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On the Interpretation of Probabilities in Generalized Probabilistic Models

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Quantum Interaction (QI 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10106))

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Abstract

We discuss generalized probabilistic models for which states not necessarily obey Kolmogorov’s axioms of probability. We study the relationship between properties and probabilistic measures in this setting, and explore some possible interpretations of these measures.

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Notes

  1. 1.

    An orthomodular lattice \(\mathcal {L}\), is defined as an orthocomplemented lattice satisfying that, for any a, b and c, if \(a\le c\), then \(a\vee (a^{\bot }\wedge c)=c\). In the Hilbert space case, projection operators are in one to one correspondence to closed subspaces. These form an orthomodular lattice with “\(\vee \)" representing the closure of the sum of two subspaces, “\(\wedge \)" its intersection, and “\((...)^{\bot }\)"representing the orthogonal complement of a given subspace. “\(\le \)" means subspace inclusion. See [23] for a detailed exposition.

  2. 2.

    It is important to notice here that different notions of “rational agent” could be used. In particular, it would be interesting to study the possibility of using Dutch Book Arguments in the generalized setting.

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Acknowledgments

The Authors acknowledge CONICET and UNLP (Argentina). We are grateful to the anonymous reviewers, whose comments have helped to improve the manuscript. This publication was also made possible through the support of grant 57919 from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. FH also is grateful to the participants of the Conference “Quantum interactions - 2016” (San Franciso, July 2016), for the stimulating and lively discussions that have enriched this work.

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Authors and Affiliations

  1. National University La Plata and CONICET IFLP-CCT, C.C. 727, 1900, La Plata, Argentina

    Federico Holik, Gustavo Bosyk & Angelo Plastino

  2. University of Buenos Aires - CONICET, Buenos Aires, Argentina

    Sebastian Fortin

Authors
  1. Federico Holik
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  2. Sebastian Fortin
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  3. Gustavo Bosyk
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  4. Angelo Plastino
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Correspondence to Federico Holik .

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Editors and Affiliations

  1. San Francisco State University, San Francisco, California, USA

    Jose Acacio de Barros

  2. University of Oxford, Oxford, United Kingdom

    Bob Coecke

  3. City University of London, London, United Kingdom

    Emmanuel Pothos

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Holik, F., Fortin, S., Bosyk, G., Plastino, A. (2017). On the Interpretation of Probabilities in Generalized Probabilistic Models. In: de Barros, J., Coecke, B., Pothos, E. (eds) Quantum Interaction. QI 2016. Lecture Notes in Computer Science(), vol 10106. Springer, Cham. https://doi.org/10.1007/978-3-319-52289-0_16

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  • DOI: https://doi.org/10.1007/978-3-319-52289-0_16

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