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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