Definition of locality in Bell experiments

Question

Continuing from my previous question on Brunner et al.'s paper; so given a standard Bell experimental setup:

where independent inputs $x,y \in \{0, 1\}$ decide the measurement performed by Alice & Bob on quantum state $S$ with outcomes $a,b \in \{-1, 1\}$, $a$ and $b$ are correlated (not independent) if:

(1) $P(ab|xy) \ne P(a|xy)P(b|xy) \ne P(a|x)P(b|y)$

Of course, there are perfectly innocent non-quantum reasons why $a$ and $b$ could be correlated; call these reasons confounding variables, some artifact of when Alice & Bob's systems interacted in the past. The set of all confounding variables we call $\lambda$. If we take into account all variables in $\lambda$, a local theory claims that $a$ and $b$ will become independent and thus $P(ab|xy)$ will factorize:

(2) $P(ab|xy,\lambda) = P(a|x,\lambda)P(b|y,\lambda)$

This equation expresses outcomes depending only on their local measurement and past variables $\lambda$, and explicitly not the remote measurement.

Question one: what is the mathematical meaning of the comma in equation (2)?

Question two: what is an example of a variable in $\lambda$?

The paper then says the following:

The variable $\lambda$ will not necessarily be constant for all runs of the experiment, even if the procedure which prepares the particles to be measured is held fixed, because $\lambda$ may involve physical quantities that are not fully controllable. The different values of $\lambda$ across the runs should thus be characterized by a probability distribution $q(\lambda)$.

Question three: why was it a set of variables before but is now only a single variable?

Question four: what is an example of a probability distribution for $q(\lambda)$ here?

We then have the fundamental definition of locality for Bell experiments:

(3) $P(ab|xy) = \int_{Λ} q(\lambda)P(a|x, \lambda)P(b|y, \lambda) d\lambda$ where $q(\lambda|x,y) = q(\lambda)$

Question five: What does the Λ character mean under the integral sign?

General question: so we have a continuous probability distribution $q(\lambda)$ over which we're integrating. Why are we multiplying the RHS of equation (2) by $q(\lambda)$ in the integrand? That would seem to make equation (3) different than equation (2). What's an example of this integral with concrete values?

Answer 1

I think that I can explain the definition through the following simple example:

Suppose that you perform two experiments in the same house in two separate rooms. In the first you measure the observable $A$ and instantaneously in the second you measure the observable $B$. The measurements are afflicted with noise, so you do not get a definite answer every time you measure but a distribution $P(A)$ and $P(B)$.

Now, as very well known that the noise is temperature dependent (we suppose that the temperature is uniform throughout the house), so when you repeat your experiment at a different teperature you observe that you obtain a different distribution. Thus, our distributions are temperature dependent and we write them as: $P(A, T)$, and $P(B, T)$.

Now, since the two rooms are remote, we know that there is no effect of each measurement on the other, thus our joint distribution must have the form: $$P(AB,T) = P(A,T) P(B,T) $$

The comma here means that the temperature is a parameter on which the distribution depends and not an observable.

Now, suppose that we don't know in which season in which we performed the experiment; but we only know the probability density function f(T) of the temperature over the seasons. The best we can do in order to forecast predictions is to perform a weighted averaging over all the temperature values, and make our predictions according to: $$ P(AB) = \int_{\mathcal{T}} f(T) P(A,T) P(B,T) dT$$ Here $ \mathcal{T}$ (corresponding to the character $\Lambda$ in the question) is the space of all temperature values. This space can be one dimensional as in our case where our experiments depend on one parameter, or it can be a multidimensional manifold parametrizing the possible parameters which can influence our experiment.

Now, the measurements are local, they were performed in distant rooms, with no correlation, except through the fact that the temperature was the same every time the observables were measured instantaneously.

The only difference of the above from your question is that the observables in the question are conditional $a|x$, $b|y$, instead of $A$ and $B$. This just mean that you did not perform a single experiment of measuring $A$, but several experiments: For example if $x$ and $y$ are binary; it means that you performed an experiment measuring the observable $A = a|0$ where you held $x$ at $0$, then performed another experiment measuring $A = a|1$, etc.

A simple example of the confounding variable $\lambda$ exemplified by the temperature in the above description, would be a binary variable, in this case its probability density is: $$q(\lambda) = \frac{1}{2}(\delta(q) + \delta(q-1))$$ ($\delta$ is the Dirac delta functions), and in this case, its range will be the whole real axis $\Lambda = \mathbb{R}$ equipped with a Lebesgue measure.