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It is well-known that quantum mechanics does not admit a noncontextual ontological model, and there are countless different proofs of it. I'm interested in the simplest proofs that can be cast as an inequality, and bonus points if there's a proof that a simpler one can't be found.

The definition of contextuality that I care about is the one by Spekkens, that is: I don't care about determinism nor require that the proof of impossibility be specifically about measurement contextuality; failure at either measurement or preparation is fine. Spekkens himself provided very simple proofs for two-dimensional Hilbert spaces, but it is not clear to me how to cast his proofs as inequalities.

Also, it's well-known that unlike nonlocality, contextuality admits state-independent proofs. It would be nice to know the simplest one in this category as well.

Of course, "simplicity" is subjective, but I hope not enough to forbid a definite answer. If you want, a criterion could be: First, dimension of the Hilbert space needed to exhibit a contradiction. Second, number of measurements needed. Or maybe the product of these numbers.

My candidates are currently Klyachko's 5-observable inequality, that is violated by 3-dimensional quantum systems, and Yu's 13-observable inequality for 3 dimensions that is violated independently of the quantum state. I have no idea if these are the best, and I find it weird that I couldn't find an inequality violated for qubits.

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In this paper: A. Cabello, S. Severini, and A. Winter, a generalisation of the notion of contextuality is presented in terms of the compatibility structure of a graph, something akin to the graph colourability arguments in Kochen Specker proofs. If you ask questions with binary outcomes, and you say that certain combinations of questions are compatible and exclusive (both answers cannot be true, or $1$) then questions are vertices of a graph, and if they are compatible and exclusive, you assign an edge. Then asking all questions and putting it into an inequality such as the Klyachko inequality, the non-contextual upper bound is for all possible graphs given by the independence number of the graph. The quantum upper bound is given the by the Lovasz theta function. So to find the smallest example of a contextual inequality which a violation by quantum mechanics, one just needs to find the smallest graph (in the number of vertices and edges) where the Lovasz theta function is larger than the independence number. This is the pentagon, or 5-Cycle graph, the one used in the Klyachko inequality.

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In which sense it is a generalization? It seems to me that they just represent the traditional notion of contextuality in terms of graphs. Also, they restrict themselves to the consideration of projective measurements, so they only prove that the Klyachko inequality is the simplest projective one; to claim more would contradict the fact that there can be violations of contextuality for qubits. – Mateus Araújo Nov 3 '11 at 17:23
It seems that their method can also be used to determine the simplest state-independent inequality, but they didn't seem to have done so. – Mateus Araújo Nov 3 '11 at 17:25

Spekkens himself and co authors developed a non-contextuality inequality for a 2-dimensional system and 2 observables here: Note that the inequality is derived in the context of a communication protocol with an additional assumption that no information about the parity of the message (a bit string) is transmitted. I sure it's impossible to get a "simpler" proof in terms of fewer dimensions and observables.

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While this is an interesting result, is not the sort of inequality I am looking for, as it uses an additional assumption that's not needed in the general case. – Mateus Araújo Nov 3 '11 at 17:52

A recent paper of Adán Cabello completely characterizes state-independent contextuality, and by doing so proves that Yu and Oh's 13-observable inequality for 3-dimensional systems is in fact the simplest state-independent noncontextual inequality possible.

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