(©Robert DiSalle, 2002)

    As we have seen, Einstein had many philosophical objections to quantum mechanics. Most of them can be considered "external" to quantum mechanics: they stem from the belief that there is, or ought to be, a deterministic account of the phenomena that quantum mechanics considers to be irreduceably statistical. But Einstein  continually attempted to find "internal" objections to the theory, problems that would reveal the inadequacy of the theory itself-- to show, in other words, that the bizarre aspects of the theory were problems with the theory, not accurate reflections of a world that is truly as bizarre as the theory says it is.

In 1935 Einstein, Podolsky, and Rosen produced an argument that quantum mechanics is "incomplete": it fails to give an account, not only of some underlying deterministic reality that some other theory might postulate, but of aspects of reality that ought to come under the purview of quantum mechanics itself.  They wrote a paper called “Can the Quantum-Mechanical Description of Physical Reality Be Considered Complete?” It begins by stating what is meant by "reality":

Criterion of Reality: “If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of reality corresponding to this quantity.” (EPR, 1935.)

Given two physical systems I and II that have interacted, they are described by a common quantum-mechanical state ψ. They are assumed to be sufficiently separated in space to make interaction impossible. In the case of two particles, ψ assigns to their positions a negligible probability of being found within some large (macroscopic) area.
 

By measuring an observable A on system I, we can predict with certainty the result of a measurement of observable P on system II.

By measuring some different observable B on system I, we can predict with certainty the result of a measurement of observable Q on system II.

But observables P and Q are non-commuting, i.e. they cannot have definite values, according to the Uncertainty Principle. (E.g. P is position, Q is momentum.)

Since the measurements on I are made without disturbing II–the two are two far apart to interact–we can conclude that the values of P and Q are both “elements of reality.”

But the quantum mechanical state ψ does not assign definite values to P and Q.

Therefore ψ does not give a complete description of physical reality.

EPR argument does not affect the soundness of quantum mechanics, “which is based on a coherent mathematical formalism covering automatically any procedure of measurement like that indicated. The apparent contradiction in fact only discloses an essential inadequacy of the customary viewpoint of natural philosophy for a rational account of physical phenomena of the type with which we are concerned in quantum mechanics.”

“A criterion like that proposed by EPR contains...an essential ambiguity when it is applied to the actual problems with which we are here concerned. The ambiguity regards the meaning of the expression, “without in any way disturbing the system.”
 

The question is not the mechanical disturbance of one system by the measurement of the other. It is, instead, “an influence on the very conditions which define the possible types of predictions regarding the future behaviour of the system.”

EPR’s argument exploits the fact the quantum mechanics predicts correlations between separated systems, provided that they are described by the same quantum-mechanical state.
 

If two particles are ejected from a central source in opposite directions (say, left and right), their respective behaviors be correlated in seemingly bizarre ways: when each passes between a pair of magnets (assuming the Stern-Gerlach arrangement), the resulting measurements of its spin will show a statistical correlation with those of the other, not apparently based on characteristics of the particular particle or the orientation of the pair of magnets through which it passes, but on the orientation of the magnets relative to the other set of magnets.

The probability that the two particles will have opposite values of spin (call them up vs. down) is governed by a formula based on the angle θ between the two magnet-pair orientations, so that probability = ½cos²(θ/2). When θ= 0 (when both pairs have the same orientation), the particles will have opposite spin values with probability =1; when the angle between the orientations =+120̊, the probability of opposite spins is only 1/4. These predictions allow that the detectors (the magnets) can be sufficiently separated so that the measurement of one of the two particles will not influence the measurement of the other.

Question: Were the spins of the two particles determined before their separation by some unknown factors-- the hidden variables? 

In other words, did the particles receive, before separating, “instructions” that determine their subsequent behavior for particular magnet orientations? The assumption underlying this question is that there is no faster-than-light signalling to "inform" one particle about the orientation of the other magnet, or the spin of the other particle. Just to be sure, we could have an experimental arrangement in which the orientations could be changed while the particles are "in flight".

Bell’s theorem  is a mathematical inequality that expresses the limits on possible correlations that must be obeyed by any hidden variable theory, i.e. the limits of possible correlations that can be produced by "instructions" prepared in advance.

Its ultimate implication is that any specifications of such hidden parameters will necessarily violate at least one of the following principles:
 

1. Locality: Systems that are spacelike separated do not influence on another. Causal influence is transmitted through space at the speed of light, or slower.

2. Separability: The complete description of the state of any system does not include any information about systems that are spacelike separated from it.

3. The predictions of quantum mechanics.

Consider the case of a pair of Stern-Gerlach magnets, with three possible orientations: vertical, + 120^o from vertical, and -120^o. In the case of the constant anticorrelation (when both detectors have the same orientation), the initial states of the particles could contain instructions that will always lead to opposite spins; a particle might thus have the instruction, spin up when the detector is vertical, down if +120^o , and down if -120^o .
 

Since in the case now under consideration θ=0, the anticorrelation can be achieved simply by instructing the other particle, down if vertical, up if +120^o. In other words, these initial states will yield the outcome predicted by quantum mechanics for the case θ=0.) 
 

But these instructions can yield opposite spins only when the detectors have the same orientation. 
 

Since the particles can't know at their creation how the magnets will be oriented relative to each other (and, in any case, the experimenters can change the relative orientations during the flight of the particles), it would have to be assumed at their creation that the orientations will be identical if the prediction were to be realized. 

In short, to match the predictions for the case θ=0, the particles would have to begin in every case with opposite instructions.

But particles instructed to agree with quantum mechanics in this case would violate the predictions of the theory in cases where the detectors had different orientations. 

In the case θ =0 there were three possible orientations, vertical, +120^o and -120^o (call them 1, 2, and 3); when θ≠0, however, there are six possible pairs of orientations: 
 

12, 21, 13, 31, 23, 32. 

If the particles have the instructions described above, which were rigged to yield opposite spins when θ=0 – i.e., up-down-down and down-up-up -then clearly in two of the six possible orientation pairs, namely 23 and 32, the two particles will have opposite spins. 

Moreover, since there could also be pairs with instructions up-up-up and down-down-down, which would always show opposite spins, two out of six is only the minimum value for the probability of opposite spin values.

By Bell's theorem, if the eventual spin measurements of particle pairs are determined by hidden initial states, then in all cases where the detectors have different orientations, spin values will be opposite with a probability of at least 1/3 -as opposed to the 1/4 predicted by quantum mechanics.
 

Experimental result: The probability is 1/4. Quantum mechanics is correct. Local-realist hidden variable theories must be wrong.
 

A more complete mathematical account of Bell's theorem is given by Rae, pp. 37-42.  To get the essential idea behind the theorem, and its philosophical significance, consider the following:
 

Quantum mechanics is incredibly well confirmed by experiments. Therefore any hidden variables theory that is to be considered scientifically viable must reproduce the predictions of quantum mechanics, including the bizarre statistical correlations.

But if it is going to satisfy the philosophical aims of any such theory, it must conform to the principle of local realism: it must assign definite states to particles that are responsible for their future behavior; there must be no changes of state without causal influence, and causal influences cannot be immediate action at a distance

A simple counting argument answers the following question: If we prepare the states of two systems in advance, i.e. give “instructions” for their behavior in possible experiments, what possible correlations can exist between the properties of the two systems?

The answer to this question is “Bell’s inequality”. 

Bell’s inequality is violated by the statistical predictions of quantum mechanics.

Moreover, experiments confirm quantum mechanics, not Bell’s inequality.