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Hidden Variables in Quantum Mechanics

Hidden variables are extra components added to try to banish counterintuitive features of quantum mechanics. There are several different variations of models that describe hidden variables and how they interact with the observable world. This Demonstration covers six possible properties that can be asked of a hidden-variable model. The formal definitions of the six properties are in Details.
There are various relationships among these properties that are described by a Venn diagram with 17 regions. Each region represents a particular combination of these hidden-variable properties. For each region, one can ask if such a hidden-variable model is possible or not.
Tour the Venn diagram. Click a particular region to see whether or not a hidden-variable model satisfying the properties defining that region can be built. The region will be highlighted in green if the answer is yes, and in red if the answer is no. A caption will also appear, describing the math that is underneath the answer. The captions are explained in Details.

SNAPSHOTS

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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.
[5] A. Brandenburger and H. J. Keisler, "Observable Implications of Unobservable Variables," preliminary report, September 2010. http://pages.stern.nyu.edu/~abranden/oiuv-09-25-10.pdf.

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