Defining the geometry of Bell inequalities

Question

Bell inequalities can be discussed in the language of geometry. In papers such as [1], there is a general flow of definitions leading to the geometric picture of Bell inequalities: $$\text{Behaviors} \to \text{Space of behaviors} \to \text{No signaling, local, quantum behaviors}.$$ where a (seemingly) standard definition of behavior (sometimes called "correlation") is:

Definition 1 (Behavior): Let $l, m, n \in \mathbb{N}$. Let $\vec{X} = \{X_i\}$ where $X_i \in \{A_0, A_1, ..., A_{m-1}\}$ are random variables with outcomes $\vec{a} = \{a_i\}$ where $a_i \in \{0,1,...,n-1\}$ for all $i \in \{1, 2, ..., l\}$. Then, we call the conditional probability $p(\vec{a} \lvert \vec{X})$ an $(l,m,n)$-behavior.

In various papers, the space of all behaviors is identified with a real vector space. Seemingly, the vector space $V$ is constructed as $$V = \{p(\vec{a} \lvert \vec{X})\}_{\vec{a},\vec{X}}. \tag{1}$$ However, I do not see how this is a real vector space or a vector space at all. Alternatively, I thought that the space of behaviors is actually all convex combinations of behaviors, denoted $C$ and defined by $$\vec{\alpha} \in C \iff \vec{\alpha} = \sum_{\vec{a}, \vec{X}} \vec{\alpha}_{\vec{a},\vec{X}} p(\vec{a} \lvert \vec{X}). \tag{2}$$

What is actually meant by the space of all behaviors in the context of Bell inequalities? That is, the space that is talked about that contains as proper subsets the local polytope of behaviors and the convex, bounded quantum behaviors.

[1] Fadel et al. 2024. "Deriving three-outcome permutationally invariant Bell inequalities ". https://arxiv.org/abs/2406.11792.

Answer 1

Behaviours are contained in a real (Euclidean) vector space. The set of behaviours is not a vector space, but rather a convex polytope contained in an ambient vector space.

Consider a standard bipartite 2-2 scenario: two parties, each one with two possible measurement choices leading to binary outcomes. The behaviours are vectors $\mathbf p\equiv ( p(ab|xy) )_{a,b,x,y\in\{0,1\} }$. In other words, these are vectors $\mathbf p\in\mathbb{R}^{2^4} $. The vector space is $\mathbb{R}^{2^4}$, not the set of behaviours itself. You can of course then say something more: for these to be behaviours you need $\sum_{a,b} p(ab|xy)=1$ for all $x,y$, which tells you that behaviours are actually contained in a $(2^4-2^2)$-dimensional affine hyperplane within $\mathbb{R}^{2^4}$.

Going another step forward, you can say that the set (not space!) of all behaviours is the convex hull of $(2^2)^{2^2}=4^4=256$ deterministic behaviours. That's because for any choice of $x,y\in\{0,1\}$ we can write $p(ab|xy)$ as a convex combination of 4 extremal deterministic behaviours, namely those characterised by $p^{(1)}(00|xy)=1$, $p^{(2)}(01|xy)=1$, $p^{(3)}(10|xy)=1$, and $p^{(4)}(00|xy)=1$.

Adding further assumptions, like locality, further constraints the corresponding set of behaviours. But the general property of these sets being convex aways remains.