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Positive Operator-Valued Measures in Quantum Decision Theory

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Quantum Interaction (QI 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8951))

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Abstract

We show that the correct mathematical foundation of quantum decision theory, dealing with uncertain events, requires the use of positive operator-valued measure that is a generalization of the projection-valued measure. The latter is appropriate for operationally testable events, while the former is necessary for characterizing operationally uncertain events. In decision making, one has to distinguish composite non-entangled events from composite entangled events. The mathematical definition of entangled prospects is based on the theory of Hilbert-Schmidt spaces and is analogous to the definition of entangled statistical operators in quantum information theory. We demonstrate that the necessary condition for the appearance of an interference term in the quantum probability is the occurrence of entangled prospects and the existence of an entangled strategic state of a decision maker. The origin of uncertainties in standard lotteries is explained.

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Author information

Authors and Affiliations

  1. Department of Management, Technology and Economics, ETH Zürich, Swiss Federal Institute of Technology, 8092, Zürich, Switzerland

    Vyacheslav I. Yukalov & Didier Sornette

  2. Bogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, 141980, Russia

    Vyacheslav I. Yukalov

  3. Swiss Finance Institute, C/o University of Geneva, 40 Blvd. Du Pont D’Arve, 1211, Geneva 4, Switzerland

    Didier Sornette

Authors
  1. Vyacheslav I. Yukalov
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  2. Didier Sornette
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Corresponding author

Correspondence to Vyacheslav I. Yukalov .

Editor information

Editors and Affiliations

  1. ETH Zürich, Zürich, Switzerland

    Harald Atmanspacher

  2. Universitätsspital Bern, Bern, Switzerland

    Claudia Bergomi

  3. University of Freiburg, Freiburg, Germany

    Thomas Filk

  4. Queensland University of Technology, Brisbane, Queensland, Australia

    Kirsty Kitto

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Yukalov, V.I., Sornette, D. (2015). Positive Operator-Valued Measures in Quantum Decision Theory. In: Atmanspacher, H., Bergomi, C., Filk, T., Kitto, K. (eds) Quantum Interaction. QI 2014. Lecture Notes in Computer Science(), vol 8951. Springer, Cham. https://doi.org/10.1007/978-3-319-15931-7_12

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  • DOI: https://doi.org/10.1007/978-3-319-15931-7_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-15930-0

  • Online ISBN: 978-3-319-15931-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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