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

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


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

► References

[1] I. E. Segal, Postulates for general quantum mechanics, Annals of Mathematics 48, 930–948 (1947).

[2] G. W. Mackey, Mathematical Foundations of Quantum Mechanics (A. Benjamin, Inc., New York, 1963).

[3] G. Ludwig, Versuch einer axiomatischen Grundlegung der Quantenmechanik und allgemeinerer physikalischer Theorien, Zeitschrift für Physik 181, 233–260 (1964).

[4] G. Ludwig, Attempt of an axiomatic foundation of quantum mechanics and more general theories. II, Comm. Math. Phys. 4, 331–348 (1967).

[5] G. Ludwig, Attempt of an axiomatic foundation of quantum mechanics and more general theories. III, Communications in Mathematical Physics 9, 1–12 (1968).

[6] G. Dähn, Attempt of an axiomatic foundation of quantum mechanics and more general theories. IV, Communications in Mathematical Physics 9, 192–211 (1968).

[7] P. Stolz, Attempt of an axiomatic foundation of quantum mechanics and more general theories. V, Communications in Mathematical Physics 11, 303–313 (1969).

[8] P. Stolz, Attempt of an axiomatic foundation of quantum mechanics and more general theories VI, Communications in Mathematical Physics 23, 117–126 (1971).

[9] B. Mielnik, Theory of filters, Communications in Mathematical Physics 15, 1–46 (1969).

[10] B. Mielnik, Generalized quantum mechanics, Communications in Mathematical Physics 37, 221–256 (1974).

[11] R. Giles, Foundations for Quantum Mechanics, Journal of Mathematical Physics 11, 2139 (1970).

[12] S. Gudder, Convex structures and operational quantum mechanics, Communications in mathematical Physics 29, 249–264 (1973).

[13] E. B. Davies and J. T. Lewis, An operational approach to quantum probability, Communications in Mathematical Physics 17, 239–260 (1970).

[14] L. Hardy, Quantum theory from five reasonable axioms, eprint arXiv:quant-ph/​0101012 (2001), quant-ph/​0101012.

[15] J. Barrett, Information processing in generalized probabilistic theories, Phys. Rev. A 75, 032304 (2007).

[16] 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.

[17] L. Masanes and M. P. Muller, A derivation of quantum theory from physical requirements, New Journal of Physics 13, 063001 (2011).

[18] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Informational derivation of quantum theory, Physical Review A 84, 012311 (2011).

[19] H. Barnum and A. Wilce, Information processing in convex operational theories, Electronic Notes in Theoretical Computer Science 270, 3 – 15 (2011).

[20] 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).

[21] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Probabilistic theories with purification, Physical Review A 81, 062348 (2010).

[22] 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).

[23] 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:/​/​​10.1088/​1367-2630/​16/​12/​123029.

[24] 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].

[25] C. M. Lee and J. H. Selby, Higher-order interference in extensions of quantum theory, Foundations of Physics 47, 89–112 (2017).

[26] M. P. Muller and C. Ududec, Structure of reversible computation determines the self-duality of quantum theory, Phys. Rev. Lett. 108, 130401 (2012).

[27] C. M. Lee and J. Barrett, Computation in generalised probabilisitic theories, New Journal of Physics 17, 083001 (2015), arXiv:1412.8671 [quant-ph].

[28] N. Brunner, M. Kaplan, A. Leverrier, and P. Skrzypczyk, Dimension of physical systems, information processing, and thermodynamics, New Journal of Physics 16, 123050 (2014).

[29] G. Chiribella and C. M. Scandolo, Operational axioms for diagonalizing states, EPTCS 2, 96–115 (2015).

[30] G. Chiribella and C. M. Scandolo, Entanglement and thermodynamics in general probabilistic theories, New Journal of Physics 17, 103027 (2015).

[31] 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].

[32] G. Kimura, K. Nuida, and H. Imai, Distinguishability measures and entropies for general probabilistic theories, Reports on Mathematical Physics 66, 175 – 206 (2010).

[33] 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).

[34] P. Janotta and R. Lal, Generalized probabilistic theories without the no-restriction hypothesis, Physical Review A 87, 052131 (2013).

[35] H. Barnum, M. A. Graydon, and A. Wilce, Composites and Categories of Euclidean Jordan Algebras, Quantum 4, 359 (2020).

[36] J. Bae, D.-G. Kim, and L.-C. Kwek, Structure of optimal state discrimination in generalized probabilistic theories, Entropy 18, 39 (2016).

[37] J. H. Selby and J. Sikora, How to make unforgeable money in generalised probabilistic theories, Quantum 2, 103 (2018).

[38] T. Heinosaari, L. Leppäjärvi, and M. Plávala, No-free-information principle in general probabilistic theories, Quantum 3, 157 (2019).

[39] E. Davies, Example Related to the Foundations of Quantum Theory, Journal of Mathematical Physics 13, 39–41 (1972).

[40] A. J. Short and J. Barrett, Strong nonlocality: a trade-off between states and measurements, New Journal of Physics 12, 033034 (2010).

[41] 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).

[42] 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).

[43] 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].

[44] A. Aleksandrova, V. Borish, and W. K. Wootters, Real-vector-space quantum theory with a universal quantum bit, Physical Review A 87, 052106 (2013).

[45] W. K. Wootters, Optimal Information Transfer and Real-Vector-Space Quantum Theory, arXiv eprints:quant-ph/​1301.2018 (2013).

[46] D. N. Finkelstein, Notes on Quaternion Quantum Mechanics, CERN publications: European Organization for Nuclear Research (CERN, 1959).

[47] K. Życzkowski, Quartic quantum theory: an extension of the standard quantum mechanics, Journal of Physics A: Mathematical and Theoretical 41, 355302 (2008).

[48] 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).

[49] 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].

[50] B. Dakić, T. Paterek, and C. Brukner, Density cubes and higher-order interference theories, New Journal of Physics 16, 023028 (2014).

[51] T. D. Galley and L. Masanes, Classification of all alternatives to the Born rule in terms of informational properties, Quantum 1, 15 (2017).

[52] 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).

[53] L. Masanes, T. Galley, and M. P. Muller, The measurement postulates of quantum mechanics are operationally redundant, Nature Communications 10, 1361 (2019).

[54] A. Wilce, Four and a half axioms for finite dimensional quantum mechanics (2009), arXiv:0912.5530 [quant-ph].

[55] L. d. Rio, J. Åberg, R. Renner, O. Dahlsten, and V. Vedral, The thermodynamic meaning of negative entropy, Nature 474, 61–63 (2011).

[56] M. Horodecki and J. Oppenheim, Fundamental limitations for quantum and nanoscale thermodynamics, Nature Communications 4, 2059 (2013).

[57] 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).

[58] P. Skrzypczyk, A. J. Short, and S. Popescu, Work extraction and thermodynamics for individual quantum systems, Nature Communications 5, 4185 (2014).

[59] 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).

[60] 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).

[61] J. H. Gallier, Advanced geometric methods in computer science, https:/​/​​ cis610/​cis610-18-sl18.pdf (2018), chapter 20: Manifolds arising from group actions.

[62] 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).

[63] R. Sanyal, F. Sottile, and B. Sturmfels, Orbitopes, Mathematika 57, 275–314 (2011).

[64] L. Vesely, Extreme points of compact convex sets, http:/​/​​users/​libor/​AnConvessa/​ext.pdf (2010),.

[65] 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).

[66] M. Pawłowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Żukowski, Information causality as a physical principle, Nature 461, 1101–1104 (2009).

[67] H.-C. Wang, Two-point homogeneous spaces, Annals of Mathematics 55, 177–191 (1952).

[68] B. H. Gross, Some applications of gelfand pairs to number theory, Bull. Amer. Math. Soc. (N.S.) 24, 277–301 (1991).

[69] 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).

[70] S. Helgason, Groups and geometric analysis: integral geometry, invariant differential operators, and spherical functions, Pure and applied mathematics No. 113 (Academic Press, Orlando, 1984).

[71] J. Wolf, Harmonic Analysis on Commutative Spaces, Mathematical Surveys and Monographs, Vol. 142 (American Mathematical Society, Providence, Rhode Island, 2007).

[72] M. B. Halima, Branching rules for unitary groups and spectra of invariant differential operators on complex Grassmannians, Journal of Algebra 318, 520–552 (2007).

[73] W. Fulton and J. Harris, Representation theory : a first course, Graduate texts in mathematics (Springer-Verlag, New York, Berlin, Paris, 1991).

[74] M. R. Sepanski, Compact Lie Groups, 1st ed. (Springer-Verlag New York, 2007).

[75] A. S. Holevo, Probabilistic and statistical aspects of quantum theory, 2nd ed., Quaderni Monographs No. 1 (Edizioni della normale, Pisa, 2011).

Cited by

[1] Martin Plávala, "General probabilistic theories: An introduction", Physics Reports 1033, 1 (2023).

[2] Thomas D. Galley, Flaminia Giacomini, and John H. Selby, "A no-go theorem on the nature of the gravitational field beyond quantum theory", Quantum 6, 779 (2022).

[3] John H. Selby, David Schmid, Elie Wolfe, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens, "Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality", Physical Review A 107 6, 062203 (2023).

[4] Giacomo Mauro D'Ariano, Marco Erba, and Paolo Perinotti, "Classicality without local discriminability: Decoupling entanglement and complementarity", Physical Review A 102 5, 052216 (2020).

[5] Paulo J. Cavalcanti, John H. Selby, Jamie Sikora, and Ana Belén Sainz, "Decomposing all multipartite non-signalling channels via quasiprobabilistic mixtures of local channels in generalised probabilistic theories", Journal of Physics A Mathematical General 55 40, 404001 (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2024-07-15 14:40:37) and SAO/NASA ADS (last updated successfully 2024-07-15 14:40:38). The list may be incomplete as not all publishers provide suitable and complete citation data.