Factorization Properties of Finite Spaces

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 Schwingers. 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.