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Einstein's Spooks and Bell's Theorem

by John G. Cramer

Alternate View Column AV-37
Keywords: nonlocality, Bell's Theorem, quantum mechanics, Einstein, Podolsky, Rosen, Copenhagen interpretation
Published in the January-1990 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted 6/23/89 and is copyrighted © 1989, John G. Cramer. All rights reserved.
No part may be reproduced in any form without the explicit permission of the author.

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Albert Einstein disliked quantum mechanics, the physical theory that deals with matter and energy at the smallest possible scale. Quantum mechanics, as developed by Heisenberg, Schrödinger, Dirac, and others, had many strange features that ran head-on into Einstein's finely honed intuition and understanding of how a proper universe ought to operate. Over the years he developed a list of objections to the various peculiarities of quantum mechanics. At the top of Einstein's list of complaints was what he called "spooky actions at a distance".

Einstein's "spookiness" is now called nonlocality, the mysterious ability of Nature to enforce correlations between separated but entangled parts of a quantum system that are out of speed-of-light contact, to reach faster-than-light across vast spatial distances or even across time itself to ensure that the parts of a quantum system are made to match. This column is about nonlocality, and how, through Bell's theorem, the nonlocality implicit in nature has been demonstrated in the laboratory.

In 1935 Einstein, with his collaborators Boris Podolsky and Nathan Rosen, published his list of objections to quantum mechanics in what has come to be known as "the EPR paper". The EPR paper lodged three main complaints against quantum mechanics, one of which was nonlocality. EPR argued that "no real change" could take place in one system as a result of a measurement performed on a distant second system, as quantum mechanics required. Einstein and his colleagues also found it unacceptable that in quantum mechanics the measurement of one variable often precludes any knowledge of a second "complementary" variable (for example momentum and position). They further objected to the intrinsic randomness and non-predictivity of quantum mechanics. They argued that the theory must be incomplete, and that it must eventually be replaced by a superior and more complete theory which would observe light's speed limits, would allow all physical quantities to have well-defined values, and would permit accurate prediction of the outcome of any quantum event.

A slow-motion uproar followed the publication of the EPR paper. The founders of quantum mechanics immediately tried to come to grips with the EPR criticisms. Niels Bohr focused this debate on the second of the EPR objections, the simultaneous "reality" of complementary variables like momentum and position. The arguments over this intellectual battleground washed back and forth for the next thirty years. During this ongoing controversy, EPR supporter David Bohm introduced the notion of a "local hidden variable" theory, a partially formulated theory that would replace quantum mechanics with a theoretical structure omitting the paradoxical features of quantum mechanics to which the EPR paper had objected. In Bohm's hidden-variable alternative to quantum mechanics, all correlation were established locally at sub-light speed. Position and momentum were permitted to have simultaneous values, values that were real, but were "hidden" and inaccessible to direct measurement.

Working physicists, however, paid little attention to hidden variable theories. Bohm's approach was far less useful than orthodox quantum mechanics for calculating the behavior of physical systems. Since it was apparently impossible to resolve the EPR/hidden-variable debate by performing an experiment, physicists tended to ignore the whole controversy. They continued to use and improve quantum mechanics without agonizing over whether there were underlying problems gnawing at the roots of the theory. The EPR objections were considered problems for philosophers and mystics, not Real Physicists.

In 1964 this perception changed. John S. Bell, a theoretical physicists working at the CERN laboratory in Geneva, proved an amazing theorem which demonstrated that certain experimental tests could distinguish the predictions of quantum mechanics from those of any local hidden-variable theory. Bell, following the lead of David Bohm, had based his calculations not on measurements of position and momentum, the focus of Einstein's arguments, but on measurements of the states of polarization of photons of light.

Before discussing Bell's theorem further, we should pause to develop some background on the polarization of light. The phenomena we refer to as visible light, infrared or ultraviolet rays, radio waves, X-rays, or gamma rays are all aspects of the same basic physics. They are travelling waves produced when electric and magnetic fields vibrate together at right angles to each other as they move through space at the speed of light. The direction in which the electric field of a light wave vibrates determines the polarization of the wave. If the electric field vibrates always in the same plane, we say that this is the plane of polarization, and that the wave has linear polarization in that plane. If the electric field vibrates so as to trace out a "cork screw" in space, a right or left handed helix as the wave travels on its straight line path, we say that the wave has left or right circular polarization. For the present discussion we will concern ourselves only with linear polarization and will use the term polarization to indicate linear polarization.

It is quite easy to measure polarization of visible light. Special optical filters, for example the lenses of Polaroid(TM) sunglasses, absorb light polarized in one direction while transmitting the light polarized in the perpendicular direction. There are also polarization-sensitive splitters (Nichol prisms) that will divide a beam of light into two beams, one, for example, with vertical polarization and the other with horizontal polarization.

Using such devices, light can be divided into two polarization components. For example, light polarized at some angle to the vertical can be split into a vertical and a horizontal polarization component. Similarly, both vertically and horizontally polarized light can be split into a +45o and -45o components of equal intensity. If a beam of unpolarized light is passed first through one polarization filter and then another, the intensity of the transmitted beam varies in accordance with Malus' Law, which states that the light intensity I(a) is proportional to the square of the cosine of the angle a between the polarization direction of the first filter and that of the second filter, i.e., I(a)=IoCos2(a), where Io is the intensity then the filters are parallel. This equation tells us that when the planes of polarization of the two filters make a right angle, the crossed filters look black and no light is transmitted. When the planes of polarization make an angle of 45o, half the light is transmitted.

In an atom, if an orbital electron is kicked into a higher orbit by an energetic photon or an electrical discharge, the electron returns to its lowest energy state by a process called a cascade, a series of quantum jumps to lower orbits, each jump producing a single light photon of a particular wavelength. A two-photon cascade in which the atom as a whole begins and ends with no net rotational motion is of particular interest, because the cascade produces a pair of photons which have correlated polarizations. When the photons from the cascade travel in opposite directions, the no-rotation restriction requires that if one of the photons is measured to have any definite polarization state, the other photon is required by quantum mechanics to have exactly the same polarization state. Such photon pairs are said to be in entangled quantum states. Experimental tests of Bell's theorem, sometimes called EPR experiments, use entangled photons from such a cascade.

EPR experiments measure the coincident arrival of two such photons at opposite ends of the apparatus, as detected by quantum-sensitive photomultiplier tubes after each photon has passed through a polarizing filter. The photomultipliers at opposite ends of the apparatus produce electrical pulses which, when they occur at the same time, are recorded as a "coincidence" or two photon event. The rate of such coincident events when the polarization directions of the two filters have the values a1 and a2 is measured. Then one or both of the angles are changed and the rate measurement is repeated, until a complete map of rate vs. angles is developed.

Bell's theorem deals with the way in which the coincidence rate of an EPR experiment falls off when the two polarizing angles a1 and a2 are not equal. Bell proved mathematically that for all local hidden-variable theories the decrease in the coincidence rate must be linear (or less) with the angular difference between the two filters. Suppose, for example, that we misalign the angles of the two polarization filters so that the angle between the polarization directions of the two filters is a=a1-a2 . We measure the coincidence rate R(a), as compared to the rate Ro when the filters are perfectly aligned. That rate drops by an amount D1=Ro-R(a). Now we double the amount of the misalignment, so that the decrease in rate is D2=Ro-R(2a). For this situation, Bell's theorem requires that D2 must be less than or equal to twice D1 (D2 <= 2D1).

This prediction of Bell's theorem is one of the so-called "the Bell inequalities". It can be thought of in the following way. Consider that the coincidence rate Ro when the polarizing filters are aligned (a=0) is a "signal", to which "noise" is added when a misalignment is introduced. If the "noise" D1 introduced by moving one filter an amount q to the right is not correlated with the "noise" D1 introduced by moving the other filter by the same angle to the left, then at most, when both sources of noise are present, the noise D2 from a 2a misalignment should be twice D1. However, the two uncorrelated noise sources may occasionally cancel, permitting D2 to be less than twice D1. Therefore, this Bell inequality states that D2 must be less than or equal to twice D1.

Quantum mechanics, on the other hand, predicts that the coincidence rate R(a1,a2) depends only on the relative angle a=a1-a2 between the two polarization directions, and that R(a) obeys Malus' Law. In other words, quantum mechanics predicts that R(a1,a2)=R(a)=RoCos2(a). Therefore, D1=Ro[1-Cos2(a)] and D2=Ro[1-Cos2(2a)]. When the misalignment angle a is fairly small, this means that D2  is about four times D1 and clearly much larger than twice D1 (i.e., D2 ~ 4D1 > 2D1). This is a clear violation of Bell's theorem because the coincidence rate, as predicted by quantum mechanics, falls off much too fast with increasing angle to be consistent with Bell's theorem, which predicts an approximately linear decrease.

When two theories make such distinctly different predictions about outcome of the same experiment, a measurement can be performed to test them. For quantum mechanics and Bell's theorem this crucial EPR experiment has now been performed a number of times. And we find that quantum mechanics always wins. The EPR experiments demonstrate very significant violations of the Bell Inequalities while confirming the predictions of quantum mechanics.

When the first experimental results from EPR experiments became available, they were interpreted as a demonstration that hidden variable theories must be wrong. This interpretation changed when it was realized that Bell's theorem assumed a local hidden variable theory, and that nonlocal hidden variable theories can also violate Bell's theorem and agree with the experimental measurements. The assumption made by Bell that had been put to the test was the assumption of locality, not hidden variables. Locality was in conflict with experiment.

Or, to put it another way, the intrinsic nonlocality of quantum mechanics has been demonstrated by the experimental tests of Bell's theorem. It has been experimentally demonstrated that nature arranges the correlations between the polarization of the two photons by some faster-than-light mechanism that violates Einstein's intuitions about the intrinsic locality of all natural processes. What Einstein called "spooky actions at a distance" are an important part of the way nature works at the quantum level. Einstein's faster-than-light spooks cannot be ignored.

Readers and writers of science fiction will want to know what the nonlocality of nature means for faster-than-light travel and faster-than-light communication. Can we harness Einstein's spooks to drive our spaceships and carry our messages? The answer, alas, seems to be no. Very clever people have tried to exploit the seeming loophole in the lightspeed barrier implied by nonlocality to design superluminal communication devices. See, for example, my column "Paradoxes and FTL Communication" [Analog, Sept-'88]. In every case, including the Calcutta scheme mentioned in that article, these have been shown to be unworkable. Nature seems to have reserved her faster-than-light telephone line for her own private conversations.

References:

EPR:
Albert Einstein, Boris Podolsky, and Nathan Rosen, Physical Review 47, 777-780 (1935).

Bell's Inequalities:
John S. Bell, Physics 1, 195-200 (1964);
John S. Bell, Reviews of Modern Physics 38, 447-452 (1966).

EPR Experiments:
Stuart J. Freedman and John F. Clauser, Physical Review Letters 28, 938-941 (1972).


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