Set of equations distinguishing between quantum mechanics, in which particle properties do not exist until they are measured, and any local deterministic theory holding that the properties exist, but are hidden until they are observed. The equations also suggest an experimental method for determining which theory more accurately reflects reality.

Consider the EPR paradox. A neutral pion is at rest in a vacuum. It decays into two photons which, by conservation of spin, have anticorrelated spins: one spin up and one spin down, for a net of zero spin. By conservation of momentum, they will head off in opposite directions at the speed of light. Once they are far away from one another (perhaps a mile or two) each photon encounters a Stern-Gerlach analyzer which is set, at random and independently of the other analyzer, to one of three measurement axes. One analyzer is manned (womaned?) by Alice; the other is manned by Bob.

Assume for the moment a local, deterministic theory - there is no "spooky action at a distance," and physical properties exist before they are measured. Operating under such a theory, each photon must somehow contain information as to its behavior should it encounter an analyzer set to any of the three positions. For the purpose of keeping our conclusions general, we'll abstain from stating any particular method in which this information might be encoded. We'll also call the three possible axes X, Y, and Z, though we do not mean to imply by this that they are mutually perpendicular. They could be oriented any-which-way.

Under these assumptions, and keeping in mind that the photons' spins are anticorrelated, there are eight possible deterministic "instruction sets" that the photons could possess, each with an associated probability of showing up in experiment:

Possibility-||Axis measured|Alice|Bob|

P1-||X|U|D|
P1-||Y|U|D|
P1-||Z|U|D|

P2-||X|U|D|
P2-||Y|U|D|
P2-||Z|D|U|

P3-||X|U|D|
P3-||Y|D|U|
P3-||Z|U|D|

P4-||X|D|U|
P4-||Y|U|D|
P4-||Z|U|D|

P5-||X|U|D|
P5-||Y|D|U|
P5-||Z|D|U|

P6-||X|D|U|
P6-||Y|U|D|
P6-||Z|D|U|

P7-||X|D|U|
P7-||Y|D|U|
P7-||Z|U|D|

P8-||X|D|U|
P8-||Y|D|U|
P8-||Z|D|U|

Now, since these are probabilities of a certain photon pair being measured (it doesn't matter what the probabilities actually ARE), we can express the probability of a certain final outcome as a sum of two of the above probabilities. For example, the probability that Alice's photon arrives to find her detector in the X position and is measured as spin up AND that Bob's photon arrives at a Y position detector and is also measured as spin up is given by

P(X,U;Y,U)=P3+P5

...the sum of the probabilities of the only two instruction sets that permit this. Because probabilities can never be non-real or negative, we can accurately state that

P3+P5 ≤ P3+P5+P2+P7

...no matter what the actual probabilities are. We'll just rewrite this to

P3+P5 ≤ (P2+P5)+(P3+P7)

...in a bit of simple arithmetic. Converting the sums of these new pairs of probabilities into the probability of the actual outcome that they represent, we get

P(X,U;Y,U) ≤ P(X,U;Z,U)+p(Y,U;Z,U)

That is, the probability that both photons will be measured to be spin up when Alice is measuring on axis X and Bob is measuring on axis Y is less than or equal to the probability of the same outcome when Alice is measuring on X and Bob on Z, plus the probability of both coming out spin up when Alice measures Y and Bob measures Z. Because we have formulated all our arguments in the most general possible terms, any local "hidden variables" theory satisfies this inequality, regardless of how it states the information is encoded and regardless of how the axes are chosen.

In contrast, the quantum theory does not satisfy this inequality for some axis choices. For instance, when all three axes are chosen to be coplanar, with X perpendicular to Y and Z between them at a 45-degree angle to both, quantum mechanics predicts

(1/2*(sin2(90/2)) ≤ (1/2*(sin2(45/2))+(1/2 *(sin2(45/2))

or

.25 ≤ .146

...which is patently false. Quantum mechanics violates the Bell inequality; thus, it is fundamentally different than any local "hidden variables" theory, providing different predictions for the same problem.

This experiment (and others like it) has recently been performed to the satisfaction of the great majority of the scientific community, once involving "Alice" and "Bob" detectors as much as seven miles apart. Almost all experimental results verify the predictions made by quantum mechanics. Reality violates the Bell inequalities; thus, all local deterministic theories fail to represent reality accurately. Any successful theory of quantum-scale phenomena must be either non-local, and thus challenge relativity directly, or non-deterministic, like modern quantum mechanics.

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