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!

References in corpus (70)

• Colloquium: Understanding Quantum Weak Values: Basics and Applications
• Contextuality for preparations, transformations, and unsharp measurements
• Probabilistic theories with purification
• Einstein, incompleteness, and the epistemic view of quantum states
• Hudson's Theorem for finite-dimensional quantum systems
• Negativity and contextuality are equivalent notions of nonclassicality
• Contextuality in Measurement-based Quantum Computation
• Anomalous Weak Values Are Proofs of Contextuality
• An experimental test of noncontextuality without unwarranted idealizations
• Preparation contextuality powers parity-oblivious multiplexing
• Quasi-probability representations of quantum theory with applications to quantum information science
• Introduction to the book "Quantum Theory: Informational Foundations and Foils"
• Quasi-quantization: classical statistical theories with an epistemic restriction
• Quantum from principles
• The completeness of quantum theory for predicting measurement outcomes
• Jarzynski-like equality for the out-of-time-ordered correlator
• The quasiprobability behind the out-of-time-ordered correlator
• Quantum Advantage in Postselected Metrology
• Contextual advantage for state discrimination
• Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations
• Quantum fluctuation theorems, contextuality and work quasi-probabilities
• Quantum theory of successive projective measurements
• Out-of-Time-Ordered-Correlator Quasiprobabilities Robustly Witness Scrambling
• Framed Hilbert space: hanging the quasi-probability pictures of quantum theory
• Anomalous weak values and contextuality: robustness, tightness, and imaginary parts
• From the Kochen-Specker theorem to noncontextuality inequalities without assuming determinism
• Certifying quantum signatures in thermodynamics and metrology via contextuality of quantum linear response
• A quasiprobability distribution for heat fluctuations in the quantum regime
• Negative quasiprobabilities enhance phase estimation in quantum-optics experiment
• Complex joint probabilities as expressions of determinism in quantum mechanics
• Quantifying quantum coherence via nonreal Kirkwood-Dirac quasiprobability
• Kirkwood-Dirac quasiprobability approach to the statistics of incompatible observables
• Quantum circuits for measuring weak values, Kirkwood--Dirac quasiprobability distributions, and state spectra
• The Characterization of Noncontextuality in the Framework of Generalized Probabilistic Theories
• State independent contextuality advances one-way communication
• Kirkwood-Dirac nonclassicality, support uncertainty and complete incompatibility
• All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed sets of operational equivalences
• Why interference phenomena do not capture the essence of quantum theory
• Conditions tighter than noncommutation needed for nonclassicality
• Uniqueness of noncontextual models for stabilizer subtheories
• Contextual advantage for state-dependent cloning
• Preparation contextuality as an essential feature underlying quantum communication advantage
• Contextual advantages and certification for maximum confidence discrimination
• Non-adaptive Measurement-based Quantum Computation and Multi-party Bell Inequalities
• Aspects of the phenomenology of interference that are genuinely nonclassical
• Quasiprobabilities in quantum thermodynamics and many-body systems
• What is nonclassical about uncertainty relations?
• A simple proof that anomalous weak values require coherence
• From statistical proofs of the Kochen-Specker theorem to noise-robust noncontextuality inequalities
• Projective measurements can probe non-classical work extraction and time-correlations
• Multivariate trace estimation in constant quantum depth
• Experimental certification of contextuality, coherence and dimension in a programmable universal photonic processor
• Contextuality without incompatibility
• Contextuality in entanglement-assisted one-shot classical communication
• A structure theorem for generalized-noncontextual ontological models
• Measuring incompatibility and clustering quantum observables with a quantum switch
• Coherence and contextuality in a Mach-Zehnder interferometer
• An invertible map between Bell non-local and contextuality scenarios
• Non-Abelian transport distinguishes three usually equivalent notions of entropy production
• Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements
• Almost Quantum Correlations are Inconsistent with Specker's Principle
• Discriminating mirror symmetric states with restricted contextual advantage
• Inequalities witnessing coherence, nonlocality, and contextuality
• Fine's theorem, noncontextuality, and correlations in Specker's scenario
• Characterizing the geometry of the Kirkwood-Dirac positive states
• A linear program for testing nonclassicality and an open-source implementation
• Contextuality with vanishing coherence and maximal robustness to dephasing
• Hypergraph framework for irreducible noncontextuality inequalities from logical proofs of the Kochen-Specker theorem
• Kirkwood-Dirac classical pure states
• Direct Measurement of Kirkwood-Rihaczek distribution for spatial properties of coherent light beam

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• 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