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01.10.2023

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

verfasst von: V. V. Kornyak

Erschienen in: Programming and Computer Software | Ausgabe 5/2023

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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.

Vorheriger Artikel On Implementation of Numerical Methods for Solving Ordinary Differential Equations in Computer Algebra Systems

Nächster Artikel Resonances and Periodic Motions of Atwood’s Machine with Two Oscillating Weights

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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.

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

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

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

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., ω_ij = –1 and relations (4.6) take form e_ie_j = –e_je_i.

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

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.

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

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.

Pattee, H.H., The complementarity principle in biological and social structures, J. Soc. Biol. Struct., 1978, vol. 1, no. 2, pp. 191–200. https://doi.org/10.1016/S0140-1750(78)80007-4CrossRef

Schwinger, J., Unitary operator bases, Proc. Natl. Acad. Sci. U. S. A., 1960, vol. 46, no. 4, pp. 570–579. https://doi.org/10.1073/pnas.46.4.570MathSciNetCrossRefMATH

Wootters, W.K. and Fields, B.D., Optimal state-determination by mutually unbiased measurements, Ann. Phys., 1989, vol. 191, no. 2, pp. 363–381.MathSciNetCrossRef

Kornyak, V.V., Quantum models based on finite groups, J. Phys.: Conf. Ser., 2018, vol. 965, p. 012023. https://doi.org/10.1088/1742-6596/965/1/012023

Kornyak, V.V., Modeling quantum behavior in the framework of permutation groups, EPJ Web Conf., 2018, vol. 173, p. 01007. https://doi.org/10.1051/epjconf/201817301007.

Kornyak, V.V., Mathematical modeling of finite quantum systems, Lect. Notes Comput. Sci., 2012, vol. 7125, pp. 79–93.CrossRef

t'Hooft, G., The Cellular Automaton Interpretation of Quantum Mechanics. Fundamental Theories of Physics, Springer, 2016. https://doi.org/10.1007/978-3-319-41285-6

Weyl, H., The Theory of Groups and Quantum Mechanics, Martino Fine Books, 2014.MATH

Durt, Th., Englert, B.-G., Bengtsson, I., and Życzkowski, K., On mutually unbiased bases, Int. J. Quantum Inf., 2010, vol. 8, no. 4, pp. 535–640.CrossRefMATH

10.

D'Ariano G. Mauro, Paris Matteo G.A., and Sacchi Massimiliano F., Quantum tomography, Adv. Imaging Electron Phys., 2003, vol. 128, pp. 206–309.

11.

Ivanovic, I.D., Geometrical description of quantal state determination, J. Phys. A: Math. General, 1981, vol. 14, no. 12, pp. 3241–3245. https://doi.org/10.1088/0305-4470/14/12/019MathSciNetCrossRef

12.

Englert, B.-G. and Aharonov, Y., The mean king’s problem: Prime degrees of freedom, Phys. Lett. A, 2001, vol. 284, no. 1, pp. 1–5.MathSciNetCrossRef

13.

Bandyopadhyay Somshubhro, Boykin P. Oscar, Roychowdhury Vwani, and Vatan Farrokh, A new proof for the existence of mutually unbiased bases, Algorithmica, 2002, vol. 34, no. 4, pp. 512–528.MathSciNetCrossRefMATH

14.

Jagannathan Ramaswamy, On generalized Clifford algebras and their physical applications, The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, Springer, 2010, pp. 465–489.

15.

Kirillov, A.A., Elementy teorii predstavlenii (Elements of the Representation Theory), Moscow: Nauka, 1978.

16.

Banks, T., Finite deformations of quantum mechanics, 2020.

17.

Kornyak, V.V., Quantum mereology in finite quantum mechanics, Discrete Contin. Models Appl. Comput. Sci., 2021, vol. 29, no. 4, pp. 347–360. https://journals.rudn.ru/miph/article/view/29428.

18.

Kornyak, V.V., Subsystems of a closed quantum system in finite quantum mechanics, J. Math. Sci., 2022, vol. 261, pp. 717–729. https://doi.org/10.1007/s10958-022-05783-2MathSciNetCrossRefMATH

19.

Kornyak, V.V., Decomposition of a finite quantum system into subsystems: Symbolic–numerical approach, Program. Comput. Software, 2022, vol. 48, pp. 293–300.MathSciNetCrossRefMATH

20.

Brierley, S., Weigert, S., and Bengtsson, I., All mutually unbiased bases in dimensions two to five, Quantum Inf. Comput., 2010, vol. 10, pp. 803–820.MathSciNetMATH

21.

Brierley, S. and Weigert, S., Constructing mutually unbiased bases in dimension six, Phys. Rev. A, 2009, vol. 79, p. 052316. https://doi.org/10.1103/PhysRevA.79.052316

22.

Colomer, M.P., Mortimer, L., Frérot, I., Farkas, M., and Acín, A., Three numerical approaches to find mutually unbiased bases using Bell inequalities, 2022.

23.

Klappenecker, A. and Rötteler, M., Constructions of mutually unbiased bases, Int. Conf. Finite Fields and Appl., Springer, 2003, pp. 137–144.

24.

Bron, C. and Kerbosch, J., Algorithm 457: Finding all cliques of an undirected graph, Commun. ACM, 1973, vol. 16, pp. 575–577.CrossRefMATH

25.

Reingold, E.M., Nievergelt, J., and Deo, N., Combinatorial Algorithms: Theory and Practice, Pearson College Div, 1977.MATH

26.

Tarjan, R.E. and Trojanowski, A.E., Finding a maximum independent set, SIAM J. Comput., 1977, vol. 6, no. 3, pp. 537–546. https://doi.org/10.1137/0206038MathSciNetCrossRefMATH

27.

Carraghan, R. and Pardalos, P.M., An exact algorithm for the maximum clique problem, Operations Res. Lett., 1990, vol. 9, no. 6, pp. 375–382. https://doi.org/10.1016/0167-6377(90)90057-CCrossRefMATH

28.

Robson, J.M., Algorithms for maximum independent sets, J. Algorithms, 1986, vol. 7, pp. 425–440.MathSciNetCrossRefMATH

29.

Östergård, P.R.J., A fast algorithm for the maximum clique problem, Discrete Appl. Math., 2002, vol. 120, no. 1, pp. 197–207. https://doi.org/10.1016/S0166-218X(01)00290-6MathSciNetCrossRefMATH

Titel: Complementarity in Finite Quantum Mechanics and Computer-Aided Computations of Complementary Observables
verfasst von: V. V. Kornyak
Publikationsdatum: 01.10.2023
Verlag: Pleiades Publishing
Erschienen in: Programming and Computer Software / Ausgabe 5/2023
Print ISSN: 0361-7688
Elektronische ISSN: 1608-3261
DOI: https://doi.org/10.1134/S036176882302010X

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