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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
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 | |||||||||||||||||||||
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| Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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| Divisions: | Faculty of Science > Computer Science Faculty of Science > Mathematics |
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| 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: |
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| 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: |
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| Version or Related Resource: | https://wrap.warwick.ac.uk/111658 | |||||||||||||||||||||
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