Quantum correlation

# Quantum correlation

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Encyclopedia
In Bell test experiments
Bell test experiments
The Bell test experiments serve to investigate the validity of the entanglement effect in quantum mechanics by using some kind of Bell inequality...

the term quantum correlation has come to mean the expectation value of the product of the outcomes on the two sides. In other words, the expected change in physical characteristics as one quantum system passes through an interaction site. In the paper that inspired the Bell tests -- John Bell's
John Stewart Bell
John Stewart Bell FRS was a British physicist from Northern Ireland , and the originator of Bell's theorem, a significant theorem in quantum physics regarding hidden variable theories.- Early life and work :...

of 1964 -- it was assumed that the outcomes A and B could each only take one of two values, -1 or +1. It followed that the product, too, could only be -1 or +1, so that the average value of the product would be:
(N++ + N-- - N+- - N-+)/Ntotal

where, for example, N++ is the number of simultaneous occurrences ("coincidences") of the outcome +1 on both sides of the experiment.

In actual experiments, though, detectors are not perfect and there are usually many null outcomes. The correlation can still be estimated using the sum of coincidences, since clearly zeros will not contribute to the average, but in practice instead of
dividing by Ntotal it has become customary to divide by the total number of observed coincidences,
(N++ + N-- + N+- + N-+)

The legitimacy of this method relies on the assumption that the observed coincidences constitute a fair sample of the emitted pairs.

Following local realist assumptions as in Bell's 1964 paper, the estimated quantum correlation will converge after a sufficient number of trials to:
QC(a, b) = ∫ dλ ρ(λ) A(a, λ)B(b, λ)

where a and b are detector settings and λ is the hidden variable
Hidden variable theory
Historically, in physics, hidden variable theories were espoused by some physicists who argued that quantum mechanics is incomplete. These theories argue against the orthodox interpretation of quantum mechanics, which is the Copenhagen Interpretation...

, drawn from a distribution ρ(λ).

The quantum correlation is the key statistic
Statistic
A statistic is a single measure of some attribute of a sample . It is calculated by applying a function to the values of the items comprising the sample which are known together as a set of data.More formally, statistical theory defines a statistic as a function of a sample where the function...

in the CHSH
CHSH inequality
In physics, the CHSH Bell test is an application of Bell's theorem, intended to distinguish between the entanglement hypothesis of quantum mechanics and local hidden variable theories. CHSH stands for John Clauser, Michael Horne, Abner Shimony and Richard Holt, who described it in a much-cited...

and some of the other "Bell inequalities", tests of which open the way for experimental discrimination between quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

on the one hand and local realism or local hidden variable theory
Local hidden variable theory
In quantum mechanics, a local hidden variable theory is one in which distant events are assumed to have no instantaneous effect on local ones....

on the other.