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Chistikov, Dmitry, Goulko, Olga, Kent, Adrian and Paterson, Michael S. (2020) Globe-hopping. Proceedings of the Royal Society A : mathematical, physical and engineering sciences, 476 (2238). doi:10.1098/rspa.2020.0038

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Official URL: http://dx.doi.org/10.1098/rspa.2020.0038

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Abstract

We consider versions of the grasshopper problem (Goulko & Kent 2017 Proc. R. Soc. A473, 20170494) on the circle and the sphere, which are relevant to Bell inequalities. For a circle of circumference 2π, we show that for unconstrained lawns of any length and arbitrary jump lengths, the supremum of the probability for the grasshopper’s jump to stay on the lawn is one. For antipodal lawns, which by definition contain precisely one of each pair of opposite points and have length π, we show this is true except when the jump length ϕ is of the form π(p/q) with p, q coprime and p odd. For these jump lengths, we show the optimal probability is 1 − 1/q and construct optimal lawns. For a pair of antipodal lawns, we show that the optimal probability of jumping from one onto the other is 1 − 1/q for p, q coprime, p odd and q even, and one in all other cases. For an antipodal lawn on the sphere, it is known (Kent & Pitalúa-García 2014 Phys. Rev. A90, 062124) that if ϕ = π/q, where q∈N, then the optimal retention probability of 1 − 1/q for the grasshopper’s jump is provided by a hemispherical lawn. We show that in all other cases where 0 < ϕ < π/2, hemispherical lawns are not optimal, disproving the hemispherical colouring maximality hypotheses (Kent & Pitalúa-García 2014 Phys. Rev. A90, 062124). We discuss the implications for Bell experiments and related cryptographic tests.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Q Science > QC Physics
Divisions: Faculty of Science > Computer Science
Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): Bell's theorem, Quantum theory -- Mathematical models
Journal or Publication Title: Proceedings of the Royal Society A : mathematical, physical and engineering sciences
Publisher: The Royal Society Publishing
ISSN: 1364-5021
Official Date: 1 June 2020
Dates:
DateEvent
1 June 2020Published
24 June 2020Available
15 May 2020Accepted
Date of first compliant deposit: 5 June 2020
Volume: 476
Number: 2238
DOI: 10.1098/rspa.2020.0038
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Copyright Holders: © 2020 The Author(s)
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
UNSPECIFIEDCentre for Discrete Mathematics and its Applications, University of WarwickUNSPECIFIED
EP/M013472/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
EP/T001011/1 [EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
UNSPECIFIEDFoundational Questions Institutehttp://dx.doi.org/10.13039/100009566
UNSPECIFIEDIndustry Canadahttp://dx.doi.org/10.13039/501100003102
UNSPECIFIEDOntario Ministry of Research, Innovation and Sciencehttp://dx.doi.org/10.13039/501100003400
Version or Related Resource: https://wrap.warwick.ac.uk/111658
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