The quantum set-up has many settings, so let's fix some definitions. I will be taking the Hilbert space approach with a minor modification that I will make explicit.

We begin with a Hilbert space $\mathcal{H}$, and let $\mathcal{L}(\mathcal{H})$ be the set of all of its projection operators to (closed, I guess? I will be focusing on the finite dimensional case anyway.) subspaces.

In that case we define a quantum probability measure as any function $p:\mathcal{L}(\mathcal{H})\rightarrow \mathbb{R}$ satisfying:

- $\forall P\in\mathcal{L}(\mathcal{H}) \,\, p(P)\geq 0$.
- $p(I)=1$.
- For every countable set of mutually orthogonal projections $\{P_i\}_i$ we have additivity: $$p(\sum_i P_i)=\sum_ip(P_i).$$

Technical Note: The content of Gleason's Theorem is that if $dim(\mathcal{H})\geq 3$ then quantum probability measures are in 1-1 correspondence with so-called density operators. In most textbooks one would find the density operator as the definition of a "state". I feel that the density operator approach is much less intuitive, so I will go ahead and treat the non-equivalence between the two approaches in low dimension as a not-very-important oddity, and take the probability measure approach as primary.

An observable, the quantum version of a random variable, is a choice of a an orthogonal decomposition of $\mathcal{H}$ as $\oplus_{i\in I}\mathcal{H}_i$, and a choice of a real number for each $i\in I$. (Usually in the literature it is said that it is a compact self-dual operator; those are of course equivalent by the spectral theorem. Sometimes is it assumed to be non-bounded, but that's crazy and I can't wrap my head around that so I'm not going to try right now.)

I will focus on low dimensions to try to understand what all this means.

# 2 Dimensions

Let $A$ and $B$ be observables, and let $A$ be $diag(a,b)$ in the orthonormal basis $u_1, u_2$ and $B$ be $diag(c,d)$ in the orthonormal basis $v_1, v_2$. If $span(u_1)=span(v_1)$ then $span(u_2)=span(v_2)$, and $p(P_{span(u_1)})=p(P_{span(v_1)})$ and $p(P_{span(u_2)})=p(P_{span(v_2)})$. If $span(u_1)\neq span(v_1),span(v_2)$, then you can choose $p$ so that $p(P_{span(u_1)})$ be anything you want, and $p(P_{span(v_1)})$ be anything you want.

So essentially, the totality of the situations you can describe are classical. Either $A$ and $B$ are two independent coin flips with whatever Bernoulli parameter you want (and the Bernoulli parameter of one says nothing about the other), or it's a single coin flip that determines the value for both $A$ and $B$.

# 3 Dimensions

It seems to me, therefore, that if there is any hope to describe something non-classical it would be if the eigendecomposition of $A$ has a summand that intersects a summand of the eigendecomposition of $B$ non-trivially.

To give an example, let's say that the standard basis is an eigenbasis for $A$, and that the eigenbasis for $B$ is $u_1, u_2, e_3$, where $span(u_1,u_2)=span(e_1,e_2)$.

It seems to me that the 3rd property of quantum probability measures only applies to determine that $p(P_{span(u_1)})+p(P_{span(u_2)})=p(P_{span(u_1,u_2)})=p(P_{span(v_1,v_2)})=p(P_{span(v_1)})+p(P_{span(v_2)})$.

In other words, I can describe this situation thusly. Flip a coin, it's heads with probability $p(P_{span(e_3)})$. If heads, great. If tails, flip two more coins; one with Bernoulli parameter $p(P_{span(u_1)})$ and the other with Bernoulli parameter $p(P_{span(v_1)})$.

This can be described classically. The possible situations are:

1st coin heads (with probability $p(Proj_{span(e_3)})$)

1st coin tails; 2nd coin heads; 3rd coin heads (with probability $p(Proj_{span(u_1)})p(Proj_{span(v_1)})$)

1st coin tails; 2nd coin heads; 3rd coin tails (with probability $p(Proj_{span(u_1)})(1-p(Proj_{span(v_1)}))$)

1st coin tails; 2nd coin tails; 3rd coin heads (with probability $(1-p(Proj_{span(u_1)}))p(Proj_{span(v_1)})$)

1st coin tails; 2nd coin tails; 3rd coin tails (with probability $(1-p(Proj_{span(u_1)}))(1-p(Proj_{span(v_1)}))$)

# Main Question

Given the definitions above and the examples I have presented: in what sense is quantum probability more expressive than classical probability? Am I misunderstanding something? And how does all of this relate to Bell's Theorem if at all?

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