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0 Factorization Properties of Finite Spaces

Boris Simkhovich, A. Mann, J. Žák

Published Aug 18, 2016• 1 citation
OpenAlex

Abstract

In 1960 Schwinger [J. Schwinger, Proc.Natl.Acad.Sci. 46 (1960) 570- 579] proposed the algorithm for factorization of unitary operators in the finite M dimensional Hilbert space according to a coprime decomposition of M. Using a special permutation operator A we generalize the Schwinger factorization to every decomposition of M. We obtain the factorized pairs of unitary operators and show that they obey the same commutation relations as Schwinger's. We apply the new factorization to two problems. First, we show how to generate two kq-like mutually unbiased bases for any composite dimension. Then, using a Harper-like Hamiltonian model in the finite dimension M = M1M2, we show how to design a physical system with M1 energy levels, each having degeneracy M2.

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In 1960 Schwinger (J Schwinger 1960 Proc. Natl Acad. Sci. 46 570–9) proposed the algorithm for factorization of unitary operators in the finite M -dimensional Hilbert space according to a coprime decomposition of M . Using a special permutation operator A we generalize the Schwinger factorization to every decomposition of M . We obtain the factorized pairs of unitary operators and show that they obey the same commutation relations as Schwinger's. We apply the new factorization to two problems. First, we show how to generate two kq -like mutually unbiased bases for any composite dimension. Then, using a Harper-like Hamiltonian model in the finite dimension M = M 1 M 2 , we show how to design a physical system with M 1 energy levels, each having degeneracy M 2 .

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Journal of Physics Conference Series•Mar 1, 2008•1 citation
In 1960 Schwinger [J. Schwinger, Proc.Natl.Acad.Sci. 46 (1960) 570- 579] proposed the algorithm for factorization of unitary operators in the finite M dimensional Hilbert space according to a coprime decomposition of M. Using a special permutation operator A we generalize the Schwinger factorization to every decomposition of M. We obtain the factorized pairs of unitary operators and show that they obey the same commutation relations as Schwinger's. We apply the new factorization to two problems. First, we show how to generate two kq-like mutually unbiased bases for any composite dimension. Then, using a Harper-like Hamiltonian model in the finite dimension M = M1M2, we show how to design a physical system with M1 energy levels, each having degeneracy M2.
In 1960 Schwinger (J Schwinger 1960 Proc. Natl Acad. Sci. 46 570–9) proposed the algorithm for factorization of unitary operators in the finite M -dimensional Hilbert space according to a coprime decomposition of M . Using a special permutation operator A we generalize the Schwinger factorization to every decomposition of M . We obtain the factorized pairs of unitary operators and show that they obey the same commutation relations as Schwinger's. We apply the new factorization to two problems. First, we show how to generate two kq -like mutually unbiased bases for any composite dimension. Then, using a Harper-like Hamiltonian model in the finite dimension M = M 1 M 2 , we show how to design a physical system with M 1 energy levels, each having degeneracy M 2 .
The Ndistinct prime numbers that make up a composite number M allow its bi-partitioning into pairs of two relatively prime factors. Each such pair defines a pair of conjugate representations. An example of such pairs of conjugate representations, each of which spans the M-dimensional space, are the kq representations, which are the most natural representations for periodic systems. Here, we emphasize their relevance to factorizations: the number of prime numbers that make up M relates directly to the number of conjugate pairs of kq representations. It is also shown how Schwinger's factorization procedure is used for reducing M-dimensional space into subspaces according to the prime numbers that make up M.
Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, Mðd; CÞ, acting on such Hilbert spaces. The notion of MUUB re°ects the equiprobable guesses of unitary operators in one basis of Mðd; CÞ when estimating a unitary operator in another. Though, for prime dimension d, the maximal number of MUUBs is known to be d2 1, there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of d. MUUBs can also exist for some d-dimensional subspace of Mðd; CÞ with the maximal number being d.
The prime number decomposition of a finite-dimensional Hilbert space reflects itself in the representations that the space accommodates. The representations appear in conjugate pairs for factorization to two relative prime factors which can be viewed as two distinct degrees freedom. These, Schwinger's quantum degrees of freedom, are uniquely related to a von Neumann lattices in the phase space that characterizes the Hilbert space and specifies the simultaneous definitions of both (modular) positions and (modular) momenta. The area in phase space for each quantum state in each of these quantum degrees of freedom, is shown to be exactly hhh, Planck's constant.
In recent years the finite phase plane has been used to accommodate spin 1/2 particles which play a decisive role as a tool in quantum computing [1]. For this, one often uses conjugate operators in the framework of the Wigner function approach. Conjugate variables are important both in classical and quantum mechanics. The best known example of such operators are the coordinate x and the momentu p [2]. In elementary quantum mechanics there are very few such pairs of conjugate operators. One of the striking results in a finite phase plane of dimension M is that there exist more than one pair of conjugate operators in the xp-plane. In this talk we show how to build 2 conjugate kq-representations (k-quasimomentum, q- quasicoordinte) [3]. These conjugate representations are then applied to Harper-like Hamiltonians, and it's shown how to design physical systems with energy spectra with any desired number of discrete energy levels, and with prescribed degeneracy. A general theory of quantum mechanics in finite phase plane was developed by Schwinger [4].