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.

[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.