A theorem of Bell, proving that certain predictions of quantum mechanics are inconsistent with the entire family of local hidden-variable theories, is generalized so as to apply to realizable experiments. A proposed extension of the experiment of Kocher and Commins, on the polarization correlation of a pair of optical photons, will provide a decisive test between quantum mechanics and local hidden-variable theories.
Einstein
, Podolsky, and Rosen (EPR) in a classic paper presented a paradox which led them to infer that quantum
mechanics is not a complete theory. They concluded that the quantum mechanical description of a physical system should
be supplemented by postulating the existence of "hidden variables, "the specification of which would predetermine the
result of measuring any observable of the system. They believed the predictions of quantum mechanics to be correct,
but only as consequences of statistical distributions of the hidden variables. Bohr
argued in reply that no paradox can be derived from
the assumption of completeness if one recognizes that quantum mechanics concerns only the interaction of microsystems
with experimental apparatus and not their intrinsic character.
There is an extensive literature purporting to prove the inconsistency of hidden-variable theories with the statistical predictions of quantum mechanics. These proofs, though mathematically valid, rest upon physically unrealistic postulates. ' Bell' succeeded in replacing these postulates by a physically reasonable condition of locality. He showed that in a Gedankenexperiment of Bohm' (a variant of that of EPR) no local hidden-variable theory can reproduce all of the statistical predictions of quantum mechanics. This result is somewhat ironical in view of Einstein's convictions' that quantum mechanical predictions concerning spatially separated systems are incompatible with his conditions for locality unless hidden variables exist.
Bell's theorem has profound implications in that it points to a decisive experimental test of the entire family of local hidden-variable theories. The aim of this paper is to propose explicitly such an experiment. For this purpose, we first present a generalization of Bell's theorem which applies to realizable experiments. Second, we indicate that neither of the experimental realizations of Bohm's Gedankenexperiment has produced evidence against local hidden-variable theories, even though the results of both are compatible with quantum mechanical predictions. Third, we show that a simple extension of one of these experiments can provide a decisive test.
Generalization of Bell's theorem. —Consider an ensemble of correlated pairs of particles moving so that one enters apparatus I, and the other apparatus II~, where a and 5 are adjustable apparatus parameters. In each apparatus a particle must select one of two channels labeled +1 and —1. Let the results of these selections be represented by A(a) and 8(b), each of which equals +I according as the first or second channel is selected.
Suppose now that a statistical correlation of A(a) and 8(b) is due to information carried by and localized within each particle, and that at some time in the past the particles constituting one pair were in contact and communication regarding this information. The information, which emphatically is not quantum mechanical, is part of the content of a set of hidden variables, denoted collectively by ~. The results of the two selections are then to be deterministic functions A(a, h) and 8(b, h). Locality reason- ably requires A(a, h) to be independent of the parameter b and 8(b, h) to be likewise independent of a, since the two selections may occur at an arbitrarily great distance from each other. Finally, since the pair of particles is generally emitted by a source in a manner physically independent of the adjustable parameters a and b, we assume that the normalized probability distribution p(h) characterizing the ensemble is independent of a and b.
Defining the correlation function P(a, b) —= j rA(a, h)B(b, h)p(h)dh, where 1 is the total h. space, we have
( P(a, b) -P(a, c) ) - J r ( A(a, h) 8(b, h) —A(a, h)8(c, h) ) p(h) dh = Jr [ A(a, h) 8(b, h) ( [1—8(b, h)8(c, h) ]p(h) dh = Jr[1-8(b, h)8(c, h) ]p(h)dh = 1-fr 8(b, h)8(c, h) p(h)dh.
Suppose that for some b' and b we have P(b', b) = l-b, where 0 ~ b -1. Experimentally interesting cases will have 6 close to but not equal to zero. Here we avoid Bell's experimentally unrealistic re- striction that for some pair of parameters b and b there is perfect correlation (i.e., 5=0). Dividing I' into two regions I"+ and I' such that 1, =(h~A(b', h) =+8(b, h)j, we have fr p(h)dh= —,'b. Hence,
J r 8(b, h) 8(c, h) p(h) dh = Jr A(b', h)8(c, h) p(h) dh-2 fr A(b', h) 8(c, h) p (h) dh -P(b', )-2f IA(b', h)8(, h) lp(h)dh=P(b', )-b,
and therefore
~P(~, )-b(P~, c) ~ -2-P(b', b)-P(b', c) (Ia)
In the experiment proposed below P(a, b) depends only on the parameter difference b a Def—ini.ng o. —= b a, P=c-b,-and y=b b', we have—
IP(o)-P(~+tI) I -2 P(y) P(P-+y)-
In principle entire measuring devices, each consisting of a filter followed by a detector, could be used for I, and Ilq, and the values +I of A(a) and 8(b) would denote detection or nondetection of the particles. Inequalities (1) would then apply directly to experimental counting rates. Unfortunately, if the particles are optical photons (as in the experiment proposed below) no practical tests of (1) can presently be performed in this way, because available photoelectric efficiencies are rather small. We shall therefore henceforth interpret A (a) = +I and 8(b) =+I to mean emergence or nonemergence of the photons from the respective filters. Also the filters will be taken to be linear polarization filters, and a and 5 will represent their orientations. It will be convenient to introduce an exceptional value ~ of the parameter a (and likewise of b) to represent the removal of a polarizer; clearly, A(~) and 8(~) necessarily equal +1. Since P(a, b) is an emergence correlation function, in order to derive an experimental prediction from (1) an additional assumptione must be made: that if a pair of photons emerges from I„ilq the probability of their joint detection is independent of a and b Then if .the flux into I„ Ilq is a constant independent of a and b, the rate of coincidence detection R(a, b) will be proportional to re[A(a)„8(b), ], where u[A(a), , 8(b), ] is the probability that A(a) =+I and 8(b) =+1. Letting RO=R(~, ~), R,(a) =R(a, ~), and R,(b) =R(~, b), making use of the evident formulas P(a, b) =w[A(a), B(b),]—w[A(a) „B(b) ]—w[A(a), B(b),] +w[A(a), B(b) ]
and
G₂(0) = 号−cost + sin²d− cos²t), G₂(6) = 号−(sin²t + 2) cost, Ga (0) = §ーcostー号cos³t.
The predictions for a 0-1-1 electric-dipole cascade (and for a 1-1-0, provided the initial statistical state of the atom is isotropic) are obtained from (3) upon replacing F₁(0) with -F2(0) where
F₂(0) = 2G,2(t)[2G₂(0)G(0) +2G₂²(0)]-1.
For sufficiently efficient polarizers one easily sees that there exist sets of relative orientations for which the quantum mechanical counting rates violate Inequality (2b). The greatest violation always occurs at a = 22.5°, B=45°, and y = 157.5° for the 0-1-0 cascade and at a = 67.5°, B=135°, and y= 112,5° for the 0-1-1 cascade. Note that in each case the four angles a, a +B, Y, and B +y which occur in Inequality (2b) characterize only two distinct relative orientations of the polarizers, namely 22.5° and 67.5°.
In an actual experiment F,(6) is less than 1, because of finite half-angle 0, and єMis never unity. Assuming the use of calcite polarizers (for which €m~ 10-5), taking eM= EM" = EM for simplicity, and using the above choices for a, В, and y, we find that for either type of cascade, the condition for violation of Inequality (2b) is
√2F, (0)+1>2/EM.
This is the essential requirement on the design of a decisive experiment. For given F;(0), (4) implies a lower limit on EM, and vice versa. Since both F₁(0) and F₂(6) are monotonically decreasing functions, a lower limit on F;(0) implies an upper limit on 0. Condition (4) and numerical evaluation of F₁(6) and F₂(9) lead to the curves shown in Fig. 1, from which one can directly read combinations of 0 and eM suitable for a decisive experiment. The experiment can bе performed with uncoated calcite polarizers (EM ≈0.92); however, if the polarizers have antireflection coatings (eм-0.95), a larger & and hence a larger counting rate can be achieved.
The authors gratefully acknowledge helpful discussions with Y. Aharonov, M. Jammer, L. Kasday, D. Nartonis, C. Papaliolios, F. Pipkin, D. Pritchard, J. L. Snider, H. Stein, and C. R. Willis.
*This paper is an expansion of ideas presented by one of us (J.F.C.) at the Spring 1969 meeting of the American Physical Society [Bull. Am. Phys. Soc. 14, 578 (1969)]. The same conclusions had been reached independently by two of us (A.S. and M.А.H.). †Present address: Department of Physics, University of California, Berkeley, Calif. 3For an excellent survey of these proofs see J. S. Bell, Rev. Mod. Phys. 38, 447 (1966). 4J. S. Bell, Physics 1, 195 (1965). 5D. Bohm, Quantum Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1951), p. 614. Albert Einstein: Philosopher-Scientist, edited by P. Schilpp (Library of Living Philosophers, Evanston, Ill., 1949), pp. 681-682. 'C. S. Wu and I. Shaknov, Phys. Rev. 77, 136 (1950); С. А. Кocher and E. D. Commins, Phys. Rev. Letters 18, 575 (1967). See also C. A. Kocher, thesis, University of California, 1967, University of California Radiation Laboratory Report No. UCRL-17587 (unpublished). BIt may appear that the assumption can be established experimentally by measuring detection rates when a controlled flux of photons of known polarization impinges on each detector. From the standpoint of hiddenvariable theories, however, these measurements are irrelevant, since the distribution of the hidden variables when the fluxes are thus controlled is almost certain to be different from the p(λ) governing our ensemble. In view of the difficulty of an experimental check, the assumption could be challenged by an advocate of hidden-variable theories in case the outcome of the proposed experiment favors quantum mechanics. How84 .86 .88 .90 .92 .94 .96 .98 1.00 EM FIG. 1. Upper limits on detector half-angle as a function of polarizer efficiency єм. To test for hiddenvariable theories, the experiment must be performed in the region below the appropriate curve-the upper curve for a 0-1-0 cascade, the lower for a 0-1-1. ever, highly pathological detectors are required to con mechanical counting rates. 9It has been shown by D. Bohm and Y. Aharonov, Phys. Rev. 108, 1070 (1957), that the WS experiment is a decisive refutation of a hypothesis studied by W. H. Furry, Phys. Rev. 49, 393, 476 (1936). 13 ОстовER 1969 10For details see M. A. Horne, thesis, Boston University, 1969 (unpublished). 11 The distribution is given in H. S. Snyder, S. Paste