How dynamics constrains probabilities in general probabilistic theories

Thomas D. Galley; Lluis Masanes

doi:10.22331/q-2021-05-21-457

How dynamics constrains probabilities in general probabilistic theories

Thomas D. Galley¹ and Lluis Masanes²

¹Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
²Department of Computer Science, University College London, Gower Street, London WC1E 6BT, United Kingdom

Published:	2021-05-21, volume 5, page 457
Eprint:	arXiv:2002.05088v3
Doi:	https://doi.org/10.22331/q-2021-05-21-457
Citation:	Quantum 5, 457 (2021).

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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.

► BibTeX data

@article{Galley2021howdynamics,
  doi = {10.22331/q-2021-05-21-457},
  url = {https://doi.org/10.22331/q-2021-05-21-457},
  title = {How dynamics constrains probabilities in general probabilistic theories},
  author = {Galley, Thomas D. and Masanes, Lluis},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {457},
  month = may,
  year = {2021}
}

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