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}*(sin^{2}(90/2)) ≤
(^{1}/_{2}*(sin^{2}(45/2))+(^{1}/_{2
}*(sin^{2}(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.