The EPR Paradox and Quantum Nonlocality (Part 1)

In 1935, Einstein along with two colleagues, Boris Podolsky and Nathan Rosen, devised a thought experiment.[1]  This has since been called the EPR experiment based on the first initials of their names.  Einstein, Podolsky and Rosen (E-P-R) imagined a quantum system (atom) which could be ruptured such that two electrons were dispatched to two differing measurement devices. Each electron would carry a property called 'spin'. Since the atom had zero spin, this meant one would have spin (+ 1/2), the other (-1/2). The diagram below illustrates this, the atom being disrupted inside the box, with its opposing spin electrons sent to the left and right.
 
A1 (+ ½ ) <----------->(- ½ )A2


Orthodox quantum mechanics forbade the simultaneous measurement of a property (say different spin states) for the same system. If you got one, you could not obtain the other. This was a direct outcome of the Heisenberg Indeterminacy Principle which stated that simultaneous quantum measurements could not be made to the same precision.

 E-P-R argued that this showed the incompleteness of quantum mechanics. It was not the 'paragon' of physical theories its apologists claimed, if such indeterminacy was fundamentally embedded within it. At the same time they conceived an 'experiment' in which both spins could be identified - with the sole assumption that both were in definite states from the instant of their parent atom's disruption.

 Then, we need only know one spin to obtain the other. Say we know or can measure one spin to be (+1/2).[2] Since the total atomic spin is zero, the other electron must have spin (-1/2) since:   (-1/2) + (+1/2) = 0

 Thus, we manage to skirt the Indeterminacy Principle, and obtain both spins simultaneously without one measurement disturbing the other. We gain completeness, but at a staggering cost. Because this simultaneous knowledge of the spins implied  that information would have had to propagate from one spin measuring device (on the left side) to the other (on the right side) instantaneously! This was interpreted to mean faster-than-light communication, which violates special relativity. In effect, a 'paradox' ensues: quantum theory attains completeness only at the expense of another fundamental physical theory - relativity.

 By this point, Einstein believed he finally had Bohr by the throat. Figuring Bohr might come up with some trick or sly explanation, Einstein went one better at the 6th Solvay Conference held in 1930, actually designing a device (see image on next page) that he was convinced would have Bohr in tears trying to find a solution. (According to reports, it very nearly did, and a number of participants insisted Neils was in a state of "shock" believing there was no real solution.)

Using his hypothetical (purely on paper) thought device, Einstein wanted to put to rest once and for all the notion that quantum mechanics was complete, or was in any way a proper science. The mechanical device contrived by Einstein was designed as a counter-example to the Heisenberg Uncertainty principle for energy and time which states:

ΔE Δ t ³  h/2π

Fig. 1: Einstein’s Thought Experiment Device

In the device, a weight scale is located and one can see it when a door (front of box) opens, with the door controlled by a clock timer. Whenever the door flaps open, even for a split second, one photon escapes and the weight difference (between original box and after) can be computed using Einstein's mass-energy equation, e.g.: m = E/ c². Thus, the difference is taken as follows:

Weight(before door opens) - weight (after - with 1 photon of mass m = E/ c²   gone)

     Thus, since the time for brief opening is known (Δ t) and the photon's mass can be deduced from the above weight difference, Einstein argued that one can in principle  find both the photon's energy and time of passage to any level of accuracy without any need for the energy-time uncertainty principle.

     In other words, the result could be found on a totally deterministic basis!

     Bohr nearly went crazy when he studied the device, and for hours worried there was no solution and maybe the wily determinist was correct after all. When Bohr did finally come upon the solution, he realized he'd hoisted the master on his own petard.

     The thing Einstein overlooked was that his very act of weighing the box translated to observing its displacement (say, dr = r2 - r1) within the Earth's gravitational field. But in Einstein's general theory of relativity, he'd found that clocks actually do run slower in gravitational fields (a phenomenon called 'gravitational time dilation') In this case, for the Earth, one would have the fractional difference in proper time, as a fraction of time passage t:

dt/ t » GM(1/r1 - 1/r2) » g(dr)/ c²

where G is the Newtonian gravitational constant, M is the Earth's mass, and g is the acceleration of gravity (g = 980 cm/ sec² in cgs) and c = 3 x 10¹⁰ cm/sec.

Let us say the box deflection (r2 - r1)was 0.001 mm = 0.0001cm, then:

dt/t ~ (980 cm/s²)(10^-4 cm)/ (3 x 10¹⁰ cm/sec )²

dt/t »  10^-22

and for an interval say t = 0.01 sec,  and:

 

dt =   (10^-22 )(0.01 sec) = 10^-24 sec

In other words, the observation would actually generate a time uncertainty of 10^-24  sec- and hence an uncertainty dE in the energy of the photon. In other words, after the displacement (r2 - r1) arising from the measurement, the clock is in a gravitational field different from the original one. (The Energy uncertainty can meanwhile be computed from the Heisenberg energy -time relation to be dE  » 10^-10 J)

Quantum theory prevails again!

 Einstein's challenges to Bohr in the aftermath were all kind of half-hearted and had nowhere near the intensity of his clock-door device work of art. Rather than join happily with other QM theorists at the last Solvay Conference in 1933 Einstein - the perpetual determinist- remained on the sidelines "feeling the same uneasiness as he had before".

He went to work separately, on a "unified field theory" while the quantum theory edifice was formulated to its present maturity without him.

 Aside:   Most people outside physics are unaware there were seven Solvay Conferences in all, in the course of which the essential underpinnings and core interpretation of quantum mechanics was thrashed out – leading to the Copenhagen Interpretation.

 In the orthodox Copenhagen (and most conservative) interpretation of quantum theory, there can be no separation of observed (e.g. spin) state until an observation or measurement is made. Until that instant (of detection) the states are in a superposition, as described above.   More importantly, the fact of superposition imposes on all quantum phenomena an inescapable ‘black box’. In other words, no information other than statistical can be extracted before observation.

2. Bell's Theorem and the Aspect Experiment

 Years later, mathematician John S. Bell asked the question: 'What if the E-P-R experiment could actually be carried out? What sort of mathematical results would be achieved?' In a work referred to as "the most profound discovery in the history of science", Bell then proceeded to place the E-P-R experiment in a rigorous and quantifiable context, which could be checked by actual measurements.

 In a landmark theoretical achievement in 1964, Bell formulated a thought experiment based on a design similar to that shown at the opening of the chapter. He made the basic assumption of locality (i.e. that no communication could occur between A1 and A2 at any rate faster than light speed). In what is now widely recognized as a seminal work of mathematical physics, he set out to show that a theory which upheld locality could reproduce the predictions of quantum mechanics. His result predicted that the above sum, S, had to be less than or equal to 2 (S £ 2). This is known as the Bell Inequality.

 To test quantum conformity to Bell's Theorem, Alain Aspect and his colleagues at the University of Paris, set up an arrangement as sketched below.[3]  In these experiments, the detection of the polarizations[4] of photons was the key. These were observed with the photons emanating from a Krypton-Dye laser and arriving at two different analyzers, e.g.

 

(P1) A1 ¯| <------------> |­ A2 (P2)

                                              

Here, the laser device is D, and the analyzers (polarization detectors) are A1 and A2 along with two representative polarizations given at each, for two photons P1 and P2. The results of these remarkable experiments disclosed apparent and instantaneous connections between the photons at A1 and A2. In the case shown, a photon (P1) in the minimum (0) intensity polarization mode, is anti-correlated with one in the maximum intensity (1) mode.

 Say, twenty successive detections are made and we obtain, at the respective analyzers (where a ‘1’ denotes spin  +1/2 detection and ‘0’ spin  (-1/ 2):

A1:   1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

A2:   0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

 

On inspection, there is a 100% anti-correlation (i.e. 100% negative correlation) between the two and an apparent nonlocal connection. In practice, the experiment was set out so that four (not two - as shown) different orientation 'sets' were obtained for the analyzers. These might be denoted: (A1,A2)I, (A1,A2)II, (A1,A2,)III, and (A1,A2)IV.

 Each result is expressed as a mathematical (statistical) quantity known as a 'correlation coefficient'.[5] The results from each orientation were then added to yield a sum S:

S = (A1,A2)I + (A1,A2)II + (A1,A2,)III + (A1,A2)IV

 In his (1982) experiments, Aspect determined the sum with its attendant indeterminacy to be:[6]   S = 2.70 ±  0.05 and in so doing experimentally validated Bell’s Inequality.

 The crucial significance of the Aspect experiment (1982) and earlier the Clauser experiment in 1969, had not been lost on physicist David Bohm.  Both experiments reinforced for him that a novel concept had to be introduced to account for these nonlocal results. The old answer of the Copenhagen theorists: 'Leave it alone!', wouldn't do any more. In essence, Bohm had become convinced that the entrenched vagueness and philosophical abdication of the standard Copenhagen Interpretation could no longer be supported.

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    [1] Einstein, Podolsky, and Rosen.: Physical  Review, 777.

[2] More technically, this is what is referred to as ‘the z-component of electron spin’, since the electron is visualized as a spinning top, with z-axis (i.e. component) in the axial or z-direction.

    [3] Aspect, Grangier, and Roger: Physical Review Letters, 91.

    [4] Polarization is the orientation in space of the electric field E, associated with light. This can be altered, subject  to the imposition of different filters and devices.

    [5] For example, if a set of data: 1, 1, 1, 1 is correlated with another set: 0.5, 0.5, 0.5, 0.5, the correlation coefficient is 1.0. The range is between 0 (no correlation) and 1.0 (perfect correlation).

    [6] Aspect, A. et al, op. cit.