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No-broadcasting characterizes operational contextuality

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

TL;DR

The paper resolves the characterization of generalized contextuality in quantum theory by proving a tight correspondence with the no-broadcasting theorem, extending this link to subtheories through pseudo-broadcasting. It shows that preparation-contextuality and measurement-contextuality map to fixed-point structures of entanglement-breaking channels, yielding concrete criteria based on broadcasting and norm-1 post-processing. By connecting contextuality to existing notions such as joint measurability, non-disturbance, and macrorealism, it clarifies the relative strength of non-contextuality as a classicality boundary. The results provide operational tools (e.g., SDP-based tests) and illuminate how contextual advantages in quantum information can be understood through information-theoretic constraints, with potential extensions to continuous-variable systems.

Abstract

Operational contextuality forms a rapidly developing subfield of quantum information theory. However, the characterization of the quantum mechanical entities that fuel the phenomenon has remained unknown with many partial results existing. Here, we present a resolution to this problem by connecting operational contextuality one-to-one with the no-broadcasting theorem. The connection works both on the level of full quantum theory and subtheories thereof. We demonstrate the connection in various relevant cases, showing especially that for quantum states the possibility of demonstrating contextuality is exactly characterized by non-commutativity, and for measurements this is done by a norm-1 property closely related to repeatability. Moreover, we show how techniques from broadcasting can be used to simplify known foundational results in contextuality.

No-broadcasting characterizes operational contextuality

TL;DR

The paper resolves the characterization of generalized contextuality in quantum theory by proving a tight correspondence with the no-broadcasting theorem, extending this link to subtheories through pseudo-broadcasting. It shows that preparation-contextuality and measurement-contextuality map to fixed-point structures of entanglement-breaking channels, yielding concrete criteria based on broadcasting and norm-1 post-processing. By connecting contextuality to existing notions such as joint measurability, non-disturbance, and macrorealism, it clarifies the relative strength of non-contextuality as a classicality boundary. The results provide operational tools (e.g., SDP-based tests) and illuminate how contextual advantages in quantum information can be understood through information-theoretic constraints, with potential extensions to continuous-variable systems.

Abstract

Operational contextuality forms a rapidly developing subfield of quantum information theory. However, the characterization of the quantum mechanical entities that fuel the phenomenon has remained unknown with many partial results existing. Here, we present a resolution to this problem by connecting operational contextuality one-to-one with the no-broadcasting theorem. The connection works both on the level of full quantum theory and subtheories thereof. We demonstrate the connection in various relevant cases, showing especially that for quantum states the possibility of demonstrating contextuality is exactly characterized by non-commutativity, and for measurements this is done by a norm-1 property closely related to repeatability. Moreover, we show how techniques from broadcasting can be used to simplify known foundational results in contextuality.
Paper Structure (12 sections, 5 theorems, 32 equations, 2 figures)

This paper contains 12 sections, 5 theorems, 32 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Operational contextuality
  3. Broadcasting
  4. Connection between operational non-contextuality and broadcasting
  5. Relation to other notions of classicality
  6. Pseudo-broadcasting and conditions for contextuality in a subtheory
  7. Conclusions
  8. Proof of Theorem \ref{['thmstates']}
  9. Proof of Theorem \ref{['thmmeasurement']}
  10. There exist tuples $(\mathcal{T},\mathcal{A})$ that are not broadcastable, but are $1 \mapsto n$-pseudo-broadcastable
  11. Proof of Theorem \ref{['thmgpt']}
  12. $1 \mapsto n$-pseudobroadcasting in the scenario investigated in mazurek2016experimental

Key Result

Theorem 1

Let $\mathcal{T} \subset \mathcal{S(H)}$. Then there is an EBC $\Lambda$ such that $\mathcal{T} \subset \mathrm{Fix}(\Lambda)$ if and only if $(\mathcal{T},\mathcal{O},\mathcal{O})$ is broadcastable.

Figures (2)

  • Figure 1: Non-contextuality of an operational theory and the corresponding quantum mechanical entities. On the left (resp. right) are the properties requested from an ontological model for states (resp. measurements). The color green refers to ontological models for full theory, yellow refers to subtheories, and red refers to models exhibiting contextuality. The binding quantum mechanical entity is given in the middle. The black connections represent known results from Refs. TavakoliUolaplavala2022incompatibilitySelby2023plavala2024contextuality. The contributions of this work are given by the blue connections.
  • Figure 2: The regions for pseudobroadcastability of the preparations and measurements considered in mazurek2016experimental. $\mu$ and $\eta$ are unitless parameters of the dephasing channels acting on the preparations and depolarizing channel acting on the measurements, respectively. The green region where the scenario is $1 \mapsto 3$-pseudobroadcastable coincides with the region where the inequality presented in mazurek2016experimental is violated up to numerical precision of our calculations.

Theorems & Definitions (5)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Theorem 5