Wolfram Research

Mathematica MathWorld Wolfram Science More »

HOME

Hidden Variables in Quantum Mechanics


	Hidden variables are extra components added to try to banish counterintuitive features of quantum mechanics. There are many different types 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. There are various relationships among these properties that are described by a Venn diagram with 21 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 or is not possible. Tour the Venn diagram. Hover over the name of a property to see its informal definition (the formal definitions are in the Details section). Click the buttons for the properties you desire. The region this defines will then be highlighted. It will be highlighted in green if hidden-variable models of this type are possible, and in red if no hidden-variable model of this type is possible. Below the diagram you can see which of five underlying theorems is responsible for the possibility or impossibility. The theorems are the three famous no-go theorems of quantum mechanics and two existence theorems. The first no-go theorem is based on the Einstein–Podolsky–Rosen argument, the second is Bell's theorem, and the third is the Kochen–Specker theorem. The two existence theorems and the presentations of the no-go theorems come from the paper [1] "A Classification of Hidden-Variable Properties". (Existence theorems 1 and 2 are theorem 3.1 and 3.2 there.)


	Contributed by: Ariel Ropek Based on work by: Adam Brandenburger and Noson Yanofsky
	Show Initialization Code

Definitions:

Formally, we consider a space

The variables

are measurements, and the variables

are associated outcomes of measurements. There might be several particles: Ann performs a measurement on her particle, Bob performs a measurement on his particle, …. Or,

might describe a case where several measurements are performed on one particle. The definitions to come apply in either case. We take each of the spaces in

to be finite, and suppose that

is a finite product.

Let

be a finite space in 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

An empirical model

and a hidden-variable model

are (empirically) equivalent 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 a given 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.

A hidden-variable model

satisfies single-valuedness if

is a singleton.

A hidden-variable model

satisfies

-independence if for all

whenever

and

then

A hidden-variable model

satisfies strong determinism if, for every

and

, whenever

there is an

such that

and similarly for

etc.

A hidden-variable model

satisfies Weak Determinism if, for every

whenever

there is a tuple

such that

and similarly for

A hidden-variable model

satisfies outcome independence if for all

whenever

and similarly with

and

interchanged, etc.

A hidden-variable model

satisfies parameter independence if for all

whenever

and similarly for

etc.

References:

A. Brandenburger and N. Yanofsky, "A Classification of Hidden-Variable Properties," Journal of Physics A: Mathematical and Theoretical, 41, 2008, 425302.

Adam Brandenburger

Noson Yanofsky

"Hidden Variables in Quantum Mechanics" from The Wolfram Demonstrations Project
http://demonstrations.wolfram.com/HiddenVariablesInQuantumMechanics/

Contributed by: Ariel Ropek

Based on work by: Adam Brandenburger and Noson Yanofsky

Report an issue »

Interact with Demonstrations using the latest version of the free Mathematica Player—Download Now

Browse all topics

	Source page:
	Email	Name (optional)
	Occupation (optional)	Organization (optional)
	I use Mathematica often occasionally never
	Country (optional) Remember me
	Note: Please do not include anything you consider confidential or proprietary. Your message and contact information may be shared with the author of any specific Demonstration for which you give feedback, but will not otherwise be published or distributed. Privacy Policy »

Wolfram Research

Site Index

RSS

Atom