bibkey: jokinenetal2024nobroadcasting
authors: Pauli Jokinen; Mirjam Weilenmann; Martin Plávala; Juha-Pekka Pellonpää; Jukka Kiukas; Roope Uola
year: 2024
title: "No-Broadcasting Characterizes Operational Contextuality"
doi: 10.1103/PhysRevLett.133.240201
url: https://arxiv.org/abs/2406.07305v1
claim: "The letter shows that measurement and preparation non-contextuality of a subtheory of quantum theory is equivalent to 1-to-n pseudo-broadcastability for all n, and conjectures that for a subtheory whose effects are generated by k measurements, 1-to-k pseudo-broadcasting already coincides with non-contextuality."
strata_touched:
  - D5/S3/QuantumContext/SingleMeasurementPseudobroadcastingRefutation
license: citation-only
triage: anchor

No-Broadcasting Characterizes Operational Contextuality

P. Jokinen, M. Weilenmann, M. Plávala, J.-P. Pellonpää, J. Kiukas and R. Uola, arXiv:2406.07305v1 (the only version); Phys. Rev. Lett. 133(24) (2024), 240201. Quotations are from the v1 TeX source.

• The macros \hT, \hA, \hS, \hL, \lh, \hi, \N and \btn{n} are expanded to $\mathcal{T}$, $\mathcal{A}$, $\mathcal{S}$, $\mathcal{L}$, $\mathcal{L}(\mathcal{H})$, $\mathcal{H}$, $\mathbb{N}$ and $1\mapsto n$.
• \tr{X} is written $\mathrm{tr}(X)$, and citation commands are omitted.

Subtheories and non-contextuality:

We are here interested in theories that consist of some convex subset of quantum states and a convex set of effects $E$, i.e. operators with $0\le E\le I$, that includes the identity operator and yes-no questions, i.e., if an effect $E$ is part of the theory, then also $I-E$ is. Such a theory is called measurement and preparation non-contextual if $\mathrm{tr}(\rho E)={\sum }_{\lambda }\mathrm{tr}(\rho {\stackrel{\lambda )tr(\sigma }{G}}_{\lambda }E)$ for all states $\rho$ and effects $E$ of the theory. Here $\{{\stackrel{\lambda \}\lambda \; and\; \{\sigma }{G}}_{\lambda }{\}}_{\lambda }$ satisfy $\mathrm{tr}(\rho {\stackrel{\lambda )\ge 0\; and\; \sum \lambda tr(\rho G}{G}}_{\lambda })=1$ for all states of the theory, and $\mathrm{tr}({\stackrel{\lambda E)\ge 0\; and\; tr(\sigma }{\sigma }}_{\lambda })=1$ for all $\lambda$ and all effects $E$ of the theory.

Pseudo-broadcasting:

we say that $(\mathcal{T},{\mathcal{A}}_1,\dots ,{\mathcal{A}}_n)$ is $1\mapsto n$\emph{-pseudo-broadcastable} if there is a trace-preserving map $\Xi :\mathcal{L}(\mathcal{H})\to \mathcal{L}({\mathcal{H}}^{\otimes n})$ such that for all $i\in \{1,\dots ,n\}$, $\rho \in \mathcal{T}$, and $\{A_{i,k}{\}}_k\in {\mathcal{A}}_i$ it holds that $\mathrm{tr}(A_{i,k}{\mathrm{tr}}_{i^c}\Xi (\rho ))=\mathrm{tr}(A_{i,k}\rho )$, $\mathrm{tr}(\Xi (\rho ){\otimes }_{i=1}^n{A}_{i,k})\ge 0$.

Theorem 5 (Theorem thmgpt):

A subtheory of quantum theory characterized by the allowed states $\mathcal{T}$ and allowed measurements $\mathcal{A}$ is measurement and preparation non-contextual if and only if $(\mathcal{T},\mathcal{A})$ is $1\mapsto n$-pseudo-broadcastable for all $n\in \mathbb{N}$.

The conjecture, in the main text:

This is likely not a coincidence since the scenario is generated by $3$ measurements and we conjecture that for a scenario generated by $k$ measurements $1\mapsto k$-pseudobroadcasting coincides with non-contextuality.

and in Appendix D:

One can also see that $\mathcal{S}(\mathcal{A})$ is a Cartesian product of $k$ simplexes if $\mathcal{A}$ is generated by $k$ independent measurements. This leads to a conjecture that if $\mathcal{A}$ is generated by $k$ measurements, then $1\mapsto k$-pseudobroadcasting is equivalent to non-contextuality. This is only a conjecture since even if $\mathcal{A}$ is generated by $k$ measurements, $\mathcal{S}(\mathcal{A})$ in general is only a subset of a Cartesian product of $k$ simplexes. We leave resolving this issue and proving or disproving the conjecture for future work.

Verified locator

Source: https://arxiv.org/abs/2406.07305v1

• URL: https://arxiv.org/abs/2406.07305v1 (the e-print is a tar archive; member Arxiv.tex, md5 cc2fa845c0b0f4155ed18dcb76e4c92e, 534 lines):
  • subtheories, non-contextuality and pseudo-broadcasting, l. 321–331;
  • Theorem 5, l. 336–338;
  • the conjecture in the main text, l. 348;
  • the conjecture in Appendix D, l. 482.
• Published version: the University of Turku repository reprint PhysRevLet-Pellonpää-2025-FINAL.pdf (13 pages) states the conjecture on p. 6 and p. 12 with the same wording.
• DOI: 10.1103/PhysRevLett.133.240201 (https://doi.org/10.1103/PhysRevLett.133.240201; Crossref lists Physical Review Letters, volume 133, issue 24, article 240201, published 9 December 2024).