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Complementarity in Finite Quantum Mechanics and Computer-Aided Computations of Complementary Observables

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Abstract

Mathematical formulation of Bohr’s complementarity principle leads to the concepts of mutually unbiased bases in Hilbert spaces and complementary quantum observables. In this paper, we consider algebraic structures associated with these concepts and their applications to constructive quantum mechanics. We also briefly discuss some computer-algebraic approaches to the problems under consideration and propose an algorithm for solving one of them.

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Notes

  1. It is assumed that variations of the complementarity principle are applicable in various fields where, for the sake of completeness of description, different mutually incompatible means are used. For instance, paper [1] overviews the application of the complementarity principle in biology, psychology, and social sciences.

  2. The need for at least two different bases (coordinate systems) to describe physical reality is a sort of manifestation of the complementarity principle.

  3. It should be noted that cyclic permutations are the simplest components into which any permutation is decomposed.

  4. Operator A is called normal if it commutes with its adjoint: AA* = A*A.

  5. The common Clifford algebra, which is defined by decomposition of a given quadratic form in an n-dimensional space into a product of linear factors, corresponds to the case of square roots of unity, i.e., ωij = –1 and relations (4.6) take form eiej = –ejei.

  6. 't Hooft uses the term “ontic” as an abbreviation for “ontological.”

  7. We use this term because the eigenvalues of operator \(\mathcal{X}\) are frequency exponents proportional to energies in accordance with the Planck formula E = hν, which is valid for periodic processes of any nature.

  8. A clique is a complete subgraph of an undirected graph that is not contained in a larger complete subgraph.

  9. After Bron and Kerbosch’s work, a number of competing algorithms appeared [26–29]; however, due to the noncritical nature of this part of our computations, it is hardly reasonable to engage in a comparative study of these different algorithms.

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  1. Joint Institute for Nuclear Research, 141980, Dubna, Moscow oblast, Russia

    V. V. Kornyak

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Translated by Yu. Kornienko

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Kornyak, V.V. Complementarity in Finite Quantum Mechanics and Computer-Aided Computations of Complementary Observables. Program Comput Soft 49, 423–432 (2023). https://doi.org/10.1134/S036176882302010X

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