Definition for the specifics of the question as well as an example of contextuality in quantum mechanics
I have a set of measurements acting on a 2 qubit state for whom the statistics of the measurement result is contextual.
By that I mean the following. My 4 measurements $A,C,a,c$ with possible value $+,-$ have the following commutation relation. $[A,C]=0$, $[A,a]=0$, $[a,c]=0$, $[C,c]=0$. You could put them in a table and say that observables in the same row or in the same column commute. \begin{equation} \begin{bmatrix} A & C \\ a & c \end{bmatrix} \end{equation} Define the context of $A$ as the other commuting observables I observe at the same time. (could be $Aa$, or $AC$)
Looking at the following expectation value, \begin{equation} <E>=<AC>+<Aa>+<ac>-<Cc> \end{equation} Trying to assume pre-existing measurement results for $A,C,a,c$ noncontextualy (without taking the measurement context into account), then we see that $<E>_{NC}\leq 2$. (To see it, set all values A=C=a=c=1, see that this implies =2 and try to change them to make the expectation larger, not change can be made that makes it larger than 2)
With notation $Z_1=\sigma_z\otimes\mathcal{I}$, we choose $A=Z_1$, $C=Z_1Z_2$, $a=Z_1X_2$, and $c=Y_1Y_2$.
For $|MM>=\cos(\frac{\pi}{8})^2|00>+\sin(\frac{\pi}{8})|11>+\cos(\frac{\pi}{8})\sin(\frac{\pi}{8})(|01>+|10>)$, we see that $<E>_{QM}=1+\sqrt{2}><E_{NC}>$.
The inequality is violated and thus the noncontextuality assumption is false, quantum mechanics is contextual.
This is a case of state-dependent contextuality. The reason QM can violate this inequality is that probablity for assignment of values to the observables is context-dependant, even when that context is other commuting observables. What this means from a statistics standpoint is that $P(A,C),P(A,a),P(a,c),P(C,c)$ do not derive from a global distribution $P(A,C,a,c)$ assigning probabilities to the various results ++++, +++-, ++-+, .... It does not derive from it in the sense that there is no $P(A,C,a,c)$ such that the distributions over contexts such as $P(A,C)$ are not marginales. ($P(A,C)\neq\sum_{a,c}P(A,C,a,c)$)
It may remind you of non-locality and that is because contextuality is a generalization of non-locality.
The question
By the commutation relations, I can consider pairs of commuting measurements on the same state, ie $AC|{\psi}\rangle$, and since the order doesn't matter, I can look at the statistics of the measurement results for this pair $(A,C)=\{++, +-, -+, --\}$, and similarly for the other pairs $(A,a)$, $(a,c)$, $(C,c)$.
Imagining I can generates $|MM>$, measure 2 randomly chosen (from A,C,a,c) commuting observables, reset the state and do it enought, I would get the following statistics.
Here is an image of the 4 distributions over the 4 contexts:
What is a set of classical data that has similar properties? It doesn't have to be the same distributions, just that it corresponds to 4 "questions" that I can ask in the same pairs as the contexts defined above, and that the statistics of the answers to the "questions" also be contextual, ie does not derive from a global distribution.
I would like to find a dataset not generated by quantum measurement. The ultimate goal is to show that a quantum circuit can just learn the |MM>, but that it would be more ressource intensive to learn that classically.
I dont have much experience posting, so let me know if there are things that need clarification