Foundadon$ of Physics Letters, Vot 7, No. 4, 1991

HAS QUANTUM NONLOCALITY BEEN EXPERIMENTALLY VERIFIED? 1

E u a n J. Squires

Department of Mathematical Sciences University of Durham Durham City DH1 SLE United Kingdom. Received July 29, 1993 Betl's theorem tells us that if we wish to preserve the results of quantmn theory, then we cmmot supplement the theory by any sort of locally determined hidden variables. The Aspect experiments tell us that the results of q u a n t u m theory, in certain relevant circumstances, are correct. Thus, some type of information about the result of an experiment nmst travel to other points of space. If we take a reasonable, simple, model of how a measurement actually produces a result, namely, the GRW collapse model, then the experiments that have so far been done, do not distinguish between instantarmous communication, which is required in the orthodox theory, and communication at the speed of light. We discuss how models which incorporate such communication might be constructed, and urge the need for experimental tests. Likely values of the relevant p a r a m e t e r s suggest that these are possible. Finally, we note that, contrary to what is generally claimed, nonlocal collapse models which agree in all circumstances with q u a n t u m theory do permit instantaneous signals to be sent over arbitrarily large distances. Key words: q u a n t u m theory, Bell's theorem, locality.

353 0894-9875/94/0800-0353507.0a,0 0 1994 Plenum Publishing Corporation

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1. T H E S I G N I F I C A N C E OF BELL'S THEOREM THE EXPERIMENTAL TESTS

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AND

Non-locality has always been a part of orthodox q u a n t u m theory. We can illustrate this by the story of a micro-wave photon. About ten billion years ago this was born in some atomic process. Its wavefunction spread across the cosmos as a spherically symmetric ripple. It has a small, but non-zero, probability of being observed on one of thousands of galm,des. When it is observed on one, then its probability of being observed on another immediately drops to zero. There is nothing whatever surprising about this as long as we are allowed to think that it really was on one galaxy, and not on any other, even before the actual observation was made. In other words, the observation only informed us about the position; it did not create it. Q u a n t u m theory, however, does not allow us to make this assumption; it does not have such things as unobserved positions. The probabilities of quantum theory are ontological, no]~ epistemic. Nevertheless, the few people who really thought about these matters, and who could not honestly accept the unreasonable degree of nonlocality associated with orthodoxy, could aJ.ways take refuge in some sort of implicit belief that there was an underlying reality in which the photon really did know where it was going to be observed, even before we discovered this ourselves. A similar situation holds in EPR-like correlated systems. There, we could always believe that, although q u a n t u m theory did not tell us anything other than the fact that the pair of electrons, say, were either in a +, - or a - , + state, they themselves really knew, in some "hidden" way, so that a measurement at one side only revealed what was already present, thus presenting no problems for locality. All this changed with Bell's theorem[l]. This showed simply that if they did know, then not only is q u a n t u m theory incomplete (because it does not have this extra information), but it is also, in some circumstances, wrong. Here, then, were two competing theories, giving different predictions for some specific cases. T h e experiments of Aspect et al.[2] showed that any model in which the particles knew their spin directions before measurement, was wrong. Here I a m ignoring the possible loopholes in these experiments, and I a m also assuming that the settings of the polarisers are effectively free (see comments at the end of section three below). Thus, we cannot escape from

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quantum nonlocality by saying that the information was in reality already present, even if in some hidden form. It follows that information about one experiment has to reach, and affect the results of, distant experiments.

2. H O W F A S T D O E S TRANSMITTED?

INFORMATION

HAVE

TO

BE

This is the obvious next question we should ask of the experiments. If something that affects the physical world travels from one place to another then it should have a measurable velocity. It is because we had quantum theory, for which such a question has no meaning, that we tended not to ask this question. However, quantum theory is an incomplete theory because it does not have within it the objectification necessary to produce a unique answer. Also, in the version we normally use to consider these sort of questions, it is non-relativistic, and therefore there is nothing wrong, in principle, about the idea of instantaneous signals. Thus quantum theory cannot be relied upon here. In any case, physics is an experimental subject; it is about how the world is, rather about how we think it ought to be, and we should not jump to conclusions without their being tested. In order to calculate the speed of the signal that is required for consistency with the Aspect et al. experiments, we need to know at what time a particular linear superposition containing two results, i.e., + and - spins, changes to give something containing only one result. Since, as we have noted, quantum theory itself does not have a mechanism for this, we need some sort of model for }low this happens. The simplest such model for this purpose is the original GRW spontaneous collapse model[3,4]. We suppose therefore that each spin is measured by means of an apparatus that produces a linear superposition of spatially separated states of a macroscopic system. Each such system contains a certain number of particles, N, which might reasonably be assumed to be of the order of 1022. Every one of these particles has a certain probability, say r -I per unit time, for jumping to an approximate position eigenstate, e.g., a gaussian with a width of about 10-5cm. When one such particle has jumped it effectively takes the whole macroscopic system with

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it[3], and we can say that the measurement has been completed, i.e., the decision between + and - has been made. The average time for this to happen will be T/N, which, in a random process, is clearly also the typical difference in time between the collapse on one side and that on the other. Thus if V is the velocity at which information can be transferred, and L is the separation between the two polarisers, there will be significant conflict with the predictions of q u a n t u m theory unless V>>

LN

--

7"

If we use the original GRW number of 1016 sec for r, and take the m a x i m u m distance of about 10 m for the Aspect experiment, then we find a value of 10 7 m per sec for the right-hand-side of this inequality. This is a very interesting number; it is close to the speed of light, and suggests that reliable tests for signalling at such a speed should be possible in fllture experiments. It is clear, however, that, contrary to what seems to be widely believed, present experiments do not distinguish between models with information being sent at the speed of light, and "orthodox" quant u m theory (where they are instantaneous). To emphasise this point we recall that we already know that the collapse cannot be too fast, i.e., much faster than the figure quoted above, otherwise there would be too m a n y photons, and too much energy, spontaneously produced from matter[3,5,6]. Hence, it is extremely unlikely that the Aspect et al. experiments would have revealed any deviations from q u a n t u m theory, even if the information does take a time L/c to travel from one side of the a p p a r a t u s to the other. In other words, at the present time

there is no experiment that distinguishes between non-local quantum theory and a reasonable modified form of quantum theory in which signals are sent at the speed of light. T h e need to improve the experiments, and so push these limits, would seem to be obvious. T h e experiments should be designed with the largest possible separation between the two sides of the apparatus, and with large, i.e., rapidly collapsing, detectors. They would look for departures from the predictions of q u a n t u m theory, in particular for departures from angular m o m e n t u m conservation in individual events. As I noted above, our subject is supposed to be guided by experiment, not prejudice, so I will not comment on the likelihood of

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finding such departures. Those who are tempted to think that they seem unbelievable should note that, in the early days of quantum theory, it was generally accepted that the conservation rules would only be true as statistical averages. They should also reflect on what it is that orthodoxy requires us to believe about the 10 l° years old microwave photon referred to earlier. The well-known quotation,....so that one body m a y act upon another at a distance through a vacuum, without the mediation of anything elae, by and through which their action m a y be conveyed f r o m one to another, is to me so great an ab.~urdity that I believe no m a n who has in philosophical matters a competent faculty of thinking can ever fall into it[7], should remind us that Isaac Newton would certainly not take us seriously if we tried to convince him of this orthodox position. Would negative results rule out completely the idea that the information is restricted to travelling at the speed of light? To answer this we need to know how slow we can allow the collapse to happen. It is not easy to obtain convincing evidence here, since we do not have any direct experimental evidence for collapse. The only thing we can say with reasonable certainty is that, if it happens, it should happen before a brain is aware of a result, because otherwise the collapse would not fulfil its purpose, and we would have no reason for believing that it should occur. I am not aware of any experiments that use this condition to provide a lower limit, although one has been suggested[8]. Since awareness is a rather ill-defined concept, and its timing subject to uncontrollable physiological factors, it is hard to see how we will ever be able to set a limit which is not several orders of magnitude bigger than the n u m b e r quoted above. Thus it may be that experiments will never be able completely to discriminate against models with finite velocity signals. However, we should be able to close the window sufficiently, that it becomes very implausible that nature has been so cunning as to choose a value for r that so carefully prevents us from seeing collapse effects.

3. A R E T H E R E SIGNALS?

MODELS

WITH

FINITE

VELOCITY

A recent p a p e r of a student of mine (Lucien Hardy[9]), suggesting ways of improving the experimental limits on the time allowed for

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signals to be sent, was rejected by a journal because, it was claimed, there are no satisfactory models involving finite velocity signals. It is fortunate for the history of physics that such an attitude has not deterred experimentalists in the past! Nevertheless, it is nice to have a model, and in fact it is not difficult to find interesting possibilities. We can indeed use a modified version of Philip Pearle's simple " G a m b l e r ' s Ruin" model[10]. This model contains the basic idea behind all the stochastic continuous localisation models. In using it we ~ e also, I believe, being true to one of the lessons which we have been taught by John Bell, to whose m e m o r y this meeting is dedicated, namely, that we should try to understand simple things, before confusing the issue with irrelevancies. We consider the standard E P R - B o h m experiment, and imagine, first, that at each detector the following simple game is played. At set intervals, At, a r a n d o m choice of, say, + or - is made. If + occurs then 1 is added to the score of P aud deleted from the score of M. and vice ver.,a if - occurs. The game begins with each score being equal (for the case when the q u a n t u m weights of the two states arc the same; more generally they would be in the ratio of these weights), say 50. Eventually one or the other score reaches 100. At this stage the g a m e ends, i.e., the m e a s u r e m e n t is complete. If it ends with the P(_~I) score at 100, then P ( M ) has won and the observed result is + ( - ) spin, etc. This is the basic Pearle model. It is not satisfactory in our case, however, because it produces no correlation between the two sides, i.e., it does not respect the entanglement of the wavefunction. We can easily modify the game to take this into account. We arrange t h a t the scores on each side take notice of b o t h games. If the information regarding the results of the games travels instantaneously, then of course the scores on the two sides will be identical and there will be perfect correlation in the final result of the measurem e n t (since we actually need anti-correlation the conventions at one of the two detectors need to be reversed of course). In this situation the model will, in all cases, give the expected, quantum-mechanical, result.. On the other hand, if the information takes a finite time, A T , to travel, then the two sets of scores will not be identical. Indeed, if A T is greater t h a n the order of lOOAt, there is even the possibility of the two sides ending the game with different results, i.e., of a violation of the conservation rules. Note that the alternative way of playing the game, namely, that we slow it down by having each

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player waiting for tile results from the other before playing again, would prevent this happening, but is surely not acceptable in the real wc~rl(1 (think about the microwave photon!). It is of interest to compare what we are saying here with the rec~,nt discussion given by Finkelstein[11]. He notes, correctly, that the standard quantum theory prediction for the probability of obtaining a set. of results al, a2, ...., at times tl, t2, •.... is given by

where Pi projects onto the eigen-subspace of the operator Ai(ti), with eigenvalue ai, p(tO) is the density matrix at time ta, and we are using Heisenberg representation. The operators are time ordered in accordance with the times when the measurements are made. Different observers will, in general, see different time orderings, and will therefore use different orderings in calculating probabilities. However, only measurements that are in ~paee-like separated regions will be subject to this change in order, and since the relevant operators will then conmmte, there will be no change in the calculated probabilty. Thus, it might be claimed, there is a relativistically invariant method of obtaining the results of quantum theory in all cases. In my opinion, however, this is not adequate. We are obtaining the results by decree, rather than by calculation, tt is when we try to find a physical theory, corresponding to the "measurements" introduced in the above discussion, rather than introducing it from outside, that we meet the problems. We could similarly decree that, in the gambler's ruin games discussed above, both players will always reach the same result. This decree, however, is not consistent with the facts; there is just no way of playing the game to make it true! (We would not try to solve a problem in arithmetic by decreeing that four plus three is equal to eight). It is of course easy to put the gambler's ruin game into the form of equations. The fact that the two players generally have different scores means, in wave function language, that there is a different wavefunction at each point of space; in other words, the wavefunction carries a space, as well as a time, label[12,13]. The need for such a space dependence, which, of course, is independent of the fact that, in Schrgdinger representation, the wavefunction is written as a function of a set of positions, is shown by Pinkelstein[11] to be a consequence of Lorentz invariance in any model where the wavefunction

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collapses. The idea here is that what exists at each space-time point is a wavefunctiom which can be thought of as a set of "information", which we might write in the form [qJ >x,t. No complete relativistically invariant theory describing a proper measurement situation has been devised at the present time. This is perhaps not too surprising since relativistic theories involving interactions are known to be fraught with problems. An a t t e m p t to generalise the stocha.stic continuous localisation theory of the nonrelativistic Schr&dinger equation (see, e.g., [14]) to the Lorentz invariant Dirac equation has been made by the author[15]. This equation clearly shows the effect of the finite velocity of signals in that charge (particle number) conservation is not maintained. The effect arises because norxnalisation integrals are not taken at a particular time (in what frame?) but along the light-cone. Thus the information at one point of space-time would be "collected" from that which existed at other points of space along the backward light-cone from the point in question. This automatically reproduces the effects noted in the simple model above, with the transmission velocity being, of course, the velocity of light. For distances, measured in light units, small compared to the average collapse time, there will be no deviations from q u a n t u m theory, but for distances much greater, all the quantum correlation will disappear completely. It is important to emphasise that in this discussion the existence of .,tocha.**ic local events, consistent with special relativity, is being assumed (see [16] for discussion). Such stochastic events can only have an effect in their forward light cones - otherwise in some Lorentz frame they would be predictable from an earlier time (even from ~ --+ eo in an infinite universe). If the settings and measurement results of the two detectors in an E P R experiment are assumed to be determined by such events then, for sufficiently large separations, it seems to be inescapable that quantum correlations must break down. On the other hand if such freedom, referred to by John Belt as "pseudo-randomness", is not available, but instead everything is fixed by some initial conditions, then the whole idea of sending information, and indeed of Bell's theorem itself, loses its significance since there is no new information to send - everything is there at the beginning.

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4. C A N W E U S E Q U A N T U M N O N L O C A L I T Y (IF IT EXISTS) TO SEND SIGNALS FASTER THAN LIGHT? T h e u s u a l a n s w e r to this question is t h a t we c a n n o t . T h e r e a s o n for this is i n d e e d a very curious f e a t u r e of q u a n t u m theory, which somehow, even in its n o n - r e l a t i v i s t i c form, seems to r e s p e c t a req u i r e m e n t of relativity. T h e a r g u m e n t relies on the fact t h a t observables are r e l a t e d to h e r m i t i a n o p e r a t o r s , which have orthogonal eigenstates. 17The s t o c h a s t i c collapse m o d e l s m a i n t a i n this i m p o r t a n t p r o p e r t y is , t h o u g h d e t e r m i n i s t i c n o n - l i n e a r m o d e l s do not, a fact which has been used as an a r g u m e n t a g a i n s t such models. T h i s u s u a l answer, however, is n o t s t r i c t l y correct. In principle. at least, it is p o s s i b l e to use t h e E P R s e t - u p to send signals i n s t a n t a neously (in some s y s t e m ) , p r o v i d e d we p o s t u l a t e a non-local collapse m e c h a n i s m . To see why this is so we s u p p o s e t h a t r , the single particle collapse time, is 1020 see a n d we also s u p p o s e the existence of a conscious b r a i n c o n t a i n i n g 1019 particles. T h e r e is no fundamental r e , o n why we s h o u l d not choose such values. T h e n it is clear t h a t this b r a i n can tell the difference b e t w e e n a s p i n - d e t e c t i o n s y s t e m for an electron, say, t h a t involves a large a p p a r a t u s for l o c a t i n g the d i r e c t i o n of m o t i o n of the electron, e.g. one with 10 24 relevant p a r t i cles, a n d one t h a t uses a s i m p l e fluorescent screen, where the collapse h a p p e n s only b e c a u s e of the brain. 19 In t h e f o r m e r case, a w a r e n e s s of t h e s p i n s t a t e of the e l e c t r o n would be i m m e d i a t e , w h e r e a s in the l a t t e r it w o u l d t a k e a b o u t ten seconds. It is now easy to see how we could use such a b r a i n to receive signals from a d i s t a n t a r m of an E P R - B o h m set-up. We would use the fluorescent screen, o b s e r v e d by the small b r a i n , at the l e f t - h a n d side, a n d the large a p p a r a t u s at the right. If no m e a s u r e m e n t is m a d e at the l a t t e r , t h e n m e a s u r e m e n t s at the l e f t - h a n d - s i d e will take ten seconds. However, i m m e d i a t e l y m e a s u r e m e n t s are m a d e on the right, then the w a v e f u n c t i o n will a l r e a d y have collapsed, a n d m e a s u r e m e n t s on t h e left will give i n s t a n t results. T h u s it w o u l d a p p e a r t h a t if we are to a c c e p t q u a n t u m nonlocality, a n d use s o m e sort of s t o c h a s t i c collapse m o d e l , t h e n i n s t a n t a n e o u s s i g n a l l i n g is, in p r i n c i p l e , possible. It m a y be i m p o s s i b l e in practice, of course, b e c a u s e of the size our b r a i n s . We n o t e also t h a t the signals w o u l d be i n s t a n t a n e o u s in some Lorentz s y s t e m , so the m o d e l w o u l d have to have a p r e f e r r e d " r e s t - f r a m e " .

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5. C O N C L U S I O N S The answer to the question posed in tile title of this paper is that we need improved experiments before we can be convinced that quantum-nonloeality is something real, and not just apparent, i.e., the effect of signals travelling at the velocity of light. Such experiments appear to be possible, and could reveal that in some circumstances conservation rules are only valid ms averages. On the other hand, if the predictions of orthodox quantum theory remain correct, we will be convinced either that there is a subtle conspiracy in nature so that the assumption of pseudo-randorrmess, implicit in all Bell-type theorems, is false, or that there is something in physics that transcends our normal ideas about space and time, and which, in principle at least, seems to allow the possibility of fa~ter-that light signals. It is an exciting thought that the study of this might dominate the physics of the 21st century as quantum theory and relativity have dominated the physics of the 20th.

REFERENCES

1. J.S.Bell, Phyaica 1 (1964) 195. 2. A.Aspect, P.Grangier, and G.Roger, Phya. Rev. Left. 49 (1982) 91. 3. G.C.Ghirardi, A.Rimini, and T.Weber, Phys. Rev. 34D (1986) 470. 4. J.S.Bell in SchrSdinger-Centenary of a Polymath, C.W.Kilmister, ed. (Cambridge University Press, Cambridge, 1987). 5. G.C.Ghirardi and A.Rimini, in Sixty-two Year3 of Uncertainty, A.Miller, ed. (Plenum, New York, 1990). 6. E.J.Squires, Phys. Left. 158 (1991) 431. 7. I.Newton, letter to R.Bentley (1692), reprinted in Newton's Philosophy of Nature, H.S.Thayer, ed. (Hafner, New York, 1953). 8. E.J.Squires, Phys. Left. 148 (1990) 381. 9. L. Hardy, "Proposal for an experiment to investigate a local collapse model", Durham preprint, 1991. 10. P.Pearle in The Wave-Particle Dualism, D.Diner et al., eds. (Reidel, Dordrecht, 1984). 11. J.Finkelstein, "Covariant collapse of the state vector and real-

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ism", San Jose State University preprint, SJSU/TP-91-8. 12. P.Eberhard in Quantum Theory and Pictures of Reality, W.Schommers, ed. (Springer, New York, 1989). 13. L.Hardy, private communication. 14. G.C.Ghirardi, P.Pearle, and A.Rimini, Phys. Rev. 42A (1990) 78. 15. E.J.Squires, "Wavefunction localisation and Lorentz inv~riealce" Durhmn preprint DTP/89/47. 16. N.Maxwell, Brit. J. Phil. Sci. 60 (1993) 341. 17. E.J.Squires, Phys. Left. 130 (1988)192. 18. N.Gisin, Helv. Phyaica Acts. 62 (1989) 363. 19. F.Aicardi, A.Borsellino, G.C.Ghirardi, and R.Grassi, Found. Phy.~. Left. 4 (1991) 109.

NOTE 1. Text of a talk given at the 1991 Cesena conference, Belt ~ Theorem and the Foundation.~ of Modern Physics.