Kirkwood-Dirac representations beyond quantum states (and their relation to noncontextuality)

Abstract

Kirkwood-Dirac representations of quantum states are increasingly finding use in many areas within quantum theory. Usually, representations of this sort are only applied to provide a representation of quantum states (as complex functions over some set). We show how standard Kirkwood-Dirac representations can be extended to a fully compositional representation of all of quantum theory (including channels, measurements and so on), and prove that this extension satisfies the essential features of functoriality (namely, that the representation commutes with composition of channels), linearity, and quasistochasticity. Interestingly, the representation of a POVM element is uniquely picked out to be the collection of weak values for it relative to the bases defining the representation. We then prove that if one can find any Kirkwood-Dirac representation that is everywhere real and nonnegative for a given experimental scenario or fragment of quantum theory, then the scenario or fragment is consistent with the principle of generalized noncontextuality, a key notion of classicality in quantum foundations. We also show that the converse does not hold: even if one verifies that all Kirkwood-Dirac representations (as defined herein) of an experiment require negativity or imaginarity, one cannot generally conclude that the experiment witnesses contextuality.

5 pages; comments welcome!

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Cited by in corpus (11)

• Properties and Applications of the Kirkwood-Dirac Distribution
• Quantum speed limit for Kirkwood-Dirac quasiprobabilities
• Geometry of sets of Bargmann invariants
• Compatibility of Quantum Measurements and the Emergence of Classical Objectivity
• Robustness of contextuality under different types of noise as quantifiers for parity-oblivious multiplexing tasks
• Quantum entanglement as an extremal Kirkwood-Dirac nonreality
• Geometry of Kirkwood-Dirac classical states: A case study based on discrete Fourier transform
• Contextuality Can be Verified with Noncontextual Experiments
• Estimation of multivariate traces of states given partial classical information
• Almost no experiments have classical Kirkwood-Dirac representations
• Hermitian Kirkwood-Dirac real operators for discrete Fourier transformations