# How do we arrive at the Bell polytope constructed starting from definition of behaviours?

While reading Valerio Scarani's book ; Bell Nonlocality I came across section 2.4 where the author tries to represent the set $$\mathcal{L}$$, of all local behaviours as a polytope. The term behavior is defined (Eqn 2.4 of the book) as the set $$\mathcal{P} = \{P(a,b\ |\ x,y)\}$$ where $$P$$ represents the probability of getting the output pair $$(a,b)$$ given the input pair $$(x,y)$$ in a Bell's Test / CHSH-like game. And the word Local has been used for any probability distribution where $$P(a,b\ |\ x,y)=P(a|x)P(b|y)$$ holds. If I understand correctly, $$\mathcal{L}$$ is a set of sets, the elements of $$\mathcal{L}$$ are not real numbers (they are sets of real numbers), hence the set $$\mathcal{L}$$ lacks well-ordering within it. The rest of the section (2.4.1) of the text talks about convexity of the set $$\mathcal{L}$$ and claims that the as the set $$\mathcal{L}$$ is the convex hull of it's extremal points, which brings me back to the same confusion. What is a point within $$\mathcal{L}\ ?$$ I thought about considering the set $$\mathcal{L}$$ to contain the probabilities $$P(a,b\ |\ x,y)$$ directly as numbers (instead of being enclosed within sets). But that approach seems to cause more problems as two different behaviors might give different prescriptions of some given $$P(a_1,b_1\ |\ x_1,y_1)$$, then which prescription of this element shall we follow?

I feel like I am misunderstanding the very definition of the word "behavior" in this context. Any help is greatly appreciated.

EDIT 1 - As per Section (2.4.1) of the aforementioned book, $$\mathcal{L}$$ is called convex because $$\forall p\in[0,1]$$ and for any choice of $$\mathcal{P}_{LV,1}\in\mathcal{L}$$ and $$\mathcal{P}_{LV,2}\in \mathcal{L}$$ the convex sum $$\mathcal{P}=p \mathcal{P}_{LV,1}+(1-p)\mathcal{P}_{LV,2}$$ also belongs to $$\mathcal{L}$$. I forgot to state this definition in the originl question.

• "...claims that the as the set L is the convex...." What does this sentence mean? Commented Mar 24 at 8:56
• @DanielFBest , I have edited the question appropriately so that the convexity of $\mathcal{L}$ is defined. Commented Mar 24 at 9:05
• No, I mean about the grammar of the sentence. "...claims that the..." ...That the what? Commented Mar 24 at 9:11
• Interesting question but I'm afraid it will be difficult to answer without being familiar with the book. Unless you copy the relevant definitions from the book
– lcv
Commented Mar 24 at 11:44

It's hard to give a precise answer without having read the book, but the correlations that you mention

$$P(a,b|x,y)$$

are usually called "boxes". The set of all possible boxes is perhaps what you are after. It is the set called $$Q$$ in this Wikipedia article

• I understand from the link provided that we can view an observed behavior (also called a box) $\mathcal{P}={P(a,b | x,y)}$. But $\mathcal{L}$ is a set of all such boxes. The wikipedia page says that we can view each behavior as a set of of vectors $(a,b,x,y)$, what I am having trouble in wrapping my head around is the fact the two different boxes that are included in $L$ may give different values of some $P(a,b | x,y)$, are we not losing that information when we make this map? Because as I understand it, our space is only spanned by the parameters, not the output probabilities. Commented Mar 24 at 14:33

If I understand correctly, $$\mathcal{L}$$ is a set of sets, the elements of $$\mathcal{L}$$ are not real numbers (they are sets of real numbers), hence the set $$\mathcal{L}$$ lacks well-ordering within it.

You are correct, $$\mathcal{L}$$ is a set of sets of non-negative real numbers in a strict mathematical sense. Of course there is no natural order in this set.

What is a point within $$\mathcal{L}$$ ?

As we said every point is a set of numbers. For the purposes of this discussion is better to picture it as the set of conditional probabilities given a probability measure, that is keeping $$P$$ fixed. Or even better just consider every point as a probability measure $$P$$, i.e. one possible way to assigning probabilities to events. This is not exactly equivalent but I think the details are irrelevant for the current discussion.

I feel like I am misunderstanding the very definition of the word "behavior" in this context

I think that for the purposes of this discussion is easier to think that a behavior is just a probability measure. Now given two probability measures, its convex sum is a probability measure. And the point of the discussion, as far as I remember, is that the convex sum of local behaviors, i.e. probability measures that fulfills your second equation, will be also a local behavior. That leads you to a polytope.