Difference between CHSH and Bell inequalities

[harpo]

Bell's Inequality, P(a,b)-P(a,d)+P(c,b)+P(c,d) is calculated as:
S = a*b - a*d + c*b + c*d <= 2.
It is valid for all values of a, b, c and d between -1 and +1
It is also valid for counts, a=a's counts/total counts of a,b,c &d.
b, c,and d are derived similarly. Negative counts are not allowed
and make Bell's Inequality invalid.

That is very different from the CHSH Inequality that is calculated as:
N = Number of each type of pairs of detections
E = (N11 + N00 - N10 -N01) / (N11 + N00 + N10 + N01)
S = E1 - E2 + E3 +E4 <= 2

How does the CHSH Inequality relate to Bell's Inequality?

 

[facenian]

That is very different from the CHSH Inequality that is calculated as:
N = Number of each type of pairs of detections
E = (N11 + N00 - N10 -N01) / (N11 + N00 + N10 + N01)
S = E1 - E2 + E3 +E4 <= 2

How does the CHSH Inequality relate to Bell's Inequality?

CHSH in not very different from Bell's inequality, the difference is that Bell's E1,E2,E3 and E4 are idealized averages while CHCH' s E_i are experimental quantities

 

[harpo]

Are you saying that the CHSH experiments actually use Bell's Inequality?

 

[Nugatory]

Are you saying that the CHSH experiments actually use Bell's Inequality?

No, CHSH experiments are done with the CHSH inequality. However, the two inequalities are derived from the same starting assumptions so an observed violation of either leads to the same conclusion: one or more of these assumptions must be incorrect.

 

[facenian]

Are you saying that the CHSH experiments actually use Bell's Inequality?

Yes, but in an indirect way. The CHSH inequality is equivalent to the inequality Bell introduced in his 1964 paper. Later Bell adpoted the CHSH form too.
I assume the Bell inequality is the one he presented in 1964 but I don't really know what you mean by Bell inequality

 

[wle]

CHSH is the inequality $$\begin{equation}
\langle A_{1} B_{1} \rangle + \langle A_{1} B_{2} \rangle + \langle A_{2} B_{1} \rangle - \langle A_{2} B_{2} \rangle \leq 2
\end{equation}$$ that holds for the expectation values $\langle A_{x} B_{y} \rangle$ assuming the outcomes are attributed values $+1$ and $-1$ and that they are produced by a local hidden variable model (i.e., assuming $\langle A_{x} B_{y} \rangle = \int \mathrm{d}\lambda \, \rho(\lambda) A_{x; \lambda} B_{y; \lambda}$ with $\rho$ independent of $x$ and $y$). These can be computed according to some theory (to compare the LHV hypothesis with e.g. quantum mechanics) or estimated by detection counts in an experiment (to compare the LHV hypothesis with reality). It's the same CHSH inequality that is being tested in either case (though you need to do some additional statistical analysis to really do the experimental test rigorously).

Bell's inequality (the inequality Bell originally derived in 1964) is a special case of CHSH that holds when you happen to have perfect correlation (or anticorrelation) for one pair of measurements. For example, substitute $\langle A_{1} B_{2} \rangle = 1$ in CHSH above and rearrange and you get $$\begin{equation}
1 - \langle A_{1} B_{1} \rangle \geq \langle A_{2} B_{1} \rangle - \langle A_{2} B_{2} \rangle \,.
\end{equation}$$ Apart from different notations and conventions (e.g., it doesn't matter which measurement outcomes you call +1 and -1) this is the same as Eq. (15) in Bell's 1964 paper. Bell's inequality is not considered suitable for experiments because perfect correlation is an impossible ideal (there is always some imprecision and noise in an experiment), but it is fine for the theoretical comparison with quantum mechanics that Bell used it for.

In general Bell inequalities are linear inequality constraints on the probabilities that hold for local hidden variable models and generally don't hold if you don't assume a local hidden variable model. There are lots of Bell inequalities that are known besides CHSH. For example, the GHZ paradox can be expressed in the form of a Bell inequality (the Mermin inequality) involving $\pm 1$-valued measurements at 3 different spatial locations: $$\begin{equation}
\langle A_{1} B_{1} C_{1} \rangle - \langle A_{1} B_{2} C_{2} \rangle - \langle A_{2} B_{1} C_{2} \rangle - \langle A_{2} B_{2} C_{1} \rangle \leq 2 \,.
\end{equation}$$ The inequality holds assuming $\langle A_{x} B_{y} C_{z} \rangle = \int \mathrm{d}\lambda \, \rho(\lambda) A_{x; \lambda} B_{y; \lambda} C_{z; \lambda}$. The left-hand side can attain the algebraic bound of 4 using the same GHZ state and measurements ($\sigma_{x}$ and $\sigma_{y}$) as in the GHZ paradox. (Also notable is that this inequality can be seen as a generalisation of CHSH: substitute e.g. $C_{1} = +1$ and $C_{2} = -1$, i.e., do only trivial "measurements" at the 3rd site that always produce the same results, and you exactly recover CHSH.)