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.
 G. W. Mackey, Mathematical Foundations of Quantum Mechanics (A. Benjamin, Inc., New York, 1963).
 G. Ludwig, Versuch einer axiomatischen Grundlegung der Quantenmechanik und allgemeinerer physikalischer Theorien, Zeitschrift für Physik 181, 233–260 (1964).
 B. Dakic and C. Brukner, Quantum Theory and Beyond: Is Entanglement Special?, in Deep Beauty - Understanding the Quantum World through Mathematical Innovation, edited by H. Halvorson (Cambridge University Press, 2011) pp. 365–392.
 L. Masanes and M. P. Muller, A derivation of quantum theory from physical requirements, New Journal of Physics 13, 063001 (2011).
 H. Barnum and A. Wilce, Information processing in convex operational theories, Electronic Notes in Theoretical Computer Science 270, 3 – 15 (2011).
 H. Barnum, R. Duncan, and A. Wilce, Symmetry, compact closure and dagger compactness for categories of convex operational models, Journal of Philosophical Logic 42, 501–523 (2013).
 A. J. P. Garner, O. C. O. Dahlsten, Y. Nakata, M. Murao, and V. Vedral, A framework for phase and interference in generalized probabilistic theories, New Journal of Physics 15, 093044 (2013).
 H. Barnum, M. P. Muller, and C. Ududec, Higher-order interference and single-system postulates characterizing quantum theory, New Journal of Physics 16, 123029 (2014). https://doi.org/10.1088/1367-2630/16/12/123029.
 H. Barnum, C. Lee, C. Scandolo, and J. Selby, Ruling out Higher-Order Interference from Purity Principles, Entropy 19, 253 (2017), arXiv:1704.05106 [quant-ph].
 M. P. Muller and C. Ududec, Structure of reversible computation determines the self-duality of quantum theory, Phys. Rev. Lett. 108, 130401 (2012).
 C. M. Lee and J. Barrett, Computation in generalised probabilisitic theories, New Journal of Physics 17, 083001 (2015), arXiv:1412.8671 [quant-ph].
 N. Brunner, M. Kaplan, A. Leverrier, and P. Skrzypczyk, Dimension of physical systems, information processing, and thermodynamics, New Journal of Physics 16, 123050 (2014).
 G. Chiribella and C. M. Scandolo, Entanglement and thermodynamics in general probabilistic theories, New Journal of Physics 17, 103027 (2015).
 H. Barnum, J. Barrett, L. Orloff Clark, M. Leifer, R. Spekkens, N. Stepanik, A. Wilce, and R. Wilke, Entropy and information causality in general probabilistic theories, New Journal of Physics 12, 033024 (2010), arXiv:0909.5075 [quant-ph].
 G. Kimura, K. Nuida, and H. Imai, Distinguishability measures and entropies for general probabilistic theories, Reports on Mathematical Physics 66, 175 – 206 (2010).
 P. Janotta, C. Gogolin, J. Barrett, and N. Brunner, Limits on nonlocal correlations from the structure of the local state space, New Journal of Physics 13, 063024 (2011).
 A. J. Short and J. Barrett, Strong nonlocality: a trade-off between states and measurements, New Journal of Physics 12, 033034 (2010).
 D. Gross, M. Muller, R. Colbeck, and O. C. O. Dahlsten, All reversible dynamics in maximally nonlocal theories are trivial, Phys. Rev. Lett. 104, 080402 (2010).
 S. W. Al-Safi and A. J. Short, Reversible dynamics in strongly non-local boxworld systems, Journal of Physics A: Mathematical and Theoretical 47, 325303 (2014).
 L. Hardy and W. K. Wootters, Limited Holism and Real-Vector-Space Quantum Theory, Foundations of Physics 42, 454–473 (2012), arXiv:1005.4870 [quant-ph].
 D. N. Finkelstein, Notes on Quaternion Quantum Mechanics, CERN publications: European Organization for Nuclear Research (CERN, 1959).
 K. Życzkowski, Quartic quantum theory: an extension of the standard quantum mechanics, Journal of Physics A: Mathematical and Theoretical 41, 355302 (2008).
 L. Masanes, M. P. Mueller, D. Perez-Garcia, and R. Augusiak, Entanglement and the three-dimensionality of the Bloch ball, Journal of Mathematical Physics 55, 122203 (2014).
 M. Krumm and M. P. Müller, Quantum computation is the unique reversible circuit model for which bits are balls, npj Quantum Information 5, 7 (2019), arXiv:1804.05736 [quant-ph].
 B. Dakić, T. Paterek, and C. Brukner, Density cubes and higher-order interference theories, New Journal of Physics 16, 023028 (2014).
 T. D. Galley and L. Masanes, Any modification of the Born rule leads to a violation of the purification and local tomography principles, Quantum 2, 104 (2018).
 L. Masanes, T. Galley, and M. P. Muller, The measurement postulates of quantum mechanics are operationally redundant, Nature Communications 10, 1361 (2019).
 F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Resource theory of quantum states out of thermal equilibrium, Physical Review Letters 111, 250404 (2013).
 F. Brandão, M. Horodecki, N. Ng, J. Oppenheim, and S. Wehner, The second laws of quantum thermodynamics, Proceedings of the National Academy of Sciences 112, 3275–3279 (2015).
 K. H. Hofmann and S. A. Morris, The structure of compact groups : a primer for students, a handbook for the expert, third edition, revised and augmented ed. (Berlin, Boston: De Gruyter, 2013).
 J. H. Gallier, Advanced geometric methods in computer science, https://www.cis.upenn.edu/ cis610/cis610-18-sl18.pdf (2018), chapter 20: Manifolds arising from group actions.
 M. D. Mazurek, M. F. Pusey, K. J. Resch, and R. W. Spekkens, Experimentally bounding deviations from quantum theory in the landscape of generalized probabilistic theories, PRX Quantum 2, 020302 (2021).
 L. Vesely, Extreme points of compact convex sets, http://www.mat.unimi.it/users/libor/AnConvessa/ext.pdf (2010),.
 L. Masanes, M. P. Muller, R. Augusiak, and D. Perez-Garcia, Existence of an information unit as a postulate of quantum theory, Proceedings of the National Academy of Sciences 110, 16373–16377 (2013).
 M. Pawłowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Żukowski, Information causality as a physical principle, Nature 461, 1101–1104 (2009).
 B. H. Gross, Some applications of gelfand pairs to number theory, Bull. Amer. Math. Soc. (N.S.) 24, 277–301 (1991).
 E. Cartan, Sur la détermination d'un système orthogonal complet dans un espace de Riemann symétrique clos, Rendiconti del Circolo Matematico di Palermo (1884-1940) 53, 217–252 (1929).
 S. Helgason, Groups and geometric analysis: integral geometry, invariant differential operators, and spherical functions, Pure and applied mathematics No. 113 (Academic Press, Orlando, 1984).
 J. Wolf, Harmonic Analysis on Commutative Spaces, Mathematical Surveys and Monographs, Vol. 142 (American Mathematical Society, Providence, Rhode Island, 2007).
 M. B. Halima, Branching rules for unitary groups and spectra of invariant differential operators on complex Grassmannians, Journal of Algebra 318, 520–552 (2007).
 Thomas D. Galley, Flaminia Giacomini, and John H. Selby, "A no-go theorem on the nature of the gravitational field beyond quantum theory", arXiv:2012.01441.
 Giacomo Mauro D'Ariano, Marco Erba, and Paolo Perinotti, "Classicality without local discriminability: Decoupling entanglement and complementarity", Physical Review A 102 5, 052216 (2020).
 Martin Plávala, "General probabilistic theories: An introduction", arXiv:2103.07469.
The above citations are from SAO/NASA ADS (last updated successfully 2021-06-16 00:46:30). The list may be incomplete as not all publishers provide suitable and complete citation data.
On Crossref's cited-by service no data on citing works was found (last attempt 2021-06-16 00:46:28).
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.