Hidden Variables in Quantum Mechanics

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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.
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Contributed by: Adam Brandenburger, Ariel Ropek, and Andrei Savochkin (April 2011)
After work by: Adam Brandenburger, H. Jerome Keisler, and Noson Yanofsky
Open content licensed under CC BY-NC-SA
Snapshots
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.
Permanent Citation