel σ -algebra on the positive “time axis” R + = [0 , ∞ ). A su ffi cientcondition for a process to be measurable is that it should be right continuous. Then onehas the following (Liptser & Shiryaev 1975):Fubini’s theorem. If a process { X t } t ≥ is measurable and R S E [ | X t | ] d t < ∞ for some S ∈ B R + , then R S | X t | d t < ∞ almost surely and E "Z S X t d t = Z S E [ X t ] d t . (22)0As a consequence of Fubini’s theorem, we can interchange the order of the expectationand the integration on the right side of (20) to obtain E [ V t ] = V − σ Z t E h V s i d s , (23)from which it follows that d E [ V t ]d t = − σ E h V t i . (24)Thus we can write d E [ V t ]d t = − σ E [ V t ] (1 + α t ) , (25)where α t = E [ V t ] E [( V t − E [ V t ]) ] , (26)and we note that α t is nonnegative. If we set γ t = R t α s d s , we can integrate (25) to obtain E [ V t ] = V + V σ ( t + γ t ) . (27)Since γ t is nonnegative, we have E [ V t ] ≤ V + V σ t , (28)and this gives (18). (cid:3) III. ASYMPTOTIC PROPERTIES OF THE VARIANCE
As a consequence of (18) one deduces that the energy variance vanishes as t goes toinfinity. More precisely, it holds that V ∞ = V ∞ = lim t →∞ V t exists, in an appropriate sense, and then we need toshow that the order of the limit and the expectation in (18) can be interchanged. If bothof these conditions hold, then we conclude from (18) that V ∞ =
0. Now, when we askwhether a limit exists, we are not asking whether the result is finite or not. Limits, if theyexist, are allowed to be infinite. The question is one of convergence. Moreover, even if arandom process converges, that does not imply that the resulting function on Ω to whichthe process converges is a random variable (that is to say, a measurable function). Sothe question is whether there exists a random variable V ∞ to which the variance processconverges for large t with probability one. If the answer is yes, then one can ask whetherthe interchange of limit and expectation is valid, and if so then we are able to concludethat the result of the collapse process is a state of zero energy variance and hence anenergy eigenstate.1To show that (18) implies V ∞ = ffi cient to have athand the version that follows below (Protter 2003). First we introduce some additionalterminology. We fix a probability space and let p ∈ R satisfy p ≥
1. A random process { X t } t ≥ is said to be bounded in L p ifsup ≤ t < ∞ E [ | X t | p ] < ∞ . (29)As usual, by the supremum we mean the least upper bound. A random process { X t } t ≥ is said to be right-continuous if it holds almost surely that lim ǫ → X t + ǫ = X t for all t ≥ If a right-continuous supermartingale { X t } t ≥ is bounded in L then lim t →∞ X t exists almost surely and defines a random variable X ∞ satisfying E [ | X ∞ | ] < ∞ . Note that in asserting that lim t →∞ X t exists almost surely we mean that lim sup t →∞ X t ( ω ) = lim inf t →∞ X t ( ω ) for all ω ∈ Ω ′ for some set Ω ′ ∈ F such that P [ Ω ′ ] =
1, and that thereexists a random variable X ∞ such that X ∞ ( ω ) = lim t →∞ X t ( ω ) for all ω ∈ Ω apart from aset of measure zero.As we shall see, the martingale convergence theorem is just the tool one needs in orderto show that the energy variance process converges to zero. In particular, since the energyvariance is bounded for all t ≥
0, we have sup ≤ t < ∞ E [ | V t | p ] < ∞ for all p ≥
1. It followsby the martingale convergence theorem that V ∞ = lim t →∞ V t exists almost surely and that E [ V ∞ ] < ∞ . To proceed further we make use of the following (see, e.g., Williams 1991):Fatou’s lemma. Let { Y k } k ∈ N be a countable sequence of nonnegative integrable random variables.Then E [lim inf k →∞ Y k ] ≤ lim inf k →∞ E [ Y k ] . If { t k } k ∈ N is a countable sequence of times such that lim k →∞ t k = ∞ , then for any process { X t } t ≥ such that X ∞ = lim t →∞ X t exists it holds that lim k →∞ X t k = X ∞ . Thus, in our case wehave lim k →∞ E [ V t k ] = k →∞ V t k = V ∞ . We know that if lim k →∞ Y k exists then it isequal to lim inf k →∞ Y k . Then by Fatou’s lemma we have E [lim k →∞ V t k ] ≤ lim k →∞ E [ V t k ]. Itfollows that E [ V ∞ ] = V ∞ = IV. TERMINAL VALUE OF THE ENERGY
Let Spec[ ˆH] denote the spectrum of the Hamiltonian. Then we have the following result,which shows that H t and V t are given at each time t ≥ Proposition 3
There exists a random variable H ∞ on ( Ω , F , P ) taking values in Spec[ ˆH] suchthat H t = E t [ H ∞ ] and V t = E t [( H ∞ − E t [ H ∞ ]) ] . Proof
Since { H t } t ≥ is bounded by the highest and lowest eigenvalues of ˆ H , we havesup ≤ t < ∞ E [ | H t | ] < ∞ and hence by the martingale convergence theorem the randomvariable H ∞ = lim t →∞ H t exists and E [ H ∞ ] < ∞ . A process { X t } t ≥ on a probability space( Ω , F , P ) is said to be uniformly integrable if, given any ǫ > δ such that E [ | X t | ( | X t | > δ )] < ǫ (30)2for all t ≥
0, where ( · ) is the indicator function. Let { M t } t ≥ be a right-continuousmartingale on a probability space ( Ω , F , P ) with filtration {F t } t ≥ . Then it is known thatthe following conditions are equivalent: (i) there exists a random variable M ∞ such thatlim t →∞ E [ | M t − M ∞ | ] =
0; (ii) there exists a random variable M ∞ satisfying E [ M ∞ ] < ∞ such that M t = E t [ M ∞ ] for all t ≥
0; (iii) { M t } t ≥ is uniformly integrable. Clearly, anybounded martingale is uniformly integrable. Since { H t } t ≥ is bounded, we have H t = E t [ H ∞ ] , (31)as claimed. We turn now to the variance, in connection with which we use the following.Monotone convergence theorem. For any increasing sequence { Y k } k ∈ N of nonnegative inte-grable random variables such that lim k →∞ Y k = Y ∞ , where Y ∞ is not necessarily integrable, itholds that lim k →∞ E [ Y k ] = E [ Y ∞ ].By use of (18) and (20), together with the monotone convergence theorem, we deduce that E "Z ∞ V s d s < ∞ . (32)Hence, it follows from (19) that V + σ Z ∞ β s d W s = σ Z ∞ V s d s . (33)If we take a conditional expectation, we obtain V + σ Z t β s d W s = σ E t Z ∞ V s d s . (34)Combining this relation with (19) we deduce that V t = σ E t Z ∞ t V s d s . (35)Next we observe that as a consequence of (13) we have H ∞ − H t = σ Z ∞ t V s d W s . (36)Taking the square of each side of this equation, forming the conditional expectation, andusing the Ito isometry, we obtain E t ( H ∞ − H t ) = σ E t Z ∞ t V s d s , (37)and therefore V t = E t ( H ∞ − E t H ∞ ) , (38)as claimed. (cid:3) H of theobservable ˆ H with respect to the initial state ˆ ρ is equal to the expectation of the terminalvalue of the energy on the completion of the reduction process. This may seem like atautology, but it is not, since the statistical interpretation of the expectation value of anobservable in quantum mechanics is an assumption, not a conclusion, of the theory.Likewise, we see that the conventional squared uncertainty V is the variance of theterminal value of the energy on the completion of the reduction process. Again, the sta-tistical interpretation of the squared uncertainty is an assumption in quantum mechanics,not a conclusion of the theory. But under the dynamics of the stochastic master equationthese properties are deduced rather than assumed.The methods used in the proof of Proposition 3 can be used to give an alternativederivation of the fact that lim t →∞ E [ V t ] = V ∞ = V t = tr ˆ ρ t ˆ H − H t , (39)where H t = tr ˆ ρ t ˆ H . Writing U t = tr ˆ ρ t ˆ H , we see that { U t } t ≥ is a bounded martingale. Itfollows by the martingale convergence theorem that { U t } → U ∞ , and as a consequencewe have { V t } → U ∞ − H ∞ , from which it follows that V ∞ = U ∞ − H + σ Z ∞ V s d W s ! . (40)Since E [ U ∞ ] = U , it follows by use of the Ito isometry that E [ V ∞ ] = U − H − σ E Z ∞ V s d s . (41)On the other hand, on account of (33) we have V = σ E Z ∞ V s d s . (42)Since U − H = V , it follows that E [ V ∞ ] =
0, and therefore V ∞ = V. DERIVATION OF THE BORN RULE
The foregoing arguments show that the dynamic approach to reduction extends to thesituation where the initial state of the system need not be pure. The Born rule is anotherexample of an assumption of quantum mechanics that can be derived from the stochasticmaster equation. As before, for the given Hamiltonian let ˆ P r denote the projection operatoron to the Hilbert subspace of energy E r . Let the number of distinct energy levels be D . Proposition 4
Under the dynamics of the energy-driven stochastic master equation, with initialstate ρ , the probability that the outcome will be a state with energy E r is given by P [ H ∞ = E r ] = tr ˆ ρ ˆ P r . (43)4 Proof
It is straightforward to check that the process { π rt } t ≥∞ defined for each value of r = , , . . . , D by π rt = tr ˆ ρ t ˆ P r is a martingale. Thus we have π rt = E t [ π r ∞ ] and hencetr ˆ ρ ˆ P r = E [tr ˆ ρ ∞ ˆ P r ] . (44)On the other hand, because the state reduces asymptotically to a random energy eigen-state, we know that tr ˆ ρ ∞ ˆ P r = ( H ∞ = E r ) , (45)and since E [ ( H ∞ = E r )] = P [ H ∞ = E r ] , (46)we are led to the Born rule (43). (cid:3) It may seem tautological to assert that the probability of the outcome E r is given by thetrace of the product of the initial density matrix and the projection operator ˆ P r , but it is not.In quantum mechanics, the Born rule is an assumption, part of the statistical interpretationof the theory. Physicists are on the whole quite comfortable with this assumption, butthat does not change the fact that there is no generally accepted “derivation” of the Bornrule as a probability law arising from within quantum theory itself. Indeed, it is oneof the features of the energy-driven stochastic reduction model that a mathematicallysatisfactory explanation for this otherwise ba ffl ing aspect of quantum theory emerges. VI. SOLUTION TO STOCHASTIC MASTER EQUATION
A solution to the energy-driven stochastic master equation (6) can be written down asfollows. We start afresh, and consider a finite-dimensional quantum system for whichthe Hamiltonian (possibly degenerate) is ˆ H and the initial state (which we regard as pre-scribed) is ˆ ρ . Let a probability space ( Ω , F , P ) be given, upon which we introduce astandard Brownian motion { B t } t ≥ and an independent random variable H taking valuesin Spec[ ˆH] with the distribution P [ H = E r ] = tr ˆ ρ ˆ P r , where ˆ P r denotes the projection op-erator on to the Hilbert subspace of energy E r . Then we introduce a so-called informationprocess on ( Ω , F , P ) denoted { ξ t } t ≥ , defined by ξ t = σ tH + B t . (47)Thus { ξ t } takes the form of a Brownian motion with a random drift, the rate of drift beingdetermined by the random variable H and the parameter σ . Processes of this type arise inthe theory of stochastic filtering (Wonham 1965, Liptser & Shiryaev 2000). In the languageof filtering theory one refers to H as the signal, B t as the noise, and ξ t as the observation.Of course, the notion of observation as it is understood in the context of filtering theoryhas no immediate connection with the notion of observation as it is usually understoodin quantum mechanics. Nevertheless, the ideas that have been developed in filteringtheory are rather suggestive, so it is worth keeping the associated terminology in mindas we proceed. Loosely speaking, one can think of “that which has been observed” in thecontext of filtering theory as equivalent to “that which has irreversibly manifested itselfin the world” in the context of a physical theory. Now, let {F t } t ≥ denote the filtrationgenerated by { ξ t } t ≥ . We have the following.5 Proposition 5
Let the operator-valued process { ˆ K t } t ≥ be defined by ˆ K t = exp h − i ~ − ˆ Ht + σ ˆ H ξ t − σ ˆ H t i . (48) Then the process { ˆ ρ t } t ≥ defined by ˆ ρ t = ˆ K t ˆ ρ ˆ K ∗ t tr [ ˆ K t ˆ ρ ˆ K ∗ t ] (49) has trace unity, is nonnegative definite, and satisfies a stochastic master equation of the form d ˆ ρ t = − i ~ − [ ˆ H , ˆ ρ t ]d t + σ (cid:16) H ˆ ρ t ˆ H − ˆ H ˆ ρ t − ˆ ρ t ˆ H (cid:17) d t + σ (cid:16) ( ˆ H − H t ) ˆ ρ t + ˆ ρ t ( ˆ H − H t ) (cid:17) d W t , (50) where H t = tr ˆ ρ t ˆ H and the process { W t } t ≥ defined byW t = ξ t − σ Z t H s d s (51) is an {F t } -Brownian motion. Remark
Here we look at the stochastic master equation from a new point of view. Insteadof regarding { W t } t ≥ as an “input” to the model, we regards { ξ t } t ≥ as the input. Then both { ˆ ρ t } t ≥ and { W t } t ≥ are defined in terms of { ξ t } t ≥ , and together they satisfy equation (50). Proof
Let us set Λ t = tr [ ˆ K t ˆ ρ ˆ K ∗ t ] . From the cyclic property of the trace we obtain Λ t = tr ˆ ρ exp (cid:16) σ ˆ H ξ t − σ ˆ H t (cid:17) . (52)By Ito’s lemma, along with (d ξ t ) = d t , which follows from (47), we haved Λ t = σ tr ˆ ρ ˆ H exp (cid:16) σ ˆ H ξ t − σ ˆ H t (cid:17) d ξ t , (53)and therefore d Λ t = σ H t Λ t d ξ t , since H t = tr ˆ ρ ˆ H exp (cid:16) σ ˆ H ξ t − σ ˆ H t (cid:17) tr ˆ ρ exp (cid:16) σ ˆ H ξ t − σ ˆ H t (cid:17) . (54)If we write (49) in the form ˆ ρ t = Λ t ˆ K t ˆ ρ ˆ K ∗ t , (55)a straightforward calculation using the Ito quotient rule then gives (50). To establishthat the process { W t } t ≥ defined by (51) is an {F t } -Brownian motion under P we use theso-called L´evy criterion. We need to show (i) that (d W t ) = d t , and (ii) that { W t } t ≥ is an {F t } -martingale under P .6The first property follows immediately as a consequence of the Ito multiplication rulesapplied to (47) and (51). To check that the second property holds we need to verify for s ≤ t that E [ W t | F s ] = W s . Let G t denote the σ -algebra generated by H and { ξ u } ≤ u ≤ t . Then F t , which is generated by { ξ u } ≤ u ≤ t alone, is a sub- σ -algebra of G t , and for all t ≥ E [ E [ · | G t ] | F t ] = E [ · | F t ] . (56)Now, by (51) it holds that E [ W t | F s ] = E [ ξ t | F s ] − σ Z t E [ H u | F s ] d u . (57)As for the first term on the right side of (57), it follows from (47) that E [ ξ t | F s ] = E [ B t | F s ] + σ t E [ H | F s ] = E [ B t | F s ] + σ t H s = E [ E [ B t | G s ] | F s ] + σ t H s = E [ B s | F s ] + σ t H s = ξ s + σ ( t − s ) H s , (58)where we use the tower property to go from the second to the third line. In the secondterm on the right side of (57), we use the fact that { H t } is a martingale to deduce that Z t E [ H u | F s ] d u = Z s H u d u + Z ts H s d u = Z s H u d u + ( t − s ) H s . (59)Thus putting together the results for the two terms on the right side of (57) we have E [ W t | F s ] = ξ s − σ Z s H u d u = W s , (60)which is what we wished to show. (cid:3) VII. INFORMATION FILTRATION
The collapse property in the case of a general state admits a remarkable interpretation inthe language of stochastic filtering. As before, let us write ˆ P r ( r = , . . . , D ) for the projec-tion operator onto the Hilbert subspace H r consisting of state vectors with eigenvalue E r .For any element | a i ∈ H r we have ˆ H | a i = E r | a i , and for the Hamiltonian we can writeˆ H = D X r = E r ˆ P r . (61)Therefore, if we set ˆ R nm t = ˆ P n ˆ ρ t ˆ P m then for the diagonal terms we haveˆ R nn t = ˆ P n ˆ ρ ˆ P n exp h σ E n ξ t − σ E n t iP Dr = p r exp h σ E r ξ t − σ E r t i , (62)where p r = tr ˆ ρ ˆ P r .7 Proposition 6
For each n the process { ˆ R nn t } t ≥ is a uniformly integrable martingale, given by ˆ R nn t = E [ ( H = E n ) | F t ] ˆ P n ˆ ρ ˆ P n tr ˆ ρ ˆ P n , (63) where E [ ( H = E n ) | F t ] = p n exp h σ E n ξ t − σ E n t iP Dr = p r exp h σ E r ξ t − σ E r t i . (64)Thus, { ξ t } carries partial information about the value of the random variable H , which isrevealed as time progresses, leading asymptotically to the outcomeˆ R nn ∞ = ( H = E n ) ˆ P n ˆ ρ ˆ P n tr ˆ ρ ˆ P n , (65)which is the L ¨uders state that results under the projection postulate in the standard theoryas a consequence of an energy measurement, with the outcome E n , given that the initialstate is ˆ ρ . In the present context there is no measurement as such. Nevertheless, the finalstate of the reduction process is a L ¨uders state. For each value of n the correspondingdiagonal element of the density matrix at time t is given by the conditional expectation ofthe indicator function ( H = E n ) given the value of ξ t .Next we present a probabilistic formulation of the fact that the state decoheres asreduction proceeds. For any operator ˆ O let us write | ˆ O | = (tr ˆ O ˆ O † ) / . Definition 3
By a potential on a filtered probability space ( Ω , F , P , F ) , we mean a strictly positiveright-continuous supermartingale { π t } t ≥ with the property that lim t →∞ E [ π t ] = . Then we have the following:
Proposition 7
For each n , m such that n , m the process { | ˆ R nm t | } t ≥ is a potential. Proof
Let n , m be such that n , m . The o ff -diagonal matrix elements of the state then takethe form ˆ R nm t = ˆ P n ˆ ρ ˆ P m exp h − i ~ − ( E n − E m ) t + σ ( E n + E m ) ξ t − σ ( E n + E m ) t iP Dr = p r exp h σ E r ξ t − σ E r t i . (66)Thus we have ˆ R nm t = ˆ P n ˆ ρ ˆ P m exp h − i ~ − ( E n − E m ) t i Φ nm t , (67)where Φ nm t = exp h σ ( E n + E m ) ξ t − σ ( E n + E m ) t iP Dr = p r exp h σ E r ξ t − σ E r t i , (68)8and a calculation shows that Φ nm t = Π nm t exp h − σ ( E n − E m ) t i , (69)where Π nm t = exp h σ ( E n + E m ) ξ t − σ ( E n + E m ) t iP Dr = p r exp h σ E r ξ t − σ E r t i . (70)We claim that for any λ ∈ R the process { µ t } t ≥ defined by µ t = exp h λξ t − λ t iP Dr = p r exp h σ E r ξ t − σ E r t i (71)is a martingale. To see that this is so, note that by use of Ito’s lemma, together with therelation d ξ t = σ H t d t + d W t , we have d µ t = ( λ − σ H t ) µ t d W t , and thus µ t = exp "Z t ( λ − σ H s ) dW s − Z t0 ( λ − σ H s ) ds . (72)Since { H t } t ≥ is bounded, we deduce that { µ t } t ≥ is a martingale. Therefore, { Π nm t } t ≥ is amartingale and { Φ nm t } t ≥ is a supermartingale. By (69) one sees that E [ Φ nm t ] = exp h − σ ( E n − E m ) t i , (73)and hence lim t →∞ E [ Φ nm t ] = , (74)so { Φ nm t } t ≥ is a potential. Finally, we observe that (cid:12)(cid:12)(cid:12) ˆ R nm t (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ˆ P n ˆ ρ ˆ P m (cid:12)(cid:12)(cid:12) Φ nm t , (75)from which we obtain lim t →∞ E h (cid:12)(cid:12)(cid:12) ˆ R nm t (cid:12)(cid:12)(cid:12) i = n , m , which is what we wished to prove. (cid:3) Thus, the potential property of the o ff -diagonal terms of the density matrix in theenergy representation captures the essence of what is meant by decoherence. We see thatthe decay of the o ff -diagonal terms of the density matrix is exponential in time, and thatthe decay rate for any particular such term is proportional to the square of the di ff erenceof the associated energy levels. In fact, we can take the representation of { Φ nm t } t ≥ as apotential a step further. A calculation making use of the Ito quotient rule shows thatd Φ nm t = − σ ( E n − E m ) Φ nm t d t + σ ( E n + E m − H t ) Φ nm t d W t . (77)9As a consequence, for each n , m such that n , m we have Φ nm t = − σ ( E n − E m ) Z t Φ nm s d s + σ Z t ( E n + E m − H s ) Φ nm s d W s . (78)Taking the limit as t goes to infinity and using the fact that Φ nm ∞ = + σ Z ∞ ( E n + E m − H s ) Φ nm s d W s = σ ( E n − E m ) Z ∞ Φ nm s d s . (79)Then by taking a conditional expectation we obtain1 + σ Z t ( E n + E m − H s ) Φ nm s d W s = σ ( E n − E m ) E t "Z ∞ Φ nm s d s , (80)from which it follows by use of (78) that Φ nm t = σ ( E n − E m ) E t "Z ∞ t Φ nm s d s . (81)This identity gives us a representation of { Φ nm t } t ≥ as a so-called type- D potential. Moreexplicitly, if we define an increasing process { A nm t } t ≥ by setting A nm t = σ ( E n − E m ) Z t Φ nm s d s , (82)then we have Φ nm t = E t [ A nm ∞ ] − A nm t , (83)which is the canonical form for a potential of type D (Meyer 1966). Thus we arrive at thefollowing: Proposition 8
The state process under energy-driven stochastic reduction is of the form ˆ ρ t = D X n = E t [ ( H = E n )] ˆ P n ˆ ρ ˆ P n tr ˆ ρ ˆ P n + D X n , m = n , m ˆ P n ˆ ρ ˆ P m exp h − i ~ − ( E n − E m ) t i Φ nm t . (84)The conditional expectation in the first term is given by (64) and the potential in the secondterm is given by (69). At time zero, the two terms combine to give the initial density matrixˆ ρ . As the collapse proceeds, the first term converges to the L ¨uders state associated withthe selected energy eigenvalue E n , and the second term tails o ff to zero. It should beemphasized that if the initial state is impure, and if the Hamiltonian is degenerate, thenthe final state will in general also be impure.0 VIII. CONCLUSION
In our development of the dynamic reduction program we have taken the view that thestate of a single system can be described by a density matrix that may or may not be pure.The initial state ˆ ρ is prescribed, and its value at time t is given by the random densitymatrix ˆ ρ t . The model is understood as describing an “objective” reduction process, sothere are no observers in the theory in the usual sense. All the same, one can ask what isknown at time t , in the sense of what has manifested itself in the world (or, let’s say, in theexperimenter’s laboratory) at that time. For this purpose it seems reasonable to adopt theview that the standard interpretation of the filtration {F t } gives an adequate answer. Thismeans that for any overall outcome of chance ω ∈ Ω , the value of any F t -measureablerandom variable X t will be “known” or will have manifested itself at (or before) time t . Inparticular, the value of ˆ ρ t itself will be known at time t , as will the value of the informationprocess ξ t . Now, it is not quite meaningful to ask how ξ t can be measured in a theory inwhich there are no measurements. Nevertheless, we are forced to the conclusion that thetheory only makes sense if ξ t is known (in the sense of having manifested itself) at time t .In fact, Diosi (2015) has arrived at what we believe to be in essence a similar conclusion,that stochastic reduction models only really make sense if the { ξ t } process can in someappropriate sense be monitored in real time. This is not the same thing as saying thatthe quantum system is being actively monitored (in the sense of Diosi 1988, Barchielli& Belavkin 1991, Barchielli 1993, Wiseman 1996, Wiseman & Diosi 2001, Barchielli &Gregoratti 2009), since the monitoring that takes place in such considerations is within aframework of standard quantum dynamics, and some form of ad hoc collapse is requiredas an additional assumption to make the infinitesimal collapses occur in response to themonitoring. But it may be that in a laboratory situation it is possible to monitor { ξ t } , orequivalently { H t } , in the passive sense implicit in the structure of the information filtrationof the models that we have here described. The class of information-based models thatcan be developed by use of the filtering techniques discussed in Sections VI and VII canbe extended to a wider set of models, in which the underlying noise is not Brownianmotion but rather a general L´evy process. Such processes, like Brownian motion, havethe property of being stationary with independent increments, but are not generallyGaussian and can be discontinuous. Providing that a condition is satisfied ensuring theexistence of exponential moments, L´evy trajectories are suitable for characterizing a wideand extraordinarily diverse family of noise processes (Brody, Hughston & Yang 2013).The development of relativistic analogues of the models considered here remains an openproblem, though it seems reasonable to conjecture that in the relativistic case the reductionprocess should lead to states for which the total mass and spin take definite values. Acknowledgments
This work was carried out in part at the Perimeter Institute for Theoretical Physics, whichis supported by the Government of Canada through Innovation, Science and EconomicDevelopment Canada and by the Province of Ontario through the Ministry of Research,Innovation and Science, and at the Aspen Center for Physics, which is supported byNational Science Foundation grant PHY-1066293. DCB acknowledges support from theRussian Science Foundation (project 16-11-10218). We are grateful to S. L. Adler, T. Benoist,I. Egusquiza, S. Gao and B. K. Meister for helpful comments and discussions.1
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