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Basics of Quantum Theory for Quantum-Like Modeling Information Retrieval

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Quantum-Like Models for Information Retrieval and Decision-Making

Abstract

This chapter contains a brief introduction to the mathematical formalism and axiomatics of quantum mechanics (QM). Recently quantum mathematics and methodology started to be widely used for modeling decision making for humans and AI-systems, including quantum-like modeling information retrieval. Experts in such areas do not go deeply into the details of quantum theory. Moreover, typically such consumers of quantum theory do not use all its components. Quantum measurement theory is the most useful for application, including information retrieval. The main issue is the quantum treatment of incompatible observables represented mathematically by noncommuting Hermitian operators. At the level of statistical data incompatibility is represented as interference of probabilities, in the form of modification of the formula of total probability by adding the interference term.

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Notes

  1. 1.

    See, for example, [1,2,3,4,5,6,7,8,9,10, 15, 27, 28, 32, 38,39,40,41, 43,44,45,46,47,48, 55, 57].

  2. 2.

    In mathematical texts one typically considers linearity with respect to the first argument. Thus a mathematician has to pay attention to this difference.

  3. 3.

    To simplify formulas, we shall not put the operator-label “hat” in the symbols denoting projectors, i.e., \(P\equiv \widehat {P}.\)

  4. 4.

    Thus states are given by elements of the unit sphere of the Hilbert space H.

  5. 5.

    Here it is convenient to use this symbol, not just write as Ψ(x, y) = ψ(x)ϕ(y).

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Author information

Authors and Affiliations

  1. Linnaeus University, International Center for Mathematical Modeling in Physics and Cognitive Sciences, Växjö, Sweden

    Andrei Khrennikov

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  1. Andrei Khrennikov
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Correspondence to Andrei Khrennikov .

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Editors and Affiliations

  1. Department of Mathematics, Brussels Free University, Center Leo Apostel for Interdisciplinary Studies, Brussels, Belgium, Brussels, Belgium

    Diederik Aerts

  2. Linnaeus University, International Center for Mathematical Modeling in Physics and Cognitive Sciences, Växjö, Sweden

    Andrei Khrennikov

  3. Department of Information Engineering, University of Padova, Padova, Italy

    Massimo Melucci

  4. Department of Mathematics, Howard University, Washington, DC, USA

    Bourama Toni

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Khrennikov, A. (2019). Basics of Quantum Theory for Quantum-Like Modeling Information Retrieval. In: Aerts, D., Khrennikov, A., Melucci, M., Toni, B. (eds) Quantum-Like Models for Information Retrieval and Decision-Making. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham. https://doi.org/10.1007/978-3-030-25913-6_4

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