Details

Formally, we consider a space

.

The variables are measurements and the variables are associated outcomes of measurements. The choice of symbols is meant to represent a situation in which Alice performs a measurement on her particle, Bob performs a measurement on his particle, and so on. We take each of the spaces in to be finite, and suppose that is a finite product.

Let be a finite space on which a hidden variable lives. The overall space is then

An empirical model is a pair , where is a probability measure on . A hidden-variable model is a pair , where is a probability measure on . A hidden-variable model realizes an empirical model if for all ,

if and only if ,

and when both are nonzero,

.

We can calculate , for , from the formula

.

From this, we see that the idea of equivalence is to reproduce the probability measure on the space by averaging under a probability measure on an augmented space , where includes a hidden variable. The measure is then subject to various conditions.

1. A hidden-variable model satisfies single-valuedness if is a singleton.

2. A hidden-variable model satisfies -independence if for all ,

.

3. A hidden-variable model satisfies parameter independence if for all , whenever ,

,

and similarly for , and so on.

4. A hidden-variable model satisfies outcome independence if for all , whenever ,

and similarly with and interchanged, and so on.

5. A hidden-variable model satisfies weak determinism if, for every , whenever , there is a tuple such that .

6. A hidden-variable model satisfies strong determinism if, for every and , whenever , there is an such that , and similarly for and , and so on.

The text that appears in the diagram:

● "E1" refers to Theorem 3.1 in [4].

● "E2" refers to Theorem 3.2 in [4].

● "EPR" refers to the famous Einstein–Podolsky–Rosen argument; see [2].

● "Bell" refers to the famous Bell's theorem; see [1].

● "Sig." refers to any empirical model that is signaling; see [3] for the definition.

References

[1] J. Bell, "On the Einstein–Podolsky–Rosen Paradox," Physics, 1, 1964 pp. 195–200.

[2] A. Einstein, B. Podolsky, and N. Rosen, "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?" Physical Review, 47, 1935 pp. 770–780.

[3] G. C. Ghirardi, A. Rimini, and T. Weber, "A General Argument against Superluminal Transmission through the Quantum-Mechanical Measurement Process," Lettere al Nuovo Cimento, 27, 1980 pp. 293–298.

[4] A. Brandenburger and N. Yanofsky, "A Classification of Hidden-Variable Properties," Journal of Physics A: Mathematical and Theoretical, 41, 2008. doi:10.1088/1751-8113/41/42/425302.