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How dynamics constrains probabilities in general probabilistic theories

Galley, TD; Masanes, L; (2021) How dynamics constrains probabilities in general probabilistic theories. Quantum , 5 (457) pp. 1-40. 10.22331/q-2021-05-21-457. Green open access

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Abstract

We introduce a general framework for analysing general probabilistic theories, which emphasises the distinction between the dynamical and probabilistic structures of a system. The dynamical structure is the set of pure states together with the action of the reversible dynamics, whilst the probabilistic structure determines the measurements and the outcome probabilities. For transitive dynamical structures whose dynamical group and stabiliser subgroup form a Gelfand pair we show that all probabilistic structures are rigid (cannot be infinitesimally deformed) and are in one-to-one correspondence with the spherical representations of the dynamical group. We apply our methods to classify all probabilistic structures when the dynamical structure is that of complex Grassmann manifolds acted on by the unitary group. This is a generalisation of quantum theory where the pure states, instead of being represented by one-dimensional subspaces of a complex vector space, are represented by subspaces of a fixed dimension larger than one. We also show that systems with compact two-point homogeneous dynamical structures (i.e. every pair of pure states with a given distance can be reversibly transformed to any other pair of pure states with the same distance), which include systems corresponding to Euclidean Jordan Algebras, all have rigid probabilistic structures.

Type: Article
Title: How dynamics constrains probabilities in general probabilistic theories
Open access status: An open access version is available from UCL Discovery
DOI: 10.22331/q-2021-05-21-457
Publisher version: https://doi.org/10.22331/q-2021-05-21-457
Language: English
Additional information: © Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften. This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license (https://creativecommons.org/licenses/by/4.0/).
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > London Centre for Nanotechnology
UCL > Provost and Vice Provost Offices > UCL SLASH
UCL > Provost and Vice Provost Offices > UCL SLASH > Faculty of Arts and Humanities
UCL > Provost and Vice Provost Offices > UCL SLASH > Faculty of Arts and Humanities > Arts and Sciences (BASc)
URI: https://discovery.ucl.ac.uk/id/eprint/10129039
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