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Generalised contextuality of continuous variable quantum theory can be revealed with a single projective measurement

Pauli Jokinen, Mirjam Weilenmann, Martin Plávala, Juha-Pekka Pellonpää, Jukka Kiukas, Roope Uola

TL;DR

This work shows that applying traditional generalized contextuality to continuous-variable quantum systems can falsely imply contextuality from a single, commuting position measurement, highlighting a clash with classicality for CV observables. It introduces two physically motivated relaxations—approximate and non-normal contextuality—to align the framework with CV statistics, and proves their equivalence in the CV setting. The authors connect contextuality to broadcasting via fixed points of entanglement-breaking channels, develop a structural analysis of the broadcasting algebra, and demonstrate that normal fixed-point sets are commutative, completely atomic von Neumann algebras in separable spaces. The results offer a refined, CV-appropriate notion of noncontextuality and identify open questions about the Heisenberg-picture fixed points and potential contextuality witnesses, with implications for foundational quantum theory and CV quantum information tasks.

Abstract

Generalized contextuality is a possible indicator of non-classical behaviour in quantum information theory. In finite-dimensional systems, this is justified by the fact that noncontextual theories can be embedded into some simplex, i.e. into a classical theory. We show that a direct application of the standard definition of generalized contextuality to continuous variable systems does not envelope the statistics of some basic measurements, such as the position observable. In other words, we construct families of fully classical, i.e. commuting, measurements that nevertheless can be used to show contextuality of quantum theory. To overcome the apparent disagreement between the two notions of classicality, that is commutativity and noncontextuality, we propose a modified definition of generalised contextuality for continuous-variable systems. The modified definition is based on a physically-motivated approximation procedure, that uses only finite sets of measurement effects. We prove that in the limiting case this definition corresponds exactly to an extension of noncontextual models that benefits from non-constructive response functions. In the process, we discuss the extension of a known connection between contextuality and no-broadcasting to the continuous-variable scenario, and prove structural results regarding fixed points of infinite-dimensional entanglement breaking channels.

Generalised contextuality of continuous variable quantum theory can be revealed with a single projective measurement

TL;DR

This work shows that applying traditional generalized contextuality to continuous-variable quantum systems can falsely imply contextuality from a single, commuting position measurement, highlighting a clash with classicality for CV observables. It introduces two physically motivated relaxations—approximate and non-normal contextuality—to align the framework with CV statistics, and proves their equivalence in the CV setting. The authors connect contextuality to broadcasting via fixed points of entanglement-breaking channels, develop a structural analysis of the broadcasting algebra, and demonstrate that normal fixed-point sets are commutative, completely atomic von Neumann algebras in separable spaces. The results offer a refined, CV-appropriate notion of noncontextuality and identify open questions about the Heisenberg-picture fixed points and potential contextuality witnesses, with implications for foundational quantum theory and CV quantum information tasks.

Abstract

Generalized contextuality is a possible indicator of non-classical behaviour in quantum information theory. In finite-dimensional systems, this is justified by the fact that noncontextual theories can be embedded into some simplex, i.e. into a classical theory. We show that a direct application of the standard definition of generalized contextuality to continuous variable systems does not envelope the statistics of some basic measurements, such as the position observable. In other words, we construct families of fully classical, i.e. commuting, measurements that nevertheless can be used to show contextuality of quantum theory. To overcome the apparent disagreement between the two notions of classicality, that is commutativity and noncontextuality, we propose a modified definition of generalised contextuality for continuous-variable systems. The modified definition is based on a physically-motivated approximation procedure, that uses only finite sets of measurement effects. We prove that in the limiting case this definition corresponds exactly to an extension of noncontextual models that benefits from non-constructive response functions. In the process, we discuss the extension of a known connection between contextuality and no-broadcasting to the continuous-variable scenario, and prove structural results regarding fixed points of infinite-dimensional entanglement breaking channels.
Paper Structure (18 sections, 21 theorems, 102 equations, 1 figure)

This paper contains 18 sections, 21 theorems, 102 equations, 1 figure.

Table of Contents

  1. I. Introduction
  2. II. Generalized contextuality in finite-dimensional discrete systems
  3. III. Generalized contextuality in continuous-variable systems
  4. Three definitions of continuous variable contextuality for quantum theory
  5. The probabilistic interpretation of measurement response functions
  6. IV. Normal ontological models
  7. Contextuality and broadcasting for quantum states
  8. Contextuality and broadcasting for quantum measurements
  9. Algebraic structure of broadcasting
  10. Proofs of Subsections "Contextuality and broadcasting for quantum states" and "Contextuality and broadcasting for quantum measurements"
  11. V. Connection between the approximate and the non-normal definition
  12. Approximate interpretation
  13. Sharp measurements and the non-normal definition
  14. Conclusions
  15. Acknowledgments
  16. ...and 3 more sections

Key Result

Proposition 6

A state $\omega \in \mathcal{S}(\mathcal{L(H)})$ is normal if and only if for all POVMs $M \in \mathcal{O}$ the map $X \mapsto \omega(M(X))$ is $\sigma$-additive.

Figures (1)

  • Figure 1: Comparison of the three definitions of contextuality non-confirming sets. The non-normal definition and the approximate definition are equivalent (Proposition \ref{['propapproxnonnormalequiv']}), outlined in green. Furthermore all three definitions are equivalent when considering contextuality non-confirming states (Proposition \ref{['propdefequivalence']}). Finally for contextuality non-confirming sets of measurements Definitions \ref{['defnonnormal']} and \ref{['defapprox']} are strictly weaker than Definition \ref{['defnormal']} (Example \ref{['exposition']}, Corollary \ref{['corpvmdiscrete']} and Proposition \ref{['proppvmapproximate']}). The difference between the definitions arises from the continuity choices of the measurement responses.

Theorems & Definitions (46)

  • Definition 1
  • Example 2
  • proof
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 6
  • Definition 7
  • Theorem 8
  • Corollary 9
  • ...and 36 more