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.
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