DDynamical state reduction in an EPR experiment
Daniel J. Bedingham
Blackett Laboratory, Imperial College, London SW7 2BZ, UK. (Dated: November 2, 2018)A model is developed to describe state reduction in an EPR experiment as a contin-uous, relativistically-invariant, dynamical process. The system under considerationconsists of two entangled isospin particles each of which undergo isospin measure-ments at spacelike separated locations. The equations of motion take the form ofstochastic differential equations. These equations are solved explicitly in terms ofrandom variables with a priori known probability distribution in the physical prob-ability measure. In the course of solving these equations a correspondence is madebetween the state reduction process and the problem of classical nonlinear filtering.It is shown that the solution is covariant, violates Bell inequalities, and does notpermit superluminal signaling. It is demonstrated that the model is not governedby the Free Will Theorem and it is argued that the claims of Conway and Kochen,that there can be no relativistic theory providing a mechanism for state reduction,are false.
PACS numbers: 03.65.Ta, 03.65.Ud, 02.50.Ey, 02.50.Cw
I. INTRODUCTION
The motivation for attempting to formulate a dynamical description of state reduction[1, 2, 3, 4, 5, 6] stems from the inherent problems of quantum measurement. In standardquantum theory the state reduction postulate is a necessary supplement to the Schr¨odingerdynamics in order that we can realise definite measurement outcomes from the potentialityof the initial state vector. The problem with this picture is that the pragmatic applicationof these two different laws of evolution is left to the judgment of the physicist rather thanbeing fixed by exact mathematical formulation. Our experience in the use of quantumtheory tells us that the state reduction postulate should not be applied to a microscopicsystem consisting of a few elementary particles until it interacts with a macroscopic objectsuch as a measuring device. This works perfectly well in practice for current experimentaltechnologies, but as we begin to explore systems on intermediate scales it is not clear whetherstate reduction should be assumed or not. A solution of the problem of measurement thusrequires that we somehow set a fundamental scale to demarcate micro and macro effectswithin the dynamical framework.The formulation of an empirical model, objectively describing the dynamics of the statereduction process is a direct approach to achieving this aim. The basic requirements wehave for such a model can be characterised as follows [7, 8]: • Measurements involving macroscopic instruments should have definite outcomes. • The statistical connections between measurement outcomes and the state vector priorto measurement should be preserved. • The model should be consistent with known experimental results. a r X i v : . [ qu a n t - ph ] J u l The task of meeting these objectives in a relativistic context has met with technical diffi-culties related to renormalization [9, 10, 11, 12, 13, 14, 15, 16]. These issues derive fromthe quantum field theoretic nature of relativistic systems. In this paper we will attempt tosidestep this problem by considering a simplified quantum system with a finite-dimensionalHilbert space free from the problem of divergences. Our aim is to elucidate the dynamicalprocess of state reduction in a relativistic context.We will consider a model describing the famous experiment devised by Einstein, Podolski,and Rosen (EPR) [17]. The experiment involves two elementary particles in an entangledstate and separated by a spacelike interval. The original purpose of EPR was to argue thatquantum mechanics is fundamentally incomplete as a theory. In order to do this they madea locality assumption stating that the two particles are not able to instantaneously influenceeach other at a distance. Theoretical and experimental advances [18, 19] have since demon-strated the remarkable conclusion that the assumption of locality is incorrect. Entangledquantum systems can indeed transmit instantaneous influence at a distance when a measure-ment is performed. Although this fact negates the EPR argument, instead it poses questionsfor our understanding of quantum measurement. In particular, the notion of instantaneousinfluence due to state reduction during measurement seems to sit uncomfortably with thetheory of relativity.A formal relativistically-covariant description of the state reduction associated with mea-surement has been given by Aharanov and Albert [20]. They show that for a consistentdescription of the measurement process, the state evolution cannot take the form of a func-tion on spacetime. The proposed solution is that state evolution should be described by afunctional on the set of spacelike hypersurfaces as conceived by Tomonaga and Schwinger.This sets the scene for understanding how to formulate a fully dynamical and relativisticdescription of the state reduction process.Relativistic dynamical reduction models have been critically investigated from the per-spective of the analysis of Aharanov and Albert by Ghirardi [21]. There, the conceptualfeatures of these models are discussed and shown to lead to a coherent picture. It is theintention of this work to extend the analysis of Ghirardi by constructing an explicit model ofcontinuous state evolution. Our model, which is described in detail in section II, is designedto highlight the peculiar nonlocal features. In sections III and IV we derive closed-formsolutions to the stochastic equations of motion. The value of this is that it enables us toexamine the nonlocal character of the stochastic noise processes. In section V we apply themethod of Brody and Hughston [22, 23] to demonstrate that the equations describing thedynamical state reduction can be viewed as a description of a classical filtering problem. Insection VI we generalise our model to consider an experiment where the experimenter canfreely choose which measurement to perform on the individual particle from an incompat-ible set of possible measurements. This leads us to a discussion of the so-called Free WillTheorem [24, 25, 26, 27, 28] of Conway and Kochen in section VII. We use our findings toargue that the axiomatic assumptions of the Free Will Theorem are too restrictive and thatthe conclusions of the theorem cannot be applied to dynamical models of state reduction.
II. THE MODEL
We consider two particles denoted and , each described by an internal isospin- degree of freedom. The choice of an isospin system avoids complication encountered whendealing with conventional spin in a covariant formulation. The initial isospin state of the R R σσ if FIG. 1:
The diagram represents an experiment to measure the states of two entangled particles. Thedashed lines are the (classical) particle trajectories where particle moves to the left and particle moves to the right. The vertical represents a timelike direction whilst the horizontal represents aspacelike direction. We suppose that within the spacetime region R , a measurement is performed onparticle . Similarly within the spacetime region R (spacelike separated from R ), a measurementis performed on particle . The initial state is defined on the spacelike hypersurface σ i . The stateadvances as described by the Tomonaga picture through a sequence of spacelike surfaces defining afoliation of spacetime. two particles is defined in spacetime on an initial spacelike hypersurface σ i as the isospinsinglet state | ψ ( σ i ) (cid:105) = √ (cid:8) | + ; − (cid:105) − | − ; + (cid:105) (cid:9) . (1)The isospin states for each particle are represented with respect to a fixed axis in isospinspace.The particle trajectories in spacetime are assumed to behave classically. The two particlesmove in separate directions away from some specific location where they have been prepared.Each particle path eventually intersects with the path of an isospin measuring device. Thisleads to a localised interaction which we assume takes place in some finite region of spacetime.We assume that the classical trajectories of the particles and measuring devices, and thefinite regions of interaction are determined. Further we assume that the two measurementregions are completely spacelike separated in the sense that every point in each region isspacelike separated from every point in the other region. We denote the two measurementregions by R and R (see figure 1).In order to describe the state evolution we use the Tomonaga picture [29, 30]. Standardunitary dynamics are described in this picture by the Tomonaga equation, δ | ψ ( σ ) (cid:105) δσ ( x ) = − iH int ( x ) | ψ ( σ ) (cid:105) , (2)where H int is the interaction Hamiltonian. Given two spacelike hypersurfaces σ and σ (cid:48) differing only by some small spacetime volume ∆ ω about some spacetime point x , the σ R a x x d ωξ d a FIG. 2:
The diagram represents a sequence of spacelike hypersurfaces advancing through the spacetimeregion R a . The gray shading within R a corresponds to the spacetime volume ω aσ . The detail showsa small spacetime region within R a where the surface σ advances through a spacetime cell at point x . Associated with the cell at point x is the incremental spacetime volume d ω and the incrementalBrownian variable d ξ ax . functional derivative is defined by δ | ψ ( σ ) (cid:105) δσ ( x ) = lim σ (cid:48) → σ | ψ ( σ (cid:48) ) (cid:105) − | ψ ( σ ) (cid:105) ∆ ω . (3)The operator H int must be a scalar in order that equation (2) has Lorentz invariant form.We must also have [ H int ( x ) , H int ( x (cid:48) )] = 0 for spacelike separated x and x (cid:48) reflecting the factthat there is no temporal ordering between spacelike separated points.In differential form equation (2) can be writtend x | ψ ( σ ) (cid:105) = − iH int ( x )d ω | ψ ( σ ) (cid:105) (4)where d x | ψ ( σ ) (cid:105) represents the infinitesimally small change in the state as the hypersurface σ is deformed in a timelike direction at point x .We specify a probability space (Ω , F , Q ) along with a filtration F ξσ of F generated bya two-dimensional Q -Brownian motion { ξ σ , ξ σ } . For each interaction region R a ( a = , )the spacelike hypersurfaces { σ } characterise the time evolution for each component of theBrownian motion. Given a foliation of spacetime, we define a “time difference” betweenany two surfaces as the spacetime volume enclosed by the surfaces within the region R a .Consider the set ( σ i , σ ) of all spacetime points between the two spacelike surfaces σ i and σ ,and consider the intersection of this set with the interaction region ( σ i , σ ) ∩ R a . We denotethe spacetime volume of ( σ i , σ ) ∩ R a by ω aσ (see the gray shaded region in figure 2). Thetwo volumes ω σ and ω σ correspond to two different time parameters for the two componentBrownian motions. This definition ensures that time increases monotonically as the futuresurface σ advances. The parameterization is covariant and has the convenience of only beingrelevant during the predefined measurement events. We define an infinitesimal increment ofthe Brownian motion d ξ ax (relating to two spacelike hypersurfaces which differ only by aninfinitesimal spacetime volume d ω at point x ) by the following:d ξ ax = 0 , for x / ∈ R a ; E Q [d ξ ax |F ξσ ] = 0 , for x to the future of σ ;d ξ ax d ξ by = δ ab δ xy d ω, for x ∈ R a , y ∈ R b , (5)where E Q [ ·|F ξσ ] denotes conditional expectation in Q . We attribute d ξ ax to the spacetimepoint x independent of any spacelike surface on which x may lie. The two-dimensionalBrownian motion is given by the sum of all infinitesimal Brownian increments belonging tothe set of points ( σ i , σ ) ∩ R a , ξ aσ = (cid:90) σσ i d ξ ax , (6)so that an increment of the process can be written ξ aσ (cid:48) − ξ aσ = (cid:90) σ (cid:48) σ d ξ ax , (7)where σ (cid:48) is to the future of σ . These increments are independent and have mean zero andvariance ω aσ (cid:48) − ω aσ as can easily be demonstrated by comparison with the conventional timeparameterization of Brownian motion.The state reduction process which occurs as the isospin state is measured can now bedescribed by extension of the Tomonaga equation (4) to include a stochastic term. We defineour evolution by d x | ψ ( σ ) (cid:105) = (cid:8) λS d ξ x − λ d ω (cid:9) | ψ ( σ ) (cid:105) for x ∈ R , d x | ψ ( σ ) (cid:105) = (cid:8) λS d ξ x − λ d ω (cid:9) | ψ ( σ ) (cid:105) for x ∈ R , d x | ψ ( σ ) (cid:105) = 0 otherwise . (8)The operators S a are isospin operators for each particle with the properties S | ± ; · (cid:105) = ± | ± ; · (cid:105) , S | · ; ± (cid:105) = ± | · ; ± (cid:105) ; (9)the parameter λ is a coupling parameter. The model explicitly describes an experiment tomeasure the isospin state of each particle in the given fixed isospin direction (the case of ageneral isospin measurement direction will be considered below). The form of equations (8)can be roughly understood by considering an incremental stage in the evolution where d ξ aσ is either positive or negative. For example, if d ξ σ is positive then the stochastic term onthe right side of the first equation in (8) will augment the + state for particle whilstdegrading the − state for particle . The opposite happens if d ξ σ is negative. Eventuallyafter a certain period of evolution one of the two eigenstates will dominate. This is analogousto the famous problem of the gambler’s ruin.The drift terms on the right side of equations (8) ensure that the state norm is a positivemartingale d x (cid:104) ψ ( σ ) | ψ ( σ ) (cid:105) = 4 λ (cid:104) ψ ( σ ) | S | ψ ( σ ) (cid:105) d ξ x for x ∈ R , d x (cid:104) ψ ( σ ) | ψ ( σ ) (cid:105) = 4 λ (cid:104) ψ ( σ ) | S | ψ ( σ ) (cid:105) d ξ x for x ∈ R . (10)We can then define a physical measure P equivalent to Q according to E P [ ·|F ξσ ] = E Q [ (cid:104) ψ ( σ f ) | ψ ( σ f ) (cid:105) · |F ξσ ] E Q [ (cid:104) ψ ( σ f ) | ψ ( σ f ) (cid:105)|F ξσ ] = E Q [ (cid:104) ψ ( σ f ) | ψ ( σ f ) (cid:105) · |F ξσ ] (cid:104) ψ ( σ ) | ψ ( σ ) (cid:105) , (11)with σ f the final surface of the state evolution we are considering. This change of measureensures that physical outcomes are weighted according to the Born rule, meeting the secondbullet-pointed criterion for dynamical state reduction stated in the introduction. Note thatthe processes ξ aσ satisfy a modified distribution under the P -measure.Our model can be interpreted as an effective model describing the interaction of the twoparticles with macroscopic measuring devices in regions R and R . In more detail wewould expect the particle states to become correlated with different states of the measuringdevices. The state reduction dynamics would be expected to have a negligible effect on theindividual spin particles, however, the effect would be rapid for a macroscopic superpositionof measuring device states. Collapse of the spin particle would then occur indirectly asa result of collapse of the macro state. In our model we have assumed that the particlestates undergo a direct collapse dynamics. This allows us to ignore the fine details of theinteraction between spin particles and measuring devices.By designating spacetime regions where collapse of the isospin state occurs we avoid theissue of setting a scale distinguishing micro and macro behaviour. Our main interest hereis to understand the dynamical process of state reduction for an entangled quantum systemin a relativistic setting. III. SOLUTION IN TERMS OF Q -BROWNIAN MOTION Working in the Q -measure where ξ aσ is a Brownian process we find the following solutionfor the unnormalised state evolution: | ψ ( σ ) (cid:105) = √ (cid:110) e λξ σ − λ ω σ e − λξ σ − λ ω σ | + ; − (cid:105) − e − λξ σ − λ ω σ e λξ σ − λ ω σ | − ; + (cid:105) (cid:111) . (12)This can easily be checked with the use of (5), (6), and (8). The state norm is given by (cid:104) ψ ( σ ) | ψ ( σ ) (cid:105) = (cid:110) e λξ σ − λ ω σ e − λξ σ − λ ω σ + e − λξ σ − λ ω σ e λξ σ − λ ω σ (cid:111) . (13)We note that although equation (12) is a solution to (8), it cannot be considered as asolution to the model since it completely disregards the important role played by the physicalmeasure P . Equation (12) enables us to generate sample outcomes, however, the physicalprobability density at a given outcome can only be determined afterwards with reference tothe state norm (a likely outcome in Q may be highly unlikely in P ).We define the characteristic function associated with ξ σ and ξ σ in the P -measure asΦ ξσ ( t , t ) = E P (cid:104) e it ξ σ e it ξ σ |F ξσ i (cid:105) (14)= E Q (cid:104) (cid:104) ψ ( σ ) | ψ ( σ ) (cid:105) e it ξ σ e it ξ σ |F ξσ i (cid:105) , (15)where we have used equation (11) and the fact that the initial state has unit norm. Notingthat ξ σ and ξ σ are independent in the Q -measure we can determine the expectation usingequation (13) to findΦ ξσ ( t , t ) = (cid:110) e iλt ω σ − t ω σ e − iλt ω σ − t ω σ + e − iλt ω σ − t ω σ e iλt ω σ − t ω σ (cid:111) . (16)The characteristic function allows us to immediately demonstrate that spacelike separatedprocesses ξ σ and ξ σ are correlated under the physical measure P : E P (cid:2) ξ aσ |F ξσ i (cid:3) = − i dd t a (cid:2) Φ ξσ ( t , t ) (cid:3)(cid:12)(cid:12) t = t =0 = 0 , E P (cid:2) ξ σ ξ σ |F ξσ i (cid:3) = − d d t d t (cid:2) Φ ξσ ( t , t ) (cid:3)(cid:12)(cid:12) t = t =0 = − λ ω σ ω σ . (17)The stochastic information at one wing of the apparatus is not independent of the stochasticinformation at the other wing. We might expect this since the results of the two measure-ments that the information dictate are correlated.Before demonstrating the state reducing properties of this model, we first show in thenext section how to express the solution (12) directly in terms of a P -Brownian motion. Thiswill allow us to generate physical sample solutions. IV. SOLUTION IN TERMS OF P -BROWNIAN MOTION Let the probability space (Ω , F , P ) be given and let G σ be a filtration of F such thatindependent P -Brownian motions B aσ ( a = , ) are specified together with random variables s a (independent of B aσ ). The Brownian motions B aσ are defined under the P -measure in thesame way in which Brownian motions ξ aσ are defined under Q -measure by equations (5)and (6). The probability distribution for the random variables s a are given by P (cid:0) s = + , s = − (cid:1) = , P (cid:0) s = − , s = + (cid:1) = . (18)We assume that s a are G σ i -measurable.Now define the random processes (c.f. [23]) ξ σ = 4 λs ω σ + B σ ,ξ σ = 4 λs ω σ + B σ . (19)Our aim is to show that these processes, defined under the P -measure, can be identified asthe Q -Brownian processes ξ aσ involved in the equations of motion for the state (8). In orderto do this we must show that their characteristic function under the P -measure is identicalto that found for the Q -Brownian processes, as given by equation (16).Again let F ξσ denote the filtration generated by { ξ σ , ξ σ } . The use of F ξσ ensures that wehave no more or less information than is given by the processes { ξ σ , ξ σ } as in the originalpresentation of the model in section II. Neither s a nor B aσ are F ξσ -measurable. The onlyinformation we have regarding the realisation of these variables is { ξ σ , ξ σ } .The characteristic function for ξ σ and ξ σ is given by equation (14),Φ ξσ ( t , t ) = E P (cid:104) e it ξ σ e it ξ σ |F ξσ i (cid:105) , but now we writeΦ ξσ ( t , t ) = E P (cid:104) e it (4 λs ω σ + B σ ) e it (4 λs ω σ + B σ ) (cid:12)(cid:12)(cid:12) F ξσ i ; s = + , s = − (cid:105) + E P (cid:104) e it (4 λs ω σ + B σ ) e it (4 λs ω σ + B σ ) (cid:12)(cid:12)(cid:12) F ξσ i ; s = − , s = + (cid:105) . (20)Noting that B σ and B σ are independent we can work directly in the P -measure to confirmthat the characteristic function is once more given by equation (16). This demonstrates thatthe processes defined by equation (19) can indeed be identified as Q -Brownian motions ξ aσ .We are now in a position to express the solution to equations (8) and (11) in termsof the P -Brownian motions B aσ , and the random variables s a . This is summarised in thefollowing subsection. The fact that the solution is expressed in terms of variables with an a priori known probability distribution in the physical measure is to be contrasted with thesolution in terms of Q -Brownian motion where physical probabilities can only be determined a posteriori with knowledge of the state norm. A. Summary of solution
The solution to the equations of motion (8) is given by the unnormalised state | ψ ( σ ) (cid:105) = √ (cid:110) e λξ σ − λ ω σ e − λξ σ − λ ω σ | + ; − (cid:105) − e − λξ σ − λ ω σ e λξ σ − λ ω σ | − ; + (cid:105) (cid:111) . (21)(This is the same solution in terms of ξ aσ as presented in equation (12), however, we nowtreat ξ aσ , not as a Q -Brownian motion, but as an information process defined in terms ofvariables with known P -distributions.) The random variables ξ aσ are given by ξ σ = 4 λs ω σ + B σ ,ξ σ = 4 λs ω σ + B σ . (22)The stochastic processes B σ and B σ are independent P -Brownian motions. The randomvariables s a take values s = +1 / , s = − / / s = − / , s =+1 / /
2. Brownian motions B aσ and random variables s a are independent.Only the processes ξ aσ are measurable.This solution is as relativistically invariant as a description of state reduction can be. Weexpect the state to depend on the spacelike surface σ we choose to query. The dependence on σ results in equation (21) from the spacetime volume variables ω aσ and the random variables B aσ . We note that neither of these variables depends on the chosen foliation of spacetime.For example, the distribution of B aσ is characterized by the spacetime volume ω aσ which inturn is determined only by the surface σ . A foliation dependence would be undesirable as itwould indicate a preferred frame in the model. The fact that there is no foliation dependenceindicates also that the choice σ has no prior physical significance. B. State reduction
In this subsection we explicitly demonstrate how the solution outlined above exhibitsstate reduction to a state of well-defined isospin. Consider the isospin operators S a . Theconditional expectation of S a for the state | ψ ( σ ) (cid:105) is given by (cid:104) S a (cid:105) σ = (cid:104) ψ ( σ ) | S a | ψ ( σ ) (cid:105)(cid:104) ψ ( σ ) | ψ ( σ ) (cid:105) . (23)From equation (21) we find choosing, for example, a = , (cid:104) S (cid:105) σ = e λξ σ − λ ω σ e − λξ σ − λ ω σ − e − λξ σ − λ ω σ e λξ σ − λ ω σ e λξ σ − λ ω σ e − λξ σ − λ ω σ + e − λξ σ − λ ω σ e λξ σ − λ ω σ . (24)Now suppose we condition on the event s = +1 / , s = − /
2. We find (cid:104) S (cid:105) σ = e λB σ +2 λ ω σ e − λB σ +2 λ ω σ − e − λB σ − λ ω σ e λB σ − λ ω σ e λB σ +2 λ ω σ e − λB σ +2 λ ω σ + e − λB σ − λ ω σ e λB σ − λ ω σ = − e − λB σ − λ ω σ e λB σ − λ ω σ e − λB σ − λ ω σ e λB σ − λ ω σ . (25)Next we use the result that lim ω σ →∞ P (cid:16) e ± λB σ − λ ω σ > (cid:17) = 0 , (26)to deduce that (cid:104) S (cid:105) σ → / ω σ → ∞ or ω σ → ∞ . These volumes increase in sizeas the surface σ passes the spacetime regions R and R respectively. Since these regionsare of finite size, ω σ and ω σ can only attain fixed maximal values. We assume that thesemaximal values are sufficiently large that the limit of equation (26) is approached with highprecision. Note that the rate at which this limit is approached can be controlled by thechoice of coupling parameter λ .A similar analysis leads to the conclusion that (cid:104) S (cid:105) σ → − /
2. Conversely, if we were tocondition on the event s = − / , s = +1 /
2, we would find (cid:104) S (cid:105) σ → − / (cid:104) S (cid:105) σ → /
2. We observe that the unmeasurable random variable s a dictates the outcome of theexperiment. Only the processes ξ aσ are known to the state; the Brownian processes B aσ actas noise terms obscuring the values s a . C. Probabilities for reduction
Here we demonstrate that the stochastic probabilities for outcomes are those predictedby the quantum state prior to the measurement event. For example, we define the + stateprojection operator on particle by P + | + ; · (cid:105) = | + ; · (cid:105) ; P + | − ; · (cid:105) = 0 , (27)and the conditional expectation of this operator for the state | ψ ( σ ) (cid:105) by (cid:104) P + (cid:105) σ = (cid:104) ψ ( σ ) | P + | ψ ( σ ) (cid:105)(cid:104) ψ ( σ ) | ψ ( σ ) (cid:105) . (28)In order to calculate the unconditional expectation of (cid:104) P + (cid:105) σ it turns out to be simpler towork in the Q -measure. We proceed as follows: E P [ (cid:104) P + (cid:105) σ |F ξσ i ] = E Q [ (cid:104) ψ ( σ ) | ψ ( σ ) (cid:105)(cid:104) P + (cid:105) σ |F ξσ i ]= E Q [ (cid:104) ψ ( σ ) | P + | ψ ( σ ) (cid:105)|F ξσ i ]= E Q (cid:104) e λξ σ − λ ω σ e − λξ σ − λ ω σ (cid:12)(cid:12)(cid:12) F ξσ i (cid:105) = . (29)From the previous subsection we know that as ω aσ → ∞ then the state of each particle tendstowards a definite isospin state and consequently the conditional expectation of P + tendsto either 0 or 1. This means that as ω aσ → ∞ we have E P [ (cid:104) P + (cid:105) σ |F ξσ i ] = E P (cid:20) n (cid:104) S (cid:105) σ = 12 o (cid:12)(cid:12)(cid:12)(cid:12) F ξσ i (cid:21) = P (cid:0) (cid:104) S (cid:105) σ = (cid:12)(cid:12) F ξσ i (cid:1) , (30)0where { E } takes the value 1 if the event E is true, and 0 otherwise. From equation (29) wecan now write P (cid:0) (cid:104) S (cid:105) σ = (cid:12)(cid:12) F ξσ i (cid:1) = = (cid:104) P + (cid:105) σ i . (31)This tells us that as the dynamics lead to a definite state for each particle then the stochasticprobability of a given outcome matches the initial quantum probability. The same is trueof other projection operators as can easily be shown. V. INTERPRETATION IN TERMS OF NONLINEAR FILTERING
In this section we use the method of Brody and Hughston [22, 23] to demonstrate thatthe problem under consideration can be interpreted as a classical nonlinear filtering problem.The method was originally applied to solve an energy-based state diffusion equation.From section IV B we understand that the F ξσ -unmeasurable random variables s a repre-sent the true outcomes for the isospin eigenvalues of each particle after the measurementprocess. Only information in the form ξ aσ = 4 λs a ω aσ + B aσ is accessible to the state where therealised value of s a is masked by the F ξσ -unmeasurable noise processes B aσ .Suppose we attempt to address the problem of finding s a directly, that is, given { ξ aσ } whatis the best estimate we can make for s a . This is a classical nonlinear filtering problem. It isstraightforward to show that the best estimate for the value of s a is given by the conditionalexpectation (cid:98) s aσ = E P (cid:2) s a | F ξσ (cid:3) . (32)The aim is now to identify (cid:98) s aσ with the quantum expectation processes (cid:104) S a (cid:105) σ .We first show that ξ aσ are Markov processes. To do this we show that P (cid:0) ξ aσ < y | ξ σ , ξ σ , · · · , ξ σ k ; ξ σ , ξ σ , · · · , ξ σ k (cid:1) = P (cid:0) ξ aσ < y | ξ σ ; ξ σ (cid:1) (33)where { σ, σ , σ , · · · , σ k } is a sequence of spacelike surfaces belonging to some spacetimefoliation such that ω σ ≥ ω σ ≥ ω σ ≥ · · · ≥ ω σ k > ,ω σ ≥ ω σ ≥ ω σ ≥ · · · ≥ ω σ k > . (34)The proof of equation (33) is more or less identical to that given by Brody and Hughston [22].We use the fact that E P [ B bσ (cid:48) B bσ (cid:48)(cid:48) ] = ω bσ (cid:48) , where ω bσ (cid:48)(cid:48) ≥ ω bσ (cid:48) for b = , . Then for ω bσ ≥ ω bσ ≥ ω bσ > B bσ and B bσ ω bσ − B bσ ω bσ are independent . (35)Furthermore, B bσ ω bσ − B bσ ω bσ = ξ bσ ω bσ − ξ bσ ω bσ , (36)1from which it follows that P (cid:0) ξ aσ < y | ξ σ , ξ σ , ξ σ , · · · ; ξ σ , ξ σ , ξ σ , · · · (cid:1) = P (cid:18) ξ aσ < y | ξ σ , ξ σ ω σ − ξ σ ω σ , ξ σ ω σ − ξ σ ω σ , · · · ; ξ σ , ξ σ ω σ − ξ σ ω σ , ξ σ ω σ − ξ σ ω σ , · · · (cid:19) = P (cid:18) ξ aσ < y | ξ σ , B σ ω σ − B σ ω σ , B σ ω σ − B σ ω σ , · · · ; ξ σ , B σ ω σ − B σ ω σ , B σ ω σ − B σ ω σ , · · · (cid:19) . (37)Now from (35) we have that ξ aσ , ξ σ , and ξ σ are each independent of B σ /ω σ − B σ /ω σ , B σ /ω σ − B σ /ω σ , etc. Equation (33) follows. The same argument shows that P (cid:0) B aσ < y | ξ σ , ξ σ , · · · , ξ σ k ; ξ σ , ξ σ , · · · , ξ σ k (cid:1) = P (cid:0) B aσ < y | ξ σ ; ξ σ (cid:1) , (38)and therefore P (cid:0) s a = ± (cid:12)(cid:12) F ξσ (cid:1) = P (cid:0) s a = ± (cid:12)(cid:12) ξ σ ; ξ σ (cid:1) . (39)Next we use a version of Bayes formula to calculate this conditional probability P (cid:0) s = ± , s = ∓ (cid:12)(cid:12) ξ σ ; ξ σ (cid:1) = P (cid:0) s = ± , s = ∓ (cid:1) ρ (cid:0) ξ σ ; ξ σ | s = ± , s = ∓ (cid:1) ρ ( ξ σ ; ξ σ ) . (40)The density function for the random variables ( ξ σ ; ξ σ ) conditional on s a is Gaussian (since B aσ is a Brownian motion under P ) and is given by ρ (cid:0) ξ σ ; ξ σ (cid:12)(cid:12) s = ± , s = ∓ (cid:1) ∝ e − ω σ ( ξ σ ∓ λω σ ) e − ω σ ( ξ σ ± λω σ ) . (41)We also have that ρ (cid:0) ξ σ ; ξ σ (cid:1) = ρ (cid:0) ξ σ , ξ σ (cid:12)(cid:12) s = + , s = − (cid:1) + ρ (cid:0) ξ σ , ξ σ (cid:12)(cid:12) s = − , s = + (cid:1) . (42)We are now in a position to calculate the conditional expectation (cid:98) s aσ given by equation (32).For example, choosing a = we have (cid:98) s σ = E P (cid:2) s | F ξσ (cid:3) = P (cid:0) s = + , s = − (cid:12)(cid:12) ξ σ ; ξ σ (cid:1) − P (cid:0) s = − , s = + (cid:12)(cid:12) ξ σ ; ξ σ (cid:1) = e λξ σ e − λξ σ − e − λξ σ e λξ σ e λξ σ e − λξ σ + e − λξ σ e λξ σ . (43)This is the same expression as that given for (cid:104) S (cid:105) σ in equation (24). This demonstrates thatthe conditional expectation (cid:98) s σ , which represents our best estimate for the random variable s given only information from the filtration F ξσ , corresponds to the quantum expectation ofthe operator S , conditional on the same information. It is remarkable that the complexityof the stochastic quantum formalism corresponds to a such a conceptually intuitive classicalanalogue.2 D D
R RRR R R σσ if u v w u v w
111 2 22 σσ FIG. 3:
A Bell test experiment for two entangled isospin particles. The dashed lines are the (classical)particle trajectories where particle moves initially to the left and particle moves initially to the right.The vertical represents a timelike direction whilst the horizontal represents a spacelike direction. At D a device is used to deflect particle towards one of several measuring devices each set up to perform anisospin measurement for a different orientation in isospin space. Spacetime regions R u , R v , . . . , R w are the different interaction regions corresponding to the different isospin orientations u , v , . . . , w .Similarly for particle . The state advances through a sequence of spacelike surfaces (bold lines) defininga foliation of spacetime. The example foliation shows particle measured before particle . VI. BELL TEST EXPERIMENTS
We now suppose that the experimenters at each wing of the apparatus can choose theorientation of their isospin measurement in isospin space. We suppose that each wing ofthe experiment now consists of several measuring devices each set up to measure the isospinvalue for different isospin orientations (see figure 3). Each particle passes through a deflectiondevice, sending it towards any one of these isospin measuring devices. The deflection devicecan be controlled by the experimenter and each experimenter makes their choice of whichisospin orientation to measure independently of the other. Furthermore, the deflection andmeasuring devices on one wing of the experiment are completely spacelike separated fromthe deflection and measuring devices on the other wing. This is essentially the experimentaldesign used by Aspect in his tests of Bell inequalities [19].We can represent the initial singlet state in terms of isospin eigenstates in a basis definedby the arbitrarily chosen measurement directions. Suppose that the chosen measurementdirections correspond to the unit isospin vectors n and n and that the angle between n and n is θ , then | ψ ( σ i ) (cid:105) = √ (cid:8) cos (cid:0) θ (cid:1) | + (cid:105) n | − (cid:105) n − i sin (cid:0) θ (cid:1) | + (cid:105) n | + (cid:105) n + i sin (cid:0) θ (cid:1) | − (cid:105) n | − (cid:105) n − cos (cid:0) θ (cid:1) | − (cid:105) n | + (cid:105) n (cid:9) , (44)where, for isospin vector operators S a , the orthonormal eigenstates satisfy n a · S a | + (cid:105) n a = | + (cid:105) n a ; n a · S a | − (cid:105) n a = − | − (cid:105) n a . (45)We denote the spacetime locations of the deflection devices as D a and the particle-measuring device interaction regions as R u a , R v a , . . . , R w a for the different measurement3directions u a , v a , . . . , w a (see figure 3). For each a , a choice of measurement direction n a ismade and only one interaction region R n a is activated. Given n and n , the equations ofmotion for the state are nowd x | ψ ( σ ) (cid:105) = (cid:8) λ n · S d ξ x − λ d ω (cid:9) | ψ ( σ ) (cid:105) for x ∈ R n , d x | ψ ( σ ) (cid:105) = (cid:8) λ n · S d ξ x − λ d ω (cid:9) | ψ ( σ ) (cid:105) for x ∈ R n , d x | ψ ( σ ) (cid:105) = 0 otherwise , (46)where the stochastic increments have the generalised propertiesd ξ ax = 0 , for x / ∈ R n a ; E Q [d ξ ax |F ξσ ] = 0 , for x to the future of σ ;d ξ ax d ξ by = δ ab δ xy d ω, for x ∈ R n a , y ∈ R n b . (47)These equations describe state reduction onto isospin eigenstates defined with respect tothe n and n directions. Again we consider these equations as effective descriptions of theparticle behaviour resulting from interactions with macroscopic measuring devices.The solution of (46) for an initial isospin singlet state is found to be | ψ ( σ ) (cid:105) = √ (cid:110) cos (cid:0) θ (cid:1) e λξ σ − λ ω σ e − λξ σ − λ ω σ | + (cid:105) n | − (cid:105) n − i sin (cid:0) θ (cid:1) e λξ σ − λ ω σ e λξ σ − λ ω σ | + (cid:105) n | + (cid:105) n + i sin (cid:0) θ (cid:1) e − λξ σ − λ ω σ e − λξ σ − λ ω σ | − (cid:105) n | − (cid:105) n − cos (cid:0) θ (cid:1) e − λξ σ − λ ω σ e λξ σ − λ ω σ | − (cid:105) n | + (cid:105) n (cid:111) . (48)As demonstrated in sections III and IV it is straightforward to show that the character-istic function associated with the Q -Brownian processes ξ σ and ξ σ (equation (14)) can bereproduced directly in the P -measure if we define ξ σ = 4 λs ω σ + B σ ,ξ σ = 4 λs ω σ + B σ , (49)where B aσ are P -Brownian motions and the random variables s a now have the joint conditionalprobability distribution P (cid:0) s = + , s = − (cid:12)(cid:12) n , n (cid:1) = cos (cid:0) θ (cid:1) , P (cid:0) s = + , s = + (cid:12)(cid:12) n , n (cid:1) = sin (cid:0) θ (cid:1) , P (cid:0) s = − , s = − (cid:12)(cid:12) n , n (cid:1) = sin (cid:0) θ (cid:1) , P (cid:0) s = − , s = + (cid:12)(cid:12) n , n (cid:1) = cos (cid:0) θ (cid:1) . (50)We assume a filtration G σ such that B aσ and s a are specified. However, since the probabilitydistribution for s and s depends on both experimenters’ choice of measurement directions,we cannot simply assume that s a are G σ i -measurable. To understand the structure of thefiltration we can treat the parameters n and n as random variables which are independentof any other random variables or processes in the system we are describing. We assume that n and n are specified by G σ in such a way that n a is G σ -measurable if and only if the4deflection event for particle a is to the past of σ . Note that within this filtration, the variable n a is associated with the entire surface σ .For a given spacetime foliation the isospin measurement on one wing of the apparatusmay be complete before the other experimenter has chosen their direction. Suppose fordefiniteness that a given foliation has R n before D (see figure 3). In order to realise theprocess ξ σ say, it is necessary to realise a definite s . Since n is not G σ -measurable forspacelike surfaces which have not crossed D , it is necessary to show that the marginaldistribution of s is independent of n .In fact we have P (cid:0) s = + (cid:12)(cid:12) n , n (cid:1) = P (cid:0) s = + , s = − (cid:12)(cid:12) n , n (cid:1) + P (cid:0) s = + , s = + (cid:12)(cid:12) n , n (cid:1) = cos (cid:0) θ (cid:1) + sin (cid:0) θ (cid:1) = , (51)as required, and similarly for other marginal probabilities. This enables us to draw valuesof s from the correct probability distribution without knowledge of n which happens inthe future for the given example foliation. In this case we require that s is G σ -measurablefor some surface σ to the past of R n (figure 3).We can define some other surface σ that is to the past of R but to the future of σ andboth particle deflection events (see figure 3). Since n , n , and s , are all G σ -measurablewe can write, for example, P (cid:0) s = + (cid:12)(cid:12) G σ (cid:1) = P (cid:0) s = + (cid:12)(cid:12) s = + ; n , n (cid:1) = P (cid:0) s = + , s = + (cid:12)(cid:12) n , n (cid:1) P (cid:0) s = + (cid:12)(cid:12) n , n (cid:1) = sin (cid:0) θ (cid:1) , (52)and similarly for other conditional probabilities. This enables us to draw values of s from thecorrect probability distribution with global knowledge of n , n , and s . We can thereforesay that s is G σ -measurable.For a different foliation where R n precedes D we would use the marginal probabilitydistribution to determine s and the conditional distribution to determine s . In any casethe joint distribution is the same. The order in which s and s are assigned has no physicalsignificance. It is simply related to our arbitrary choice of spacetime foliation within thecovariant Tomonaga picture of state evolution. We also stress that the random variables s a were introduced to facilitate solution of the dynamical equations. They are not part ofthe physical model as originally presented. The purpose of the argument presented here issimply to show that the picture of state evolution is consistent and does not require priorknowledge of the experimenter’s decisions. A. State reduction
State reduction follows from the solution in the same way as shown in section IV B.For example, given n and n we condition on the event s = +1 / s = +1 /
2. Theunnormalised expectation of the spin operator for particle is found from equation (48) tobe (cid:104) ψ ( σ ) | n · S | ψ ( σ ) (cid:105) = e λB σ +2 λ ω σ e λB σ +2 λ ω σ (cid:110) cos (cid:0) θ (cid:1) (cid:16) e − λB σ − λ ω σ − e − λB σ − λ ω σ (cid:17) + sin (cid:0) θ (cid:1) (cid:16) − e − λB σ − λ ω σ e − λB σ − λ ω σ (cid:17)(cid:111) , (53)5and the state norm is (cid:104) ψ ( σ ) | ψ ( σ ) (cid:105) = e λB σ +2 λ ω σ e λB σ +2 λ ω σ (cid:110) cos (cid:0) θ (cid:1) (cid:16) e − λB σ − λ ω σ + e − λB σ − λ ω σ (cid:17) + sin (cid:0) θ (cid:1) (cid:16) e − λB σ − λ ω σ e − λB σ − λ ω σ (cid:17)(cid:111) . (54)Using equation (26) we then find that as ω σ → ∞ , (cid:104) n · S (cid:105) σ = (cid:104) ψ ( σ ) | n · S | ψ ( σ ) (cid:105)(cid:104) ψ ( σ ) | ψ ( σ ) (cid:105) → . (55)As expected the isospin of particle in the direction n tends to the value . A similarcalculation shows that (cid:104) n · S (cid:105) σ → as ω σ → ∞ , along with similar results for other givenvalues of s a .It is also straightforward to show thatlim ω σ ,ω σ →∞ (cid:104) ( n · S )( n · S ) (cid:105) σ = with probability sin (cid:0) θ (cid:1) , − with probability cos (cid:0) θ (cid:1) , (56)such that E P (cid:20) lim ω σ ,ω σ →∞ (cid:104) ( n · S )( n · S ) (cid:105) σ (cid:12)(cid:12)(cid:12)(cid:12) F ξσ i (cid:21) = − cos θ = − n · n . (57)This agrees with the result predicted by standard quantum theory and is confirmed by Belltest experiments. B. Parameter independence
The parameter independence condition states that the probability of a given outcomefor an isospin measurement on one wing of the experiment is independent of the chosenmeasurement direction on the other wing. This is an important feature since if the modelwere parameter dependent we could transmit messages at superluminal speeds.Parameter independence can be stated as follows: P (cid:18) lim ω σ →∞ (cid:104) n · S (cid:105) σ = + (cid:12)(cid:12)(cid:12)(cid:12) F ξσ i ; n , n (cid:19) = P (cid:18) lim ω σ →∞ (cid:104) n · S (cid:105) σ = + (cid:12)(cid:12)(cid:12)(cid:12) F ξσ i ; n (cid:19) , (58)and similarly for 1 ↔
2. In order to prove this relation we define projection operators P + n a by P + n a | + (cid:105) n a = | + (cid:105) n a ; P + n a | − (cid:105) n a = 0 . (59)In the limit that ω σ → ∞ we can write P (cid:0) (cid:104) n · S (cid:105) σ = + (cid:12)(cid:12) F ξσ i ; n , n (cid:1) = E P (cid:2) (cid:104) P + n (cid:105) σ (cid:12)(cid:12) F ξσ i ; n , n (cid:3) = E Q (cid:2) (cid:104) ψ ( σ ) | P + n | ψ ( σ ) (cid:105) (cid:12)(cid:12) F ξσ i ; n , n (cid:3) = E Q (cid:104) cos (cid:0) θ (cid:1) e λξ σ − λ ω σ e − λξ σ − λ ω σ (cid:12)(cid:12)(cid:12) F ξσ i ; n , n (cid:105) + E Q (cid:104) sin (cid:0) θ (cid:1) e λξ σ − λ ω σ e λξ σ − λ ω σ (cid:12)(cid:12)(cid:12) F ξσ i ; n , n (cid:105) = cos (cid:0) θ (cid:1) + sin (cid:0) θ (cid:1) = . (60)6The probability of a given outcome for particle is independent of n as required. VII. THE FREE WILL THEOREM
The Free Will Theorem of Conway and Kochen [24, 25] asserts that if an experimenter isfree to make decisions about which directions to orient their apparatus in a spin measure-ment, then the response of the spin particle cannot be a function of information content inthe part of the universe that is earlier than the response itself. The conclusion of Conwayand Kochen is that this rules out the possibility of being able to formulate a relativisticmodel of dynamical state reduction. It is claimed that a classical stochastic process whichdictates a definite spin measurement outcome must be considered to be information as de-fined within the theorem. The theorem then states that the particle’s response cannot bedetermined by this classical information, undermining the construction of dynamical modelsof state reduction. We do not reproduce the proof of the theorem here (it can be found in[24, 25]). In order to understand that the conclusion of Conway and Kochen is inappropriateit will suffice to analyse the three axioms of the Free Will Theorem with reference to themodel outlined in this paper.The first axiom SPIN specifies the existence of a spin-1 particle for which measurementsof the squared components of spin performed in three orthogonal directions will always yieldthe results 1, 0, 1 in some order. The second axiom TWIN asserts that it is possible to forman entangled pair of spin-1 particles in a combined singlet state such that if measurementsof the components of squared spin were performed in the same direction for each particlethey would yield identical results. These two axioms follow directly from the quantummechanics of spin particles. A situation is considered where experimenters at spacelikeseparated locations D and D can each choose the orthogonal set of directions in whichto measure the components of squared spin for each particle. (The proof of the Free WillTheorem makes use of the Peres configuration of 33 directions for which it can be shownthat it is impossible to find a function on the set of directions with the property that itsvalue for any orthogonal set of directions is always 1, 0, 1 in some order.) Although we haveconsidered a different spin system in this paper, the similarities between the experimentalset-ups allow us to evaluate the applicability of the Free Will Theorem to dynamical statereduction.The third axiom MIN (in the latest version of the proof [25]) states that the particleresponse at R n (using our notation where it is understood that the choice of spin measure-ment direction n corresponds to an orthogonal triple of directions) is independent of thechoice of measurement direction at D and similarly that the particle response at R n isindependent of the choice of measurement direction at D . Information is defined in the con-text of MIN in such a way that any information which influences the measurement outcomeat R n is independent of n and any information which influences the measurement outcomeat R n is independent of n . We can immediately see that this definition of information doesnot apply to the classical stochastic processes ξ aσ considered in our model. As highlightedabove, ξ aσ can be expressed in terms of a random variable s a whose value corresponds tothe eventual spin measurement outcome, and a physical Brownian motion process B aσ whichacts as a noise term, obscuring the value of s a . The realised value of s a indeed dependson the choice of measurement direction at the opposite wing of the experiment in the wayshown in section VI. Since the process ξ aσ influences the measurement outcome in a waywhich depends critically on the realised value of s a , it does not satisfy the definition of MIN7information. Furthermore, there is no reason why the mechanism of state reduction outlinedin this paper cannot be applied to any spin system including the TWIN SPIN system usedto prove the Free Will Theorem.More generally we are able to see that the MIN axiom need not be satisfied whilst stillmaintaining independence from any specific inertial frame. Viewing state evolution in theTomonaga picture we must choose a foliation of spacetime to provide a framework for aconsistent narrative of the state evolution. Covariance enters with the fact that all choicesof foliation are equivalent; the state can be defined on any spacelike hypersurface. For afoliation where R n happens before D , the state will collapse across the entire hypersurfaceas it crosses R n , to a new state consistent with the isospin measurement direction n . Inthis way the response of particle is independent of the choice of measurement direction at D (which happens later in the evolution) but the response of particle depends (via thecollapsed state) on the random variable θ . The opposite interpretation can be made for afoliation where R n is before D . Thus the MIN axiom should read that either the particleresponse at R n is independent of the choice of measurement direction at D or the particleresponse at R n is independent of the choice of measurement direction at D , the differencebeing a matter of interpretation. With this modification the proof of the Free Will Theoremno longer holds.We stress that the choice of spacetime foliation is analogous to an arbitrary gauge choice.It allows us to form a global covariant picture of state evolution without reference to anyindividual observer’s frame. VIII. CONCLUSIONS
We have argued that the principles of quantum mechanics are in need of modificationif we hope to find a unified description of micro and macro behaviour. We have seenthat alternatives to quantum dynamics can feasibly be constructed despite the apparentinvulnerability of standard quantum theory when faced with experimental evidence. Itmay even be possible to test new theories against standard quantum theory in the nearfuture [31, 32].We have demonstrated a continuous state reduction dynamics describing the measure-ment of two spacelike separated spin particles in an EPR experiment. The correlationbetween measured outcomes for the two particles, particularly when the experimenters arefree to choose the orientations of their spin measurements, offers an interesting challenge fordynamical models of state reduction. We have seen that the use of the physical probabilitymeasure induces a corresponding correlation between the stochastic processes to which theparticle states are coupled. State evolution is covariantly described using the Tomonagapicture with no dependence on any chosen frame and no possibility for superluminal com-munication. The results of measurements agree with standard quantum theory, in particularfor the purpose of performing a test of Bell inequalities for the system.The value of this model is to show that the state reduction process can indeed be de-scribed by a relativistically-invariant stochastic dynamics (contrary to the claims of Conwayand Kochen). We have shown how to solve the dynamical equations and this has led to newinsight into the structure of the filtration. In the physical measure, the covariantly-definedstochastic processes are seen to be constructed from a random variable which relates directlyto the measurement outcome and a noise process which obscures the random variable, mak-ing it inaccessible from the point of view of the state dynamics. This allows us to reinterpret8the problem of solving the stochastic equations of motion as a nonlinear filtering problemwhereby the aim is to form a best estimate of the hidden random variable based only oninformation contained in the observable processes. It is hoped that these insights might helpto indicate ways in which we might tackle state reduction dynamics in relativistic quantumfield systems.
Acknowledgements
I would like to thank Dorje Brody and Lane Hughston for a series of useful discussionsessions. I would also like to thank the Theoretical Physics Group at Imperial College wherethis work was carried out. [1] P. Pearle, Phys. Rev.
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