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Programming and Computer Software

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

Vladimir V. Kornyak

Published Oct 1, 2023• 1 citation
Publisher

Abstract

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.

DOI
10.1134/s036176882302010x

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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.
Complementary observables in quantum mechanics may be viewed as Frobenius structures in a dagger monoidal category, such as the category of finite dimensional Hilbert spaces over the complex numbers. On the other hand, their properties crucially depend on the discrete Fourier transform and its associated quantum torus, requiring only the finite fields that underlie mutually unbiased bases. In axiomatic topos theory, the complex numbers are difficult to describe and should not be invoked unnecessarily. This paper surveys some fundamentals of quantum arithmetic using finite field complementary observables, with a view considering more general axiom systems.
The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are reviewed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned. KEY WORDS: mutually unbiased bases; d-dimensional Hilbert space; Galois fields and rings; maximally entangled states.
Complementarity is a very old concept in quantum mechanics, however the rigorous definition is not so old. Complementarity of orthogonal bases can be formulated in terms of maximal Abelian algebras and this may lead to avoid commutativity of the subalgebras. In some sense this means that quantum information is treated instead of classical (measurement) information. The subject is to extend to the quantum case some features from the classical case. This includes construction of complementary subalgebras. The Bell basis has also some relation. Several open questions are discussed.
Mutually unbiased bases encapsulate the concept of complementarity---the impossibility of simultaneous knowledge of certain observables---in the formalism of quantum theory. Although this concept is at the heart of quantum mechanics, the number of these bases is unknown except for systems of dimension being a power of a prime. We develop the relation between this physical problem and the mathematical problem of finding the number of mutually orthogonal Latin squares. We derive in a simple way all known results about the unbiased bases, find their lower number, and disprove the existence of certain forms of the bases in dimensions different than power of a prime. Using the Latin squares, we construct hidden-variable models which efficiently simulate results of complementary quantum measurements.