The distance separating quantum theory from reality

Volume 99A, number 4

PHYSICS LETTERS

28 November 1983

THE DISTANCE SEPARATING QUANTUM THEORY FROM REALITY Trevor W. MARSHALL Department of Mathematics, University of Manchester, ManchesterM13 9PL, UK Received 29 September 1983

A local realistic model of optical cascades, involvingjust one angular hidden variable, is found to give a coincidence curve which differs from the quantum-theoretic curve by an average of 1.5% for currently realizable experiments. The results indicates that it is extremely difficult, though not impossible, to use such experiments to discriminate between quantum theory and local realistic theories. However, in no experiment performed to date has sufficient accuracy been achieved to make such a discrimination.

Quantum theory, with its non-local theory of measurement, gives certain predictions, for correlations between measurements on spatially separated systems, which differ widely from the predictions of all local realistic theories [1 ]. For the purpose of this article, the description of nature provided by this latter family of theories will be called, simply, "reality". The biggest contradiction between quantum theory and "reality" is provided in the Bohm version [2] of the EinsteinPodolsky-Rosen Gedankenexperiment [3], where an object with definite angular momentum decays into two objects whose angular momenta are measured after separation. It is currently widely believed that certain experiments involving the 0 - 1 - 0 photon cascade of atomic calcium have provided a realization of this Gedankenexperiment. In such experiments, two light signals of different frequencies ("Red photons" and "green photons") are analysed by two widely separated polarizer-detector devices. If the two devices register simultaneously (to within about 10 -8 s), we say that a "coincidence" has occurred. There is a certain background rate of "accidental" coincidences, resulting from the "red photon" from one atom arriving at the same time as the "green photon" from a different atom. After allowing for this background rate, we are left with a residual set of "true" coincidences, which are presumed to be observations of a "red photon" and a "green photon" coming from the same atomic 0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland

cascade process. We count the number of true coincidences per unit time, R(O1, 02) , as a function of the angles, 01 and 0 2, at which the two polarizers are set. According to the current quantum-theoretic analysis, this quantity is given by the expression RQ(01, 02) = ~1,,0 r / l r / 2 e l+ e2+ (1 + f c o s 20) (0 = 01 - 0 2 ) ,

(1)

where R 0 is the rate at which pairs of "photons" enter the lens systems collecting light for the detecting devices, ~71 and 772 are the "quantum efficiencies" of these devices (typically about 0.2), e~ and e~ are polarizers efficiencies (typically close to unity), and f is another efficiency factor (dose to but always less than unity) depending on the polarizer efficiencies and on the lens apertures. With the polarizers removed, the coincidence rate is denoted R( oo, oo), and the quantum theoretic expression for this quantity is RQ( °°, o9 =R0~1,7 2 .

(2)

The quotient of these two expressions, namely RQ(01' 02)_ 1 + +,. + / c o s 20) PQ(01' 02) = RQ(OO, oo) - 4el e2tl

(3)

is then interpreted as the joint probability of the two "photons" passing through the two polarizers. For simplicity we now confine our attention to the ideal experiment, for which e~ = e~ = f = 1. The experi163

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ments actually realized come close enough to these values to make the following analysis valid, namely that no local model can give (3) as joint probability. Indeed the spatial separation of the two polarizers means that a pair of light signals characterized, at the moment of emission, by the hidden variable set X, passes the first polarizer with a probability depending on 01 and X, and the second with a probability depending on 02 and k. Let us call these WI(01, ~.) and W2(02, ~.). Then, for given ),, the joint probability is a simple product of these, so that, overall, the local theory gives an expression of the form

PR(01,02) =fo(x)Wl(O 1, x) w2(02, x) d~,,

(4)

where 0~< W1 ~< 1,0 ~< W2 <~ 1,0~

fp(x)

dX = 1

(5)

4

done, and in one series of atomic cascade experiments [4], the discrepancy between experiment and "local realism" is reported to be forty standard deviations. It has been pointed out, however [5], that the scope of local realistic theories has been quite artificially restricted by assuming that the interactions of the light signal, first with the polarizer and then with the detector, are independent. This means that the "red photon" passes its polarizer with probability W1(01, )k) and is detected with probability r/l, the joint probability of polarization-plus-detection being the simple product of these. If we discard this assumption (for a discussion of the physical implications see ref. [5]), then eq. (4) still applies, but now W(O,)0 is to be interpreted as the probability of a signal ), being detected after passing through a polarizer set at 0. We must now compare PR(01, 02) with ¢

The discrepancy between all expressions of this type and PQ is considerable. For example, if we take ~, to be a linear polarization, and use the normal macroscopic expression (Malus' law) for the intensity reduction on passing through a perfect polarizer, then W(0, ),) = ½[1 + cos 2(0 - )O].

PR(01,

0 2 ) = 1 + 1 COS

20,

(7)

as against eQ(01, o2) = ~ + 1 cos 2 0 .

(8)

We may define the distance between these two curves as

[PR(0) -- PQ(0)] 2 0 = 0.0883 ...,

a0) (9)

which is more than 17% of the maximum OfPQ. A similar discrepancy, traditionally shown by the Bell inequality [1] is found for all such realist theories, and it is necessary to show only approximate agreement with the quantum-theoretic curve in order to refute all theories of this family. This has now been 164

1

PQ(01, 02) = ~r/lr/2(1 + cos 20).

(10)

We could, as before, discuss the distance between t various PR and PQ. It is more convenient to define

PR(O1,02) = (rll•2)-leR(

(6)

We now assume that ~. is the same for both the "red photon" and the "green photon", and that all values of X (between 0 and n) are equally probable. Then, for this model,

28 November 1983

01, 02),

(11)

fp(x)Hl(O 1, ~,)/t2(02, ~,) dX,

(12)

w(o, x ) ,

(13)

where /t(0, x) = ,7 -1 so that

O<.H(O,X)~
5.

(14)

We may then compare quantum theory with reality by looking at d(P~,PQ) for various p(X) and H(O, X). It should, however be stressed that, when we compared (7) and (8), PR and PQ were both probabilities, whereas now the genuine probabilities are PR and P~. Thus neither PQ nor P~ is a probability, and in the latter case this is reflected in the relaxed inequality (14) applied to H as opposed to (5) which was applied to W. It would be nice to know the minimum value, for all p(),) satisfying (5) and all H(O, X) satisfying (14), of the quantity d(P~, PQ). This would indeed be "the distance separating quantum theory from reality". For the moment the task of finding such a minimum is too formidable. What we can do is to restrict p(X) and H(O, ),) to a certain family of functions and minimize

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d(P~, PQ) within that family. In this way we obtain an upper bound for the minimum. First let us assume that X is a single angular variable uniformly distributed over the interval (0, n). It may be thought of as a generalized polarization variable. Let us assume further than H 1 and H 2 are given by the same functions of their arguments, and that H(O, X) =H(O - X),

(15)

where H(O - k) is an even function with period lr. Then ~r/2

, eR(01, o2) = ~1 f

28 November 1983

Table 1 0 (deg)

P~(O)

PQ(O)

H(O)

0 10 20 30 40 50 60 70 80 90

0.5178 0.4925 0.4383 0.3643 0.2823 0.2005 0.1276 0.0684 0.0279 0.0081

0.5000 0.4849 0.4415 0.3750 0.2934 0.2066 0.1250 0.0585 0.0151 0.0000

1.2761 1.2057 1.1139 0.8590 0.6526 0.0000 0.0000 0.0000 0.0000 0.0000

H(X)H(O + X) dX =e~t(0 )

7r/2

(0 = 01

-

02).

(16)

We now seek to minimize d(P~, PQ) with respect to H(X) subject to the constraint of (14). An analysis using standard techniques of the Calculus of Variations shows that the minimizing function, H(X), must either be zero or must satisfy the non-linear integral equation 7r/2 [PI~(X + q~) + P ~ ( X - 4) - ~ - ~ cos 24 cos 22t] f ' , -7r/2 X H(~) d~ = 0 . (17) More precisely, there must be a range or ranges of values of X, between 0 and n/2, within which H(X) is zero, and, in the rest of the interval, H(X) must satisfy the integral equation. We know already [5] that the distance may be made quite small with the function H(X) = ½(1 + x/~ cos 2X + ½ cos 4X),

(18)

On carrying out a numerical optimization programme, a minimum value of

d(P~, PQ) = 0.0094 ...

was obtained with the values a = 1.83,/31 = 2.72966, 132 = -0.29802,/33 = 0.15490, ~34 = -0.10892,/35 = 0.07462. The corresponding values o f P ~ , PQ and H are displayed in table 1. We note that the average separation of the two curves is 1.9% of the maximum. For realizable experiments, the efficiency parameter f i n expression (3) is always less than unity, and this makes the predictions of these local theories even closer to those of quantum theory. For example, if we put P Q ( 0 1 , 0 2 ) = ¼ [1 + 0.955 cos 20],

!

P~(O) = ~-(1 + cos 20 + ~1 cos 4 0 ) ,

(19)

and t

d(P R, PQ) = 1/32Vt2 = 0.0220 ....

(20)

This function is zero for X = 3rr/8 and takes rather small values between ~, = 3rt/8 and X = rr/2. We are therefore encouraged to seek a minimum among the family of functions 5 H(X) = ½ ~ & cos[(2n - 1)aX] ( X ~ 7r/2~), n=l

=0

(a/2~ ~< x ~<~/2). (21)

(23)

corresponding to the polarizers and lens systems used in the Aspect series of experiments [4,6,7], then we obtain

d(P R , PQ) = 0.0072 ....

for which

(22)

(24)

with the values a = 1.78,/31 = 2.65629,/32 = -0.24688, /33 = 0.16947,/34 = -0.13588,/35 = 0.09033. The average separation of the two curves is then 1.5% of the maximum. Only in the Aspect series of experiments [4,6,7] has it been possible to gather sufficient counting statistics to discriminate between quantum theory and local theories of the above family. However, none of their results so far published allows us to make the necessary comparisons. The detailed published analysis of data is confined to the values of 0(rr/8 and 3rr/8) for which the Bell inequalities give the greatest divergence, and it is now a simple matter to fired local theories giving 165

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exactly the quantum-theoretic coincidence rates for these values of 0 (see ref. [5]). It is therefore an extremely difficult, though not impossible task to use optical cascade experiments as a test of such local theories. This is made all the more difficult if serious efforts are made to exclude certain collective effects involving the radiation from neighbouring atoms, which are certainly a significant feature in the Aspect series of experiments [8], but which have not so far been taken into account in the analysis of their data. It should now be clear that the experiments so far realized are very far from achieving what has been claimed, namely the refutation of Einstein, Podolsky and Rosen. Considerable improvement in the signal-to. noise ratio, combined with longer experimental runs to gather the necessary counting statistics, will be required before a proper test can be made. This is all the more important, bearing in mind that these experiments have already been interpreted to give support to some very strange philosophical positions concerning the nature of physical reality, in some cases without even waiting for the experiments to be performed.

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28 November 1983

I am grateful to my colleagues Eric Watson and Eric Wild for providing me with the Calculus of Variations arguments leading to the family (21) of trial functions, and to Ian Gladwell for helpful advice on the choice and implementation of the numerical optimization programme.

References [1] J.F. Clauser and A. Shimony, Rep. Prog. Phys. 41 (1978) 1881. [2] D. Bohm, Quantum theory (Prentice Hall, Englewood Cliffs, NJ, 1951) pp. 614-622. [3] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47 (1935) 777. [4] A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 49 (1982) 91. [5] T.W. Marshall, E. Santos and F. Selleri, Phys. Lett. 98A (1983) 5. [6] A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 47 (1981) 460. [7] A. Aspect, J, Dalibard and C. Roger, Phys. Rev. Lett. 49 (1982) 1804. [8] T.W. Marshall, E. Santos and F. Selleri, submitted to Lett, Nuovo Cimento.