Matrix Realignment Method for Calculating the Partial Transposes and the Partial Traces of Density Matrices

Motivated by Peres´s work [A. Peres, Phys. Rev. Lett. 77: 1413-1415 (1996)] and Horodeckis´ work [M, Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 223: 1-8 (1996)], we develop a very simple method to directly calculate the partial transposes and the partial traces of density matrices in C^M otimes C^N and C² otimes C² otimes C^N composite quantum systems, which is based a realignment matrix constructed from the original density matrices.