LETTER TO THE EDITOR: Prime decomposition and correlation measure of finite quantum systems

Under the name prime decomposition (PD), a unique decomposition of an arbitrary N-dimensional density matrix into a sum of separable density matrices with dimensions determined 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 by the discrete Heisenberg group . The PD isomorphism is unitarily implemented and it is shown to be co-associative and to act on as comultiplication. Density matrices with complete PD are interpreted as group-like elements of . To quantify the distance of from its PD a trace-norm correlation index is introduced and its invariance groups are determined.