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.
Highlights
One way in which quantum mechanics differs strongly from classical mechanics is the existence of incompatible observables; if we want to think that measurements reveal properties of a system, we must reconcile this with the fact that there exist pairs of properties that cannot be simultaneously measured
By considering the same observables appearing in different contexts, that is, measured alongside different sets of other observables, Bell [1], as well as Kochen and Specker [2] showed that measurements could not be thought of as revealing underlying properties of the system in a way that was independent of the context in which the observable was measured
Discussions of contextuality often focus on scenarios in which an element of a operational theory such as quantum mechanics manifests itself in two different contexts, such as two different decompositions of a density matrix; or an observable being measured in two different ways, alongside different sets of co-measurable observables
Summary
One way in which quantum mechanics differs strongly from classical mechanics is the existence of incompatible observables; if we want to think that measurements reveal properties of a system, we must reconcile this with the fact that there exist pairs of properties that cannot be simultaneously measured. By considering the same observables appearing in different contexts, that is, measured alongside different sets of other observables, Bell [1], as well as Kochen and Specker [2] showed that measurements could not be thought of as revealing underlying properties of the system in a way that was independent of the context in which the observable was measured This property of quantum mechanics is referred to as contextuality. The notion of grouping states by the sets of events that they assign nonzero probability is not entirely novel: this viewpoint emerges naturally in the setting of classical and quantum probability theory from considerations of what it means for different parties to hold different but compatible beliefs about a system such as those of Brun, Finkelstein and Mermin [4] or Caves, Fuchs and Schack [5] This notion of possibilistic noncontextuality is strictly weaker than that of probabilistic contextuality. We demonstrate that similar results can be proven using only the weaker, possibilistic, notion of noncontextuality
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