Contextuality under weak assumptions

Abstract

The presence of contextuality in quantum theory was first highlighted by Bell, Kochen and Specker, who discovered that for quantum systems of three or more dimensions, measurements could not be viewed as deterministically revealing pre-existing properties of the system. More precisely, no model can assign deterministic outcomes to the projectors of a quantum measurement in a way that depends only on the projector and not the context (the full set of projectors) in which it appeared, despite the fact that the Born rule probabilities associated with projectors are independent of the context. A more general, operational definition of contextuality introduced by Spekkens, which we will term 'probabilistic contextuality', drops the assumption of determinism and allows for operations other than measurements to be considered contextual. Even two-dimensional quantum mechanics can be shown to be contextual under this generalised notion. Probabilistic noncontextuality represents the postulate that elements of an operational theory that cannot be distinguished from each other based on the statistics of arbitrarily many repeated experiments (they give rise to the same operational probabilities) are ontologically identical. In this paper, we introduce a framework that enables us to distinguish between different noncontextuality assumptions in terms of the relationships between the ontological representations of objects in the theory given a certain relation between their operational representations. This framework can be used to motivate and define a 'possibilistic' analogue, encapsulating the idea that elements of an operational theory that cannot be unambiguously distinguished operationally can also not be unambiguously distinguished ontologically. We then prove that possibilistic noncontextuality is equivalent to an alternative notion of noncontextuality proposed by Hardy. Finally, we demonstrate that these weaker noncontextuality assumptions are sufficient to prove alternative versions of known 'no-go' theorems that constrain ψ-epistemic models for quantum mechanics.

Contextuality and the Single-Qubit Stabilizer Subtheory

“…There are two leading definitions of contextuality: traditional contextuality [1][2][3][4][5] and generalized contextuality [6,7]. In this paper, we show that, generalized contextuality [6] is present even in the single-qubit stabilizer subtheory of quantum theory, a fact missed by previous work [21][22][23].…”

“…Contextuality [1][2][3][4][5][6][7], which includes the better-known concept of Bell non-locality as a special case, is often regarded as the fundamental non-classical property of quantum theory.…”

Contextuality and truth-value assignment

“…Along these lines, the probability function P can be considered as the degree of belief that the corresponding proposition is true (or it is expecting to be true). 4 Since in the considered case [[X 1 ∨ X 2 ]] v = 1 and [[X 1 ∧ X 2 ]] v = 0, the probability function mapping the conjunction X 1 ∨ X 2 to the interval [0, 1] can be written by the sum of the probabilities P[X 1 ∨ X 2 ] = P[X 1 ] + P[X 2 ] = 1. So, were the pre-existing truth values of the propositions X 1 and X 2 to be such that v(x 12 ) = 1, the interference pattern P[R|X 1 ∨ X 2 ] (i.e., the probability of finding the particle at a certain region R on the screen) in the two-slit set-up with none of the detectors present at the slits would be the sum of one-slit patterns P[R|X 1 ] and P[R|X 2 ], namely,…”

“…For an overview of different aspects of contextuality in quantum theory and beyond, see[2,3]. Also, see a review of the framework of ontological models in[4,5].…”

What quantum measurements measure

“…Some authors make it clear that it is hidden variables versions of quantum mechanics which are contextual, but many omit that qualification; for a recent (but hardly unique) example, see[27].…”