Prime decomposition and correlation measure of finite quantum systems

Abstract

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 $\mathcal{E}$ is introduced and its invariance groups are determined.

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

References in corpus (2)

• Quantifying Entanglement
• On the Improvement of Frequency Stardards with Quantum Entanglement

Cited by in corpus (4)

• Variations on a theme of Heisenberg, Pauli and Weyl
• Hall Normalization Constants for the Bures Volumes of the n-State Quantum Systems
• An expectation value expansion of Hermitian operators in a discrete Hilbert space
• Factorizations and Physical Representations