Prime decomposition and correlation measure of finite quantum systems

Details Prime decomposition and correlation measure of finite quantum systems Prime decomposition and correlation measure of finite quantum systems

1998-06-02

arxiv.org/abs/quant-ph/9806007v2

J.Phys.A32:L63-L69,1999

Physics

Quantum Physics

9 pages LaTeX. Revised version: changes in the terminology, updates in refs

Scientific paper

10.1088/0305-4470/32/5/001

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 correlation index $\cal E$ is introduced and its invariance groups are determined.

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