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Formalizing contextuality in sequential scenarios

Kim Vallée, Damian Markham

TL;DR

This work develops a comprehensive framework to study contextuality in sequential scenarios, where instruments can update the system state across a sequence. It defines a hidden-variable model with preparation, response, and transfer functions under a no-backward-in-time signalling constraint, and shows that non-contextuality in these sequential contexts is equivalent to the existence of ND+OD or ND+OI HVMs under appropriate conditions. The authors prove that the non-contextual empirical models form a convex polytope, extend the contextual fraction to sequential settings, and provide a constructive mapping from standard measurement-scenario frameworks to sequential ones that preserves non-contextuality. They apply these results to canonical quantum-contextuality examples (e.g., KCBS, Peres-Mermin) and discuss connections to broader operational frameworks and potential future directions, including noise-robust approaches and causal-network interpretations.

Abstract

This paper provides a framework for characterizing sequential scenarios, allowing for the identification of contextuality given empirical data, and then provides precise operational interpretations in terms of the possible hidden variable model explanations. Sequential scenarios are different in essence from non-local scenarios and standard frameworks for contextuality as each instrument is allowed to change the state as it enters subsequent instruments. Thus, it is necessary to formulate the possible state update in any hidden variable model description. Here we explore such hidden variable models for sequential scenarios, and we develop on the notion of no-disturbance: an instrument $A$ does not disturb another instrument $B$ if the statistics of $B$ are independent of whether $A$ was measured or not. We define non-contextuality inequalities for the sequential scenario, and show that violation implies that the data cannot be explained by a hidden variable model that is both deterministic and not disturbing in this sense. We further provide a translation from standard contextuality frameworks to ours, providing sequential versions which carry over the same inequalities and measures of contextuality, but now with the sequential interpretations stated.

Formalizing contextuality in sequential scenarios

TL;DR

This work develops a comprehensive framework to study contextuality in sequential scenarios, where instruments can update the system state across a sequence. It defines a hidden-variable model with preparation, response, and transfer functions under a no-backward-in-time signalling constraint, and shows that non-contextuality in these sequential contexts is equivalent to the existence of ND+OD or ND+OI HVMs under appropriate conditions. The authors prove that the non-contextual empirical models form a convex polytope, extend the contextual fraction to sequential settings, and provide a constructive mapping from standard measurement-scenario frameworks to sequential ones that preserves non-contextuality. They apply these results to canonical quantum-contextuality examples (e.g., KCBS, Peres-Mermin) and discuss connections to broader operational frameworks and potential future directions, including noise-robust approaches and causal-network interpretations.

Abstract

This paper provides a framework for characterizing sequential scenarios, allowing for the identification of contextuality given empirical data, and then provides precise operational interpretations in terms of the possible hidden variable model explanations. Sequential scenarios are different in essence from non-local scenarios and standard frameworks for contextuality as each instrument is allowed to change the state as it enters subsequent instruments. Thus, it is necessary to formulate the possible state update in any hidden variable model description. Here we explore such hidden variable models for sequential scenarios, and we develop on the notion of no-disturbance: an instrument does not disturb another instrument if the statistics of are independent of whether was measured or not. We define non-contextuality inequalities for the sequential scenario, and show that violation implies that the data cannot be explained by a hidden variable model that is both deterministic and not disturbing in this sense. We further provide a translation from standard contextuality frameworks to ours, providing sequential versions which carry over the same inequalities and measures of contextuality, but now with the sequential interpretations stated.

Paper Structure

This paper contains 40 sections, 9 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sequential scenarios
  3. Definitions
  4. Hidden variable model
  5. Preparation function
  6. Response function
  7. Transfer function
  8. No-backward-in-time signalling
  9. Hidden variable model
  10. Quantum theory as a special case
  11. Operational considerations and HVM restrictions
  12. Outcome determinism
  13. No-disturbance
  14. Example: Fair coin flip
  15. Example: Non-invasive instrument
  16. ...and 25 more sections

Key Result

Lemma 1

For a given sequence $S$, if $o_S \in O_{S}$ is a consistent outcome then there is a unique $o_{\bar{S}} \in O_{\bar{S}}$ associated with it.

Figures (1)

  • Figure 1: Example of a sequence of size $N$, where the hidden variables are represented in green, and the classical outcomes in red. First there is a state preparation $\rho$ where the hidden variable $\lambda_0$ is sampled with probability $\mu_\rho(\lambda_0)$. This hidden variable is then an input to the instrument $A_1$. It has two results: one is a classical outcome $a_1$ given with probability $\xi_{A_1}(a_1\mid \lambda_0)$, and the second a new hidden variable $\lambda_1$. This new hidden variable is sampled from the whole space $\Lambda$ with probability $\Gamma_{A_1}(\lambda_1\mid \lambda_0, a_1)$. The new hidden variable produced is forwarded to the next instrument $B_2$, and it produces a classical outcome $b_2$ as well as a hidden variable $\lambda_2$. This continues until the end of the sequence.

Theorems & Definitions (38)

  • Definition 1: Instrument
  • Definition 2: Instrument no-disturbance
  • Definition 3: Sequential scenario
  • Example 1: Sequential KCBS
  • Definition 4: (Sequential) Empirical behaviour
  • Definition 5: Compatibility of marginals
  • Definition 6: No-Backwards-in-Time-Signalling
  • Definition 7: NBTS sequential hidden variable model
  • Definition 8: Outcome determinism
  • Definition 9: No-disturbance
  • ...and 28 more