ON THE EINSTEIN PODOLSKY ROSEN PARADOX* I. Introduction

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2 196 J. S. BELL Vol. 1, No. 3 A (a, A) = ±1, 5(2, A) = ±1. (1) The vital assumption [2] is that the result B for particle 2 does not depend on the setting a, of the magnet for particle 1, nor A on 6. If p(a) is the probability distribution of A then the expectation value of the product of the two components o l.a and a 2 ' is P{a, t) =Jd\p{\)A{a, A)B(&,\) (2) This should equal the quantum mechanical expectation value, which for the singlet state is < a 1 * a <j 2 ' o > = - a * 6. (3) But it will be shown that this is not possible. Some might prefer a formulation in which the hidden variables fall into two sets, with A dependent on one and B on the other; this possibility is contained in the above, since A stands for any number of variables and the dependences thereon of A and B are unrestricted. In a complete physical theory of the type envisaged by Einstein, the hidden variables would have dynamical significance and laws of motion; our A can then be thought of as initial values of these variables at some suitable instant. III. Illustration The proof of the main result is quite simple. Before giving it, however, a number of illustrations may serve to put it in perspective. Firstly, there is no difficulty in giving a hidden variable account of spin measurements on a single particle. Suppose we have a spin half particle in a pure spin state with polarization denoted by a unit vector p. Let the hidden variable be (for example) a unit vector A with uniform probability distribution over the hemisphere A p > 0. Specify that the result of measurement of a component o a is sign X a ', (4) where a' is a unit vector depending oh a and p in a way to be specified, and the sign function is + 1 or -1 according to the sign of its argument. Actually this leaves the result undetermined when A * a _r 0* but as the probability of this is zero we will not make special prescriptions for it. Averaging over A the expectation value is < a- a > = 1-2d'/n, (5) where d* is the angle between a' and p. until Suppose then that a' is obtained from a by rotation towards p 1 - = cosd (6) 77 where 0 is the angle between a and p. Then we have the desired result < o - a > = cos 6 (7) So in this simple case there is no difficulty in the view that the result of every measurement is determined by the value of an extra variable, and that the statistical features of quantum mechanics arise because the, value of this variable is unknown in individual instances.

3 Vol. l t No. 3 ON THE EINSTEIN PODOLSKY ROSEN PARADOX 197 Secondly, there is no difficulty in reproducing, in the form (2), the only features of (3) commonly used in verbal discussions of this problem: P{a,a) = -Pd-a^-l (g) P{a, ti = 0 if a-1> = 0 ) For example, let A now be unit vector A, with uniform probability distribution over all directions, and take A (a, X) = sign a X,g* B(a, b) = -sign 6- A ) This gives P& t) =-1 + ± 9, (10) where 6 is the angle between a and 6, and (10) has the properties (8). For comparison, consider the result of a modified theory [6] in which the pure singlet state is replaced in the course of time by an isotropic mixture of product states; this gives the correlation function - 1 a. t (11) 3 It is probably less easy, experimentally, to distinguish (10) from (3), than (11) from (3). Unlike (3), the function (10) is not stationary at the minimum value - l(at d = 0). It will be seen that this is characteristic of functions of type (2). Thirdly, and finally, there is no difficulty in reproducing the quantum mechanical correlation (3) if the results A and B in (2) are allowed to depend on o and a respectively as well as on a and 6. For example, replace a in (9) by a', obtained from a by rotation towards o until Q* = cos \$, n where 8 1 is the angle between a' and o. However, for given values of the hidden variables, the results of measurements with one magnet now depend on the setting of the distant magnet, which is just what we would wish to avoid. IV. Contradiction The main result will now be proved. Because p is a normalized probability distribution, d\p(k) = 1, (12) / - and because of the properties (1), P in (2) cannot be less than - 1. It can reach - 1 at a = b only if A(a, A) = -B(a, A) (13) except at a set of points A of zero probability. Assuming this, (2) can be rewritten P(% t) = -Jdkptib A{a, A) A(% A). (14)

4 198 J. S. BELL Vol. 1, No. 3 It follows that c is another unit vector P(a, t) -P(a, c) = -Jdkp(k) [A(l, A) A(\$, A) -A(a, A) A(c, A)] = Jdkp(k) A(a, A) A{% A) [A(t, A) A(c, A) -1] using (1), whence P(a, 3) -P(ic) <JdXp(X)[l-A(\$, X)A(Z, A)] The second term on the right is P(b, c), whence 1 + P(3, c) > P(a, 3) - P(a } 3) (15) Unless P is constant, the right hand side is in general of order 6-c for small 6-c. Thus P(o, c) cannot be stationary at the minimum value (-1 at o = c) and cannot equal the quantum mechanical value (3). Nor can the quantum mechanical correlation (3) be arbitrarily closely approximated by the form (2). The formal proof of this may be set out as follows. We would not worry about failure of the approximation at isolated points, so let us consider instead of (2) and (3) the functions P(a, 3) and -a b where the bar denotes independent averaging of P(a r 7 b') and -a * o over vectors a and o within specified small angles of a and o. Suppose that for all a and o the difference is bounded by : Then it will be shown that e cannot be made arbitrarily small. Suppose that for all a and b \P(a, 3) + a*- t\ < (16) 3 - a - 3 \ < 8 (17) Then from (16) From (2) where P(t, 3) + a- 3 < + 8 (18) P{a, 3) = JdXp(X) A(a, A) 5(3, A) (19) A (a, A) < 1 and \B(&, A) < 1 (20) From (18) and (19), with a = 3, dxp{x) [A(t, A) 3(3, A) + 1] < e + 8 (21) From (19) P(a, 3) - P(a, c) = JdXp(X) [A(a, A) B(t, A) - A(a, A) (c\ A)] = Jdkpik) A(a, A) B(t, A) [1 + A(&, A) i(c, A)] -Jdkp(k) A{a, A) B(c, A) [1 + A{%, A) Bit, A)]

5 Vol. 1, No. 3 ON THE EINSTEIN PODOLSKY ROSEN PARADOX 199 Using (20) then \F{a\ t) -P(a, c) <yda«(a)[l + A(t, A) 5(c, A)] Then using (19) and 21) / d\p(\)[l + A(t, A) Bit, A)] P(^ t) - P(l c)\ < 1 + P(t c) + + S Finally, using (18), a - c - a Si - 2(c + s) < I - S (6 + 8) or 4(f + S)> a«c-a-/5 + -c-l (22) Take for example a c = 0, a 6 = 6 c = l/\/2 Then 4(<r + S) > V2"- 1 Therefore, for small finite S; e cannot be arbitrarily small. Thus, the quantum mechanical expectation value cannot be represented, either accurately or arbitrarily closely, in the form (2). V. Generalization The example considered above has the advantage that it requires little imagination to envisage the measurements involved actually being made. In a more formal way, assuming [7] that any Hermitian operator with a complete set of eigenstates is an "observable", the result is easily extended to other systems. If the two systems have state spaces of dimensionality greater than 2 we can always consider two dimensional subspaces and define, in their direct product, operators o x and a 2 formally analogous to those used above and which are zero for states outside the product subspace. Then for at least one quantum mechanical state, the "singlet" state in the combined subspaces, the statistical predictions of quantum mechanics are incompatible with separable predetermination. VI. Conclusion In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant. Of course, the situation is different if the quantum mechanical predictions are of limited validity. Conceivably they might apply only to experiments in which the settings of the instruments are made sufficiently in advance to allow them to reach some mutual rapport by exchange of signals with velocity less than or equal to that of light. In that connection, experiments of the type proposed by Bohm and Aharonov [6], in which the settings are changed during the flight of the particles, are crucial. / am indebted to Drs. M. Bander and J. K. Perring for very useful discussions of this problem. The first draft of the paper was written during a stay at Brandeis University; I am indebted to colleagues there and at the University of Wisconsin for their interest and hospitality.

6 200 J. S. BELL Vol. 1, No. 3 References 1. A. EINSTEIN, N. ROSEN and B. PODOLSKY, Phys. Rev. 47, 777 (1935); see also N. BOHR, Ibid, 48, 696 (1935), W. H. FURRY, Ibid. 49, 393 and 476 (1936), and D. R. INGLIS, Rev. Mod. Phys. 33, 1 (1961). 2. "But on one supposition we should, in my opinion, absolutely hold fast: the real factual situation of the system S 2 is independent of what is done with the system S 1, which is spatially separated from the former." A. EINSTEIN in Albert Einstein, Philosopher Scientist, (Edited by P. A. SCHILP) p. 85, Library of Living Philosophers, Evanston, Illinois (1949). 3. J. VON NEUMANN, Mathematishe Grundlagen der Quanten-mechanik. Verlag Julius-Springer, Berlin (1932), [English translation: Princeton University Press (1955)]; J. M. JAUCH and C. PIRON, Helv. Phys. Acta 36, 827 (1963). 4. J. S. BELL, to be published. 5. D. BOHM, Phys. Rev. 85, 166 and 180 (1952). 6. D. BOHM and Y. AHARONOV, Phys. Rev. 108, 1070 (1957). 7. P. A. M. DIRAC, The Principles of Quantum Mechanics (3rd Ed.) p. 37. The Clarendon Press, Oxford (1947).

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