Prime decomposition and correlation measure of finite quantum systems
Demosthenes Ellinas, Floratos, Emmanuel G · CERN Document Server (European Organization for Nuclear Research) · 1999
Under the name prime decomposition (pd), a unique decomposition of an arbitrary $N$-dimensional density matrix $\\rho$ into a sum of seperable density matrices with dimensions given by the coprime factors of $N$ is introduced. For a class of density matrices a complete tensor product factorization is achieved. The construction is based on the Chinese Remainder Theorem and the projective unitary representation of $Z_N$ by the discrete Heisenberg group $H_N$. The pd isomorphism is unitarily implemented and it is shown to be coassociative and to act on $H_N$ as comultiplication. Density matrices with complete pd are interpreted as grouplike elements of $H_N$. To quantify the distance of $\\rho$ from its pd a trace-norm entaglement index $\\cal E$ is introduced and its invariance groups are determined.