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I've been reading the info about contextual scenarios given in this answer, as well as the outline of the main ideas as presented in section II of (Leifer and Duarte 2020). Following the notation of said paper, I understand a contextuality scenario as a tuple $\mathfrak C=(X,\mathcal M,\mathcal N)$ with $X$ a set of outcomes, $\mathcal M$ a set of measurement contexts, and $\mathcal N$ a set of maximal partial measurement contexts.

Without getting too much into the details, my understanding is that each $M\in\mathcal M$ represents the subset of possible outcomes corresponding to some measurement scenario. For example, they mention the Specker triangle as the contextuality scenario with $X=\{a,b,c\}$, $\mathcal M=\{\{a,b\},\{a,c\},\{b,c\}\}$ and $\mathcal N=\varnothing$. This (I think) represents a situation where one can choose between one of three possible ways to perform a measurement, and each measurement setting can give one of two possible outcomes. For example, I might decide to perform a measurement that can give as outcome either $a$ or $b$, or instead a measurement that can give either $b$ or $c$.

The Specker triangle contextuality scenario has no value function, meaning no function $v:X\to\{0,1\}$ that can be understood as a deterministic assignment of an outcome for each context $M\in\mathcal M$. Furthermore, as mentioned in the paper, it also does not have a quantum model, meaning (I think) it cannot be understood as a possible situation arising from measuring a quantum system.

This brings me to my question: what does such an example of a (should I say noncontextual?) contextuality scenario represent? A noncontextual scenario that has a quantum model I can at least understand within the QM framework: the absence of the value function is due to the disturbance caused the act of measurement of a quantum system. Is there a way to interpret situations that don't even have a quantum model?

For example, is it sensible to think of the Specker triangle scenario as arising from a situation where the choice of context is fed to a black box, which then just replies in a specific deterministic way? This would amount to some "contextual value function" such as $\{a,b\}\mapsto a$, $\{a,c\}\mapsto a$, and $\{b,c\}\mapsto b$. I feel like this sort of "trivial" situation is not what people have in mind in this context (ha!), but I can't quite put my finger on why.

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Some opinion/comment: So, historically, what people found at first were examples of quantum realizations of graphs that represented statistical data that could not explain by noncontextual hidden variable models. This is what happens for instance in the seminal paper by Kochen and Specker with the large 117 graph proof for instance (regardless of the fact that they did not really use the word noncontextual for their models). But what people did later was to realize that in fact this property (of some statistical data being noncontextual or not) should not really be so linked to quantum theory. Now, what do I mean by this is that rigorous mathematical formalisms are built so that one can talk about contextuality regardless of the operational-probabilistic theory being considered.

As such, everything crucially depends on the way you define quantum realization of the experiment, as well as a scenario that is described by your graphs: specific degenerate examples may exist where one should not be capable of implementing the laboratory structure needed for the scenario. And as such, whenever you build a mathematical framework that is less attached to physical principles one kind of has the ability to go beyond physical constraints. The point of designing specific contextual boundaries to be respected for your measurements is that those impose rigorous constraints that allow you to differentiate between noncontextual/contextual data.

Addressing the questions:

meaning (I think) it cannot be understood as a possible situation arising from measuring a quantum system.

Yes, the point is not that one cannot do projective measurements on a quantum system, clearly. What is impossible is to impose the contextual structure of the graph on a quantum experiment. This answers also the question;

what does such an example of a (should I say noncontextual?) contextuality scenario represent?

It says that quantum theory itself has a structure that does not allow experiments to have this contextual constraint on complete measurement procedures. This is part of the interesting developments I think one can draw from a theory-independent and operational-probabilistic framework. Note that one can still make sense operationally on this structure as follows, which from what I understand is from where the Specker's parable comes from: You can think of a task such where one has three boxes that may or may not have a gem and contexts represent jointly opening the boxes, but with the constraint that if you make a measurement in the context it will with certainty have a gem. What this example shows is that for sure the gem/boxes scenario must not behave as a quantum model (terminology from the Leifer-Duarte paper). But nevertheless it is something that can be thought and rigorously put, and that can be realized by a non-deterministic model where one assigns $\omega(v)=1/2$ to all vertices-- with the terminology from the paper, there is a model for the Specker triangle, but there is no noncontextual neither quantum model. (Also, I think you should say measurement scenario, or even contextuality scenario. Noncontextuality scenario is not something common as far as I know).

Is there a way to interpret situations that don't even have a quantum model? For example, is it sensible to think of the Specker triangle scenario as arising from a situation where the choice of context is fed to a black box, which then just replies in a specific deterministic way? This would amount to some "contextual value function" such as $\{a,b\}↦a$, $\{a,c\}↦a$, and $\{b,c\}↦b$. I feel like this sort of "trivial" situation is not what people have in mind in this context (ha!), but I can't quite put my finger on why.

In the setting of these scenarios, I think that one can always considering asking questions as I have described early. (I don't know much about query, black-box, and computer science terminology but I will give it a shot here because I think you are basically correct): One can think in a protocol for which one wants to describe the location of a gem inside $n$ boxes using a probabilistic model by having access to a black-box that can be queried each time at most by two boxes and for which there is always a gem observed when done so. For $n=3$ no probabilistic quantum model exists (in the sense given by the paper).

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  • $\begingroup$ Thanks for the question, hopefully someone will post another answer and we'll learn more from it as well. $\endgroup$
    – R.W
    Commented Mar 30, 2022 at 15:51
  • $\begingroup$ One thing I would like to know is if there is a similar thing that is possible to happen in the framework of Bell scenarios. Are there Bell scenarios for which there are only post quantum correlations? I think not. Moreover, it would be interesting to see, for a question that might have a positive answer, Bell polytopes for which Quantum = Local polytopes but there are post quantum non-local correlations. $\endgroup$
    – R.W
    Commented Mar 30, 2022 at 15:54
  • $\begingroup$ Just also another reminder that this approach described by Leifer and Duarte is sort of a mix between two other graph approaches to contextuality (one in which probabilities sum to one, corresponding to the $\mathcal{M}$ part, and another in which probabilities don't need to add to one, corresponding to the $\mathcal{N}$ part). $\endgroup$
    – R.W
    Commented Mar 30, 2022 at 15:58

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