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Home , EPR paradox

Remarks on Bell’s Inequality

Zhengjun Cao1, Lihua Liu2,∗

Abstract. entanglement is of great importance to and computation. So far, all experimental demonstrations of entanglement are designed to check Bell’s inequality which is based on Bell’s formulation for EPR paradox. In this note, we specify the assumptions needed in Bell’s mathematical argument. We then show the contradictions among these assumptions. As a result, it becomes very easy to see that Bell’s inequality is trivial. Keywords. Bell’s formulation for EPR paradox, Bell’s inequality, quantum cryp- tography, quantum computation.

1 Introduction

The question of interpreting the mathematical formulation of has given rise to a variety of different answers from people of different philosophical backgrounds [3, 13, 14]. The most popular interpretation of quantum mechanics is known as the Copenhagen interpretation. It says that a measurement causes an instantaneous collapse of the describing the quantum system, and the system after the collapse is random. did not believe in the idea of genuine in nature, the main argument in the Copenhagen interpretation. In his view, quantum mechanics is incomplete and suggests that there should be ‘hidden’ variables responsible for random measurement results. In 1935, Einstein, Podolsky and Rosen [9] proposed a and condensed the philosophical discussion into a physical argument. In 1964, J. Bell [2] proposed his mathematical formulation for EPR paradox. He constructed a well-known inequality and showed that it was incompatible with the statistical predictions of quantum mechanics. In the past decades, Bell’s formulation has been frequently questioned. The vast majority of opponents are non-academic writers. On the contrary, many experiments have been performed by professionals to test the Bell’s inequality [1, 6–8, 10]. Today, Bell’s inequality is overwhelming in the sciences of and computation [4, 5, 11, 12]. In this note, we put the controversy about Bell’s formulation for EPR paradox aside. We concen- trate on the assumptions needed in Bell’s mathematical argument. We then show the contradictions among these assumptions. Consequently, it is easy to see that Bell’s inequality is trivial.

0 1Department of Mathematics, Shanghai University, Shanghai, China. 2Department of Mathematics, Shanghai Maritime University, Shanghai, China. ∗ [email protected]

1 2 Review of Bell’s formulation and arguments

2.1 The formulation for EPR paradox Consider a pair of one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern-Glerlach magnets, on −→ −→ −→ −→ −→ selected components of the spins σ 1 and σ 2. If measurement of the component σ 1 · a , where a is some unit vector, yields the value +1 then, according to quantum mechanics, measurement of −→ −→ σ 2 · a must yield the value −1 and vice versa. Let this more complete specification be effected by means of parameters λ. Without loss of generality, we write as if λ were a single continuous parameter. The result A of measuring −→σ · −→a −→ 1 is then determined by −→a and λ, and the result B of measuring −→σ · b in the same instance is −→ 2 determined by b and λ, and −→ [Assumption 1] A(−→a , λ) = ±1,B( b , λ) = ±1 (1)

If ρ(λ) is the distribution of λ then the expectation value of the product of the two −→ −→ −→ −→ components σ 1 · a and σ 2 · b is −→ Z −→ [Definition] P (−→a , b ) = ρ(λ)A(−→a , λ)B( b , λ) dλ (2)

This should equal the quantum mechanical expectation value, which for the is −→ −→ −→ −→ −→ −→ h σ 1 · a , σ 2 · b i = − a · b But it will be shown that this is not possible.

2.2 The argument for Bell’s inequality Since ρ is a normalized , R ρ(λ) dλ = 1. Because of the properties (1), −→ P in (2) cannot be less than −1. It can reach −1 at −→a = b only if

A(−→a , λ) = −B(−→a , λ) (3) except at a set of points λ of zero probability. Assuming this, (2) can be rewritten −→ Z −→ P (−→a , b ) = − ρ(λ)A(−→a , λ)A( b , λ)d λ (4)

It follows that −→c is another unit vector −→ Z −→ P (−→a , b ) − P (−→a , −→c ) = − ρ(λ)[A(−→a , λ)A( b , λ) − A(−→a , λ)A(−→c , λ)] dλ (5) Z −→ −→ = ρ(λ)A(−→a , λ)A( b , λ)[A( b , λ)A(−→c , λ) − 1] dλ

−→ −→ using (1), whence |P (−→a , b ) − P (−→a , −→c )| ≤ R ρ(λ)[1 − A( b , λ)A(−→c , λ)] dλ. The second term on −→ the right is P ( b , −→c ), whence −→ −→ |P (−→a , b ) − P (−→a , −→c )| ≤ 1 + P ( b , −→c ) (6)

2 2.3 The argument for the wanted contradiction −→ −→ Let us consider the functions P (−→a , b ) and −−→a · b , where the bar denotes independent aver- −→ −→ −→ aging of P (−→a 0, b 0) and −−→a 0 · b 0 over vectors −→a 0 and b 0 within specified small angles of −→a and −→ −→ b . Suppose that for all −→a and b the difference is bounded by : −→ −→ |P (−→a , b ) + −→a · b | ≤  (7) −→ Suppose that for all −→a and b −→ −→ |−→a · b − −→a · b | ≤ δ Then we have −→ −→ |P (−→a , b ) + −→a · b | ≤  + δ (8) From (2) −→ Z −→ P (−→a , b ) = ρ(λ)A(−→a , λ)B( b , λ) dλ (9) where −→ |A(−→a , λ)| ≤ 1 and |B( b , λ)| ≤ 1 (10) −→ From (8) and (9), with −→a = b ,

Z −→ −→ ρ(λ)[A( b , λ)B( b , λ) + 1] dλ ≤  + δ (11)

From (9),

−→ Z −→ P (−→a , b ) − P (−→a , −→c ) = ρ(λ)[A(−→a , λ)B( b , λ) − A(−→a , λ)B(−→c , λ)] dλ Z −→ −→ = ρ(λ)A(−→a , λ)B( b , λ)[1 + A( b , λ)B(−→c , λ)] dλ Z −→ −→ − ρ(λ)A(−→a , λ)B(−→c , λ)[1 + A( b , λ)B( b , λ)] dλ

Using (10) then

−→ Z −→ Z −→ −→ |P (−→a , b ) − P (−→a , −→c )| ≤ ρ(λ)[1 + A( b , λ)B(−→c , λ)] dλ + ρ(λ)[1 + A( b , λ)B( b , λ)] dλ

Then using (9) and (11) −→ −→ |P (−→a , b ) − P (−→a , −→c )| ≤ 1 + P ( b , −→c ) +  + δ

Finally, using (8) −→ −→ |−→a · −→c − −→a · b | − 2( + δ) ≤ 1 − b · −→c + 2( + δ) −→ −→ √ √ Take for example −→a · −→c = 0, −→a · b = b · −→c = 1/ 2. Then 4( + δ) ≥ 2 − 1. Therefore, for small finite δ,  cannot be arbitrarily small. Thus, the quantum mechanical expectation value cannot be represented, either accurately or arbitrarily closely, in the form (2).

3 3 Remarks on Bell’s arguments −→ Remark 1. The equality (3) holds on the condition −→a = b . Likewise, the equality (4) holds −→ −→ on the condition −→a = b , too. Note that the equality (5) holds on the condition −→a = b = −→c . −→ Thus, the inequality (6) holds on the condition −→a = b = −→c , too. Consequently, the inequality (6) could be rewritten as |P (−→a , −→a ) − P (−→a , −→a )| ≤ 1 + P (−→a , −→a ) By the definition of P and Assumption 1, we find the inequality (6) is as trivial as that 0 ≤ 1. −→ −→ Remark 2. The inequality (11) holds on the condition −→a = b . But it later takes −→a · b = −→ √ −→ −→ √ b · −→c = 1/ 2. Apparently, the assumption −→a = b contradicts the assumption −→a · b = 1/ 2. That means Bell’s mathematical argument is totally illogical.

4 Conclusion

In this note, we remark that Bell’s argument for EPR paradox is illogical because it had confused some assumptions. Besides, we show that Bell’s inequality is a trivial one. For readers’ convenience, we append Bell’s original paper to this note.

References

[1] A. Aspect et al., Phys. Rev. Lett. 47, 460 (1981), 49, 91 (1982); 49, 1804 (1982) [2] J. Bell, On the Einstein-Podolsky-Rosen paradox, Physics, 1, 195-200 (1964) [3] D. Bohm, Quantum Theory, Prentice-Hall (1951) [4] D. Bouwmeester, JW Pan, K. Mattle, M. Eibl, H. Weinfurter, A. Zeilinger, Experimental , Nature vol. 390, 11, pp.575 (1997) [5] I. Bengtsson, K. Zyczkowski, Geometry of Quantum States, An Introduction to , Cambridge University Press (2006) [6] J.F. Clauser and M. A. Horne, Experimental consequences of objective local theories, Phys. Rev. D 10, 526-535 (1974) [7] J.F. Clauser, M.A. Horne, A. Shimony and R. A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23, 880-884 (1969) [8] J.F. Clauser and A. Shimony, Bell’s theorem: experimental tests and implications, Reports on Progress in Physics 41, 1881 (1978) [9] A. Einstein, B. Podolsky, N. Rosen, Can quantum-mechanical description of physical be considered complete, Phys. Rev. 47(10), 777-780 (1935) [10] P. Grangier, G. Roger and A. Aspect, Experimental evidence for a anticorrelation effect on a beam splitter: a new light on single-photon interferences, Europhysics Letters 1, 173-179 (1986) [11] M.B. Plenio, S. Virmani, An introduction to entanglement measures, Quant. Inf. Comp. 7, 1 (2007) [12] E.G. Steward, Quantum Mechanics: Its Early Development and the Road to Entanglement, Imperial College Press (2008) [13] M. Tegmark, Parallel Universes, Scientific American, May (2003) [14] R. Tsekov, Bohmian Mechanics versus Madelung Quantum Hydrodynamics, arXiv.0904.0723 (2009)

4 Physics Vol. 1, No. 3, pp. 195-290, 1964 Physics Publishing Co. Printed in the United States

ON THE EINSTEIN PODOLSKY ROSEN PARADOX*

]. S. BELLt Department of Physics, University of Wisconsin, Madison, Wisconsin

(Received 4 November 1964)

I. Introduction

THE paradox of Einstein, Podolsky and Rosen [1] was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional vari­ ables were to restore to the theory and locality [2]. In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics. It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential dif­ ficulty . There have been attempts [3] to show that even without such a separability or locality require­ ment no "hidden variable" interpretation of quantum mechanics is possible. These attempts have been examined elsewhere [ 4] and found wanting. Moreover, a hidden variable interpretation of elementary quan­ tum theory [S] has been explicitly constructed. That particular interpretation has indeed a grossly non­ local structure. This is characteristic, according to the result to be proved here, of any such theory which reproduces exactly the quantum mechanical predictions.

II. Formulation With the example advocated by Bohm and Aharonov [6], the EPR argument is the following. Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern-Gerlach magnets, on selected components of the

Spins d l and a2 , If measurement Of the component d I ' a, where a is some unit vector, yields the value + 1 then, according to quantum mechanics, measurement of a2 ·a must yield the value -1 and vice versa. Now we make the hypothesis [2], and it seems one at least worth considering, that if the two measure­ ments are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other. Since we can predict in advance the result of measuring any chosen compo­ nent of a2 , by previously measuring the same component of d 1 , it follows that the result of any such measurement must actually be predetermined. Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state. Let this more complete specification be effected by means of parameters A. It is a matter of indiffer­ ence in the following whether A. denotes a single variable or a set, or even a set of functions, and whether the variables are discrete or continuous. However, we write as if A were a single continuous parameter.

The result A of measuring a1 ·a is then determined by a and A., and the result B of measuring 7, 2 • b in the same instance is determined by b and A., and

*Work supported in part by the U.S. Atomic Energy Commission tan leave of absence from SLAC and CERN

195 196 ]. S. BELL Vol. 1, No. 3

A (l:i, ,\) = ± 1, B (b, ,\) = ± 1. (1)

The vital assumption [2] is that the result B for particle 2 does not depend on the setting ;;, of the magnet for particle 1, nor A on b. If p (,\) is the probability distribution of ,\ then the expectation value of the product of the two com- ~ ~ ~ 7 ponents o 1 • a and o 2 • D is

P (t;, b) = .fn p (,\)A (~, ,\) B (b, ,\) (2)

This should equal the quantum mechanical expectation value, which for the singlet state is

(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 ,\ stands for any number of vari­ ables 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 ,\ can then be thought of as initial values of these variables at some suitable instant.

111. 111 us trot ion 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 ,\ with uniform probability distribution over the hemisphere A · p > 0. Specify that the result of measurement of a component ; · ;; is

-> -> I sign ,\ · a , (4)

where -.;i is a unit vector depending on ~ 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'_.= 0, but as the probability of this is zero we will not make special prescriptions for it. Averaging over ,\ the expectation value is

< ; · ~ > = 1 - 2 e' Irr , (5)

where () 1 is the angle between t;' and p. Suppose then that "ii' is obtained from ; by rotation towards P until 2 ()' 1 - cos () (6) TT

-> -> where () is the angle between a and p. Then we have the desired result

< ; . ;; > = cos () (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. Vol. 1, 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) comm only used in verbal discussions of this problem:

...... (...... P (a, a) = - P a, - a) = - 1 l (8) P

For example, let A now be unit vector A, with uniform probability distribution over all directions, and take

A(a, A) = sign a. A t (9) B (a, b) = - sign b · A \

This gives

P

(11)

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 e = 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 b and a respectively as well as on a and b. For ex­ ample, replace a in (9) by a', obtained from a by rotation towards b until

2 I 1 - - e = cos e, Tr where e' is the angle between a' and b. 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(A.) = l, (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, >..) = - B (a, A) (13) except at a set of points A of zero probability. Assuming this, (2) can be rewritten

P(a, b) = - fa>..p(A.) A(a, A) A(b, A). (14) 198 ]. S. BELL Vol. 1, No. 3

It follows that c is another unit vector

P(a, b) -P(;, c) = - fi1i.p(A) [A(a, A) A(b, A) -A(a, A) A(c, ,\))

= fi1i. p(A) A(a, ,\) A(b, ,\) [A (b, ,\) A(c, ,\) -1) using (1), whence

1 P(a, "h) -P(a, c) 1 s f

The second term on the right is P (b, c), whence

1 + p <"h, c) 2 1P

Unless p is constant, the right hand side is in general of order I b-cI for small I b-cI . Thus p (b, c) cannot be stationary at the minimum value (-1 at b = 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, 6) and _;;;. Ii

. d d . f p ( .... , 71) d ->1 71 ->1 d 71 . • where the bar d enotes 1n epen ent averaging o a, o an -a · o over vectors a an o within sp~c- ified small angles of a and b. Suppose that for all ;; and b the difference is bounded by €: - 1p

Then it will be shown that € cannot be made arbitrarily small. Suppose that for all a and b

(17)

Then from (16) IPC.i, t,) +;;;·"his (+a (18)

From (2) p

where I A

From (18) and (19), with ;; = b, I d,\p(A) [A(b, A) B(b, A) + 1) s ( + 8 (21) From (19) p 'B

= f:.>.p<1i.> .4<8, >.) 'B.p(A) A.)'Bet,,>.)] Vol. 1, No. 3 ON THE EINSTEIN PODOLSKY ROSEN PARADOX 199

Using (20) then

1?c:;, b) -?c:;, ;;)1 .'.S ;:11... ru,) [1 + A"ci, A) 8c;;, A)]

+ /aAp(A) [1 + A(b, A) B(b, A)]

Then using (19) and 21)

Finally, using (18),

I :; · ;; - :; · "b I - 2 (E + o) < 1 - "b · ;; + 2 (E + o) or

~ ~ ~ 7 7 ~ 4 (E + 0) 2: I a · c - a · b I + b • c - 1 (22)

Take for example a· (; = 0, a· b = b · (; = l/ y'2 Then

4 CE + o) ;::, v2 - 1

Therefore, for small finite o, € cannot be arbitrarily small. Thus, the quantum mechanical expectation value cannot be represented, either accurately or arbitrar­ ily 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 oper­ ator with a complete set of eigenstates is an "", 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 dimen­ sional subspaces and define, in their direct product, operators d1 and d 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 set­ ting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal involved must pr~pagate 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 suffi­ ciently 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. I am indebted to Drs. M. Bander and]. 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. 200 ]. 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).