Factorization Properties of Finite Spaces

TL;DR

This paper generalizes Schwinger's factorization of unitary operators in finite-dimensional Hilbert spaces to arbitrary decompositions of the dimension M, enabling new basis constructions and physical system designs.

Contribution

It introduces a generalized factorization method for unitary operators in finite spaces applicable to any decomposition of M, extending Schwinger's original approach.

Findings

Constructed mutually unbiased bases for any composite dimension.

Demonstrated physical system design with specified energy level degeneracies.

Extended factorization to all decompositions of finite space dimension.

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.