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.




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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:


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



Reference request for Deterministic $\subset$ Random $\subset$ Quantum
The way I interpret your question is: deterministic = pure state on an abelian C${}^\*$-algebra, random = arbitrary state on an abelian C${}^\*$-algebra, quantum = pure state on an arbitrary C${}^\*$-algebra. There's one further level of generality, arbitrary state on an arbitrary C${}^\*$-algebra, which gives you statistical ensembles of quantum states.


(Note that under this interpretation you do not have "random $\subset$ quantum", unless your "quantum" includes ensembles.)


