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Decoupling local classicality from classical explainability: A noncontextual model for bilocal classical theory and a locally-classical but contextual theory

Sina Soltani, Marco Erba, David Schmid, John H. Selby

TL;DR

The paper addresses whether locally-classical theories must be classically explainable by constructing an explicit ontological model for bilocal classical theory (BCT) and by presenting latent classical theories (LCTs) that evade such an embedding. It shows that BCT is classically explainable, providing a constructive ontological map from BCT to standard classical theory and establishing consistency via linearity, diagram preservation, and probability/determinacy preservation. Conversely, it proves that latent classical theories generally lack an ontological model, using a non-full-rank latent factor to derive a contradiction with probability preservation, thereby demonstrating that local classicality does not guarantee classical explainability. Together, these results illuminate the nuanced relationship between compositional properties (like bilocal tomography) and classical explainability, and highlight the limitations of local tomography in constraining foundational notions of classicality within OPTs.

Abstract

We construct an ontological model for the theory known as bilocal classical theory doi.org/10.1103/PhysRevA.102.052216. To our knowledge, this is only the second time that an ontological model has been constructed for an entire theory, rather than just for some particular scenarios within a theory. This result refutes a conjecture from doi.org/10.1103/PhysRevA.102.052216 which suggested that there might be no local-realist ontological model for bilocal classical theory. Moreover, it is the first time that an ontological model has been constructed for a theory that fails to be locally tomographic, showing that the assumption of local tomography underpinning the structure theorem in doi.org/10.22331/q-2024-03-14-1283 is a genuine limitation of the theorem. This demonstrates that in general there is no tension between failures of local tomography and classical explainability (i.e., generalised noncontextuality). In fact, bilocal classical theory is in many ways more simply understood via the underlying ontological model than it is within its original formulation (much as how odd-dimensional stabiliser subtheories can be more simply understood via Spekkens' toy theory). Furthermore, this result naturally leads to the question, does every locally-classical theory admit of an ontological model? By constructing a concrete counterexample, we show that this is not the case. Our findings demonstrate that there is no straightforward relationship between theories being locally-classical, and them being classically-explainable. This shows that the fundamental status of compositional properties (such as local tomography) is not a technical side-issue, but a central and unavoidable question for a coherent understanding even of classicality itself.

Decoupling local classicality from classical explainability: A noncontextual model for bilocal classical theory and a locally-classical but contextual theory

TL;DR

The paper addresses whether locally-classical theories must be classically explainable by constructing an explicit ontological model for bilocal classical theory (BCT) and by presenting latent classical theories (LCTs) that evade such an embedding. It shows that BCT is classically explainable, providing a constructive ontological map from BCT to standard classical theory and establishing consistency via linearity, diagram preservation, and probability/determinacy preservation. Conversely, it proves that latent classical theories generally lack an ontological model, using a non-full-rank latent factor to derive a contradiction with probability preservation, thereby demonstrating that local classicality does not guarantee classical explainability. Together, these results illuminate the nuanced relationship between compositional properties (like bilocal tomography) and classical explainability, and highlight the limitations of local tomography in constraining foundational notions of classicality within OPTs.

Abstract

We construct an ontological model for the theory known as bilocal classical theory doi.org/10.1103/PhysRevA.102.052216. To our knowledge, this is only the second time that an ontological model has been constructed for an entire theory, rather than just for some particular scenarios within a theory. This result refutes a conjecture from doi.org/10.1103/PhysRevA.102.052216 which suggested that there might be no local-realist ontological model for bilocal classical theory. Moreover, it is the first time that an ontological model has been constructed for a theory that fails to be locally tomographic, showing that the assumption of local tomography underpinning the structure theorem in doi.org/10.22331/q-2024-03-14-1283 is a genuine limitation of the theorem. This demonstrates that in general there is no tension between failures of local tomography and classical explainability (i.e., generalised noncontextuality). In fact, bilocal classical theory is in many ways more simply understood via the underlying ontological model than it is within its original formulation (much as how odd-dimensional stabiliser subtheories can be more simply understood via Spekkens' toy theory). Furthermore, this result naturally leads to the question, does every locally-classical theory admit of an ontological model? By constructing a concrete counterexample, we show that this is not the case. Our findings demonstrate that there is no straightforward relationship between theories being locally-classical, and them being classically-explainable. This shows that the fundamental status of compositional properties (such as local tomography) is not a technical side-issue, but a central and unavoidable question for a coherent understanding even of classicality itself.

Paper Structure

This paper contains 21 sections, 3 theorems, 116 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Operational probabilistic theories
  3. Classical theory
  4. Bilocal classical theory
  5. Ontological models and classical explainability
  6. Maps between different OPTs
  7. Ontological models for strongly causal OPTs
  8. An ontological model for bilocal classical theory
  9. Consistency checks
  10. Linearity preservation
  11. Diagram preservation
  12. Probability preservation
  13. Determinacy preservation
  14. Bilocal classical theory admits an ontological model
  15. A locally-classical theory that is not classically-explainable
  16. ...and 6 more sections

Key Result

Proposition 1.1

Every transformation $t$ from system $n$ to $m$ in BCT can be uniquely expressed as a conical combination of normalised atomic transformations.

Figures (1)

  • Figure 1: Venn diagram incorporating our results. One example of a theory is included for each region, with the exception of a single region that is empty (since the only theory that is both locally tomographic and locally-classical is CT, which is, however, also classically-explainable). Our main results, Theorem \ref{['thm:BCT-NCOM']} and Theorem \ref{['thm:counterexample']}, allow us to populate the two regions of locally-classical theories which are non-locally tomographic.

Theorems & Definitions (8)

  • Proposition 1.1
  • proof
  • Definition 1.2: Maps between two OPTs
  • Definition 1.3: Ontological models of strongly causal OPTs
  • Definition 2.1: An ontological model for BCT
  • Theorem 2.2
  • Definition 3.1: Latent classical theories (LCTs)
  • Theorem 3.2