I hope this post is on topic as a reference request.

I have seen somewhere the idea of (and saw it written just like this):

$$\text{Classical }\subset\text{ Random }\subset\text{ Quantum }.$$

I am writing a note and want to use this idea and am just wondering is there a good reference where such matters are discussed. To explain what the above means in my context see below the fold.

I have a situation where I have a non-commutative $\mathrm{C}^*$-algebra $A\subset B(\mathsf{H})$, which for the purposes of this question can be taken to be finite dimensional. Let $\mathsf{H}=\ell^2(X)\oplus\ell^2(Y)$.

$X$ is the set of classical objects, and each $\mathbb{C}e_x$ is $A$-invariant. Therefore if I am measuring $e_x$ (in the sense of quantum mechanics) with a projection $p\in A$ I am getting yes (1) or no (0) with probabilities 0/1 or 1/0.

With the help of the Born Rule I can take elements of the projective Hilbert space, superpositions, $P(\ell^2(X))$ and consider these as random objects. As $A$ restricted to $\ell^2(X)$ is commutative, it doesn't matter in which order I make measurements, but there is a probabilistic aspect.

$A$ restricted to $\ell^2(X)$ is commutative but suppose the non-commutativity kicks in for $\ell^2(Y)$. Suppose $Y=\{y_1,y_2\}$, and suppose $A$ restricted to $\ell^2(Y)$ is $M_2(\mathbb{C})$. If we measure $e_{y_1}+e_{y_2}\in P(\mathsf{H})$, with say, where $$p=\frac12\left(\begin{array}{cc}1 & e^{-i\pi/4}\\ e^{i\pi/4}&1\end{array}\right),$$ first with $p$ (record and collapse), then $p^T$ (record and collapse), then $p$ again, and calculating probabilities using inner products, we find that the probability that we find $p=1$ then $p^T=1$ then $p=0$ --- impossible for random objects --- is non-zero.

So $\text{Classical }\subset\text{ Random }\subset\text{ Quantum }$ is captured here by:

$$P(\mathbb{C}e_x)\subset P(\ell^2(X))\subset P(\mathsf{H}).$$

I hope this post is on topic as a reference request.

I have seen somewhere the idea of (and saw it written just like this):

$$\text{Deterministic }\subset\text{ Random }\subset\text{ Quantum }.$$

I am writing a note and want to use this idea and am just wondering is there a good reference where such matters are discussed. To explain what the above means in my context see below the fold.

I have a situation where I have a non-commutative $\mathrm{C}^*$-algebra $A\subset B(\mathsf{H})$, which for the purposes of this question can be taken to be finite dimensional. Let $\mathsf{H}=\ell^2(X)\oplus\ell^2(Y)$.

$X$ is the set of deterministic objects, and each $\mathbb{C}e_x$ is $A$-invariant. Therefore if I am measuring $e_x$ (in the sense of quantum mechanics) with a projection $p\in A$ I am getting yes (1) or no (0) with probabilities 0/1 or 1/0.

With the help of the Born Rule I can take elements of the projective Hilbert space, superpositions, $P(\ell^2(X))$ and consider these as random objects. As $A$ restricted to $\ell^2(X)$ is commutative, it doesn't matter in which order I make measurements, but there is a probabilistic aspect.

$A$ restricted to $\ell^2(X)$ is commutative but suppose the non-commutativity kicks in for $\ell^2(Y)$. Suppose $Y=\{y_1,y_2\}$, and suppose $A$ restricted to $\ell^2(Y)$ is $M_2(\mathbb{C})$. If we measure $e_{y_1}+e_{y_2}\in P(\mathsf{H})$, with say, where $$p=\frac12\left(\begin{array}{cc}1 & e^{-i\pi/4}\\ e^{i\pi/4}&1\end{array}\right),$$ first with $p$ (record and collapse), then $p^T$ (record and collapse), then $p$ again, and calculating probabilities using inner products, we find that the probability that we find $p=1$ then $p^T=1$ then $p=0$ --- impossible for random objects --- is non-zero.

So $\text{Deterministic }\subset\text{ Random }\subset\text{ Quantum }$ is captured here by:

$$P(\mathbb{C}e_x)\subset P(\ell^2(X))\subset P(\mathsf{H}).$$