A structure theorem for generalized-noncontextual ontological models

Abstract

It is useful to have a criterion for when the predictions of an operational theory should be considered classically explainable. Here we take the criterion to be that the theory admits of a generalized-noncontextual ontological model. Existing works on generalized noncontextuality have focused on experimental scenarios having a simple structure: typically, prepare-measure scenarios. Here, we formally extend the framework of ontological models as well as the principle of generalized noncontextuality to arbitrary compositional scenarios. We leverage a process-theoretic framework to prove that, under some reasonable assumptions, every generalized-noncontextual ontological model of a tomographically local operational theory has a surprisingly rigid and simple mathematical structure -- in short, it corresponds to a frame representation which is not overcomplete. One consequence of this theorem is that the largest number of ontic states possible in any such model is given by the dimension of the associated generalized probabilistic theory. This constraint is useful for generating noncontextuality no-go theorems as well as techniques for experimentally certifying contextuality. Along the way, we extend known results concerning the equivalence of different notions of classicality from prepare-measure scenarios to arbitrary compositional scenarios. Specifically, we prove a correspondence between the following three notions of classical explainability of an operational theory: (i) existence of a noncontextual ontological model for it, (ii) existence of a positive quasiprobability representation for the generalized probabilistic theory it defines, and (iii) existence of an ontological model for the generalized probabilistic theory it defines.

lots of diagrams!

References in corpus (15)

• Quantum Circuits Architecture
• Transforming quantum operations: quantum supermaps
• Negativity and contextuality are equivalent notions of nonclassicality
• Anomalous Weak Values Are Proofs of Contextuality
• Preparation contextuality powers parity-oblivious multiplexing
• Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations
• Framed Hilbert space: hanging the quasi-probability pictures of quantum theory
• Pre- and Post-selection paradoxes and contextuality in quantum mechanics
• Uniqueness of noncontextual models for stabilizer subtheories
• Nonnegative subtheories and quasiprobability representations of qubits
• Dilation of states and processes in operational-probabilistic theories
• Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality
• Contextuality without incompatibility
• Solvable Criterion for the Contextuality of any Prepare-and-Measure Scenario
• Causality in Higher Order Process Theories

Cited by in corpus (16)

• Kirkwood-Dirac representations beyond quantum states (and their relation to noncontextuality)
• Quantum speed limit for Kirkwood-Dirac quasiprobabilities
• Contextuality in anomalous heat flow
• No-broadcasting characterizes operational contextuality
• Shadows and subsystems of generalized probabilistic theories: when tomographic incompleteness is not a loophole for contextuality proofs
• Bell Nonlocality in Classical Systems Coexisting with other System Types
• Every non-signalling channel is common-cause realizable
• Statistical signatures of quantum contextuality
• Sufficiency of the counterfactual account of Lüders' rule to rule out ontological models of quantum mechanics
• Contextuality Can be Verified with Noncontextual Experiments
• Minimal operational theories: classical theories with quantum features
• Theory-independent monitoring of the decoherence of a superconducting qubit with generalized contextuality
• Almost no experiments have classical Kirkwood-Dirac representations
• Contextuality from the vacuum
• Typicality of Contextuality
• Decoupling local classicality from classical explainability: A noncontextual model for bilocal classical theory and a locally-classical but contextual theory