I am looking for an example (or definition) of a quantum probability experiment (if there is such a thing). Ideally it should have these properties:

1. Be purely mathematical; no mention of physics or other empirical sciences;

2. in the example, all variables should be replaced by constants that are as small or simple as possible without collapsing back into classical probability;

3. it should state what are the analogues of the ingredients of a classical probability experiment: sample space $\Omega$, outcome $\omega\in\Omega$, event $E\subseteq\Omega$, probability $\mathbb P(E)$, random variable $X:\Omega\rightarrow V$ where $V=\mathbb R$ or something else.

4. show explicitly how $\sigma$-additivity or finite additivity $\mathbb P(\sqcup_i A_i)=\sum_i \mathbb P(A_i)$ fails, or is replaced by some other rule, as the case may be. Use of measure theory is welcome.

5. be short enough to fit in an MO answer.

I looked at Greg Kuperberg's draft article here and answer here, and while I had trouble extracting what I am describing above, it made me hopeful that such a thing might be possible.

I am looking for an example (or definition) of a quantum probability experiment (if there is such a thing). Ideally it should have these properties:

1. Be purely mathematical; no mention of physics or other empirical sciences;

2. in the example, all variables should be replaced by constants that are as small or simple as possible without collapsing back into classical probability;

3. it should state what are the analogues of the ingredients of a classical probability experiment: sample space $\Omega$, outcome $\omega\in\Omega$, event $E\subseteq\Omega$, probability $\mathbb P(E)$, random variable $X:\Omega\rightarrow V$ where $V=\mathbb R$ or something else.

4. show explicitly how $\sigma$-additivity or finite additivity $\mathbb P(\sqcup_i A_i)=\sum_i \mathbb P(A_i)$ fails, or is replaced by some other rule, as the case may be. Use of measure theory is welcome.

5. be short enough to fit in an MO answer.

I looked at Greg Kuperberg's draft article here and answer here, and while I had trouble extracting what I am describing above, it made me hopeful that such a thing might be possible.