Classification of all alternatives to the Born rule in terms of informational properties

Thomas D. Galley and Lluis Masanes

Department of Physics and Astronomy, University College London,Gower Street, London WC1E 6BT, United Kingdom

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The standard postulates of quantum theory can be divided into two groups: the first one characterizes the structure and dynamics of pure states, while the second one specifies the structure of measurements and the corresponding probabilities. In this work we keep the first group of postulates and characterize all alternatives to the second group that give rise to finite-dimensional sets of mixed states. We prove a correspondence between all these alternatives and a class of representations of the unitary group. Some features of these probabilistic theories are identical to quantum theory, but there are important differences in others. For example, some theories have three perfectly distinguishable states in a two-dimensional Hilbert space. Others have exotic properties such as lack of bit symmetry, the violation of no simultaneous encoding (a property similar to information causality) and the existence of maximal measurements without phase groups. We also analyze which of these properties single out the Born rule.

The nature of measurement is one of the most debated questions in the foundations of quantum mechanics. In this work we explore alternatives to the way measurements are mathematically described in quantum mechanics. We make an exhaustive classification of all such alternatives in terms of group theory. We then analyze the information-processing properties of all these alternative theories, and conclude that quantum mechanics is the only one which satisfies an informational property known as bit symmetry.

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[2] Yurii V. Brezhnev, "The Born rule as a statistics of quantum micro-events", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476 2244, 20200282 (2020).

[3] Marius Krumm and Markus P. Müller, "Quantum computation is the unique reversible circuit model for which bits are balls", npj Quantum Information 5 1, 7 (2019).

[4] Thomas D. Galley and Lluis Masanes, "Any modification of the Born rule leads to a violation of the purification and local tomography principles", Quantum 2, 104 (2018).

[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 and Theoretical 55 40, 404001 (2022).

[6] Lluís Masanes, Thomas D. Galley, and Markus P. Müller, "The measurement postulates of quantum mechanics are operationally redundant", Nature Communications 10 1, 1361 (2019).

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[8] Ciarán M. Lee and John H. Selby, "A no-go theorem for theories that decohere to quantum mechanics", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474 2214, 20170732 (2018).

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[10] Lev Vaidman, Jerusalem Studies in Philosophy and History of Science 567 (2020) ISBN:978-3-030-34315-6.

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[12] Pei Wang, "Derive the Born’s rule from environment-induced stochastic dynamics of wave functions in an open system", The European Physical Journal Plus 135 11, 927 (2020).

[13] Thomas D. Galley and Lluis Masanes, "How dynamics constrains probabilities in general probabilistic theories", Quantum 5, 457 (2021).

[14] 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).

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