Optimization of realignment criteria and its applications for multipartite quantum states

By combining a parameterized Hermitian matrix, the realignment matrix of the bipartite density matrix, and multiple rows and columns from vectorization of reduced density matrices, the authors of Shen (Phys. Rev. A 92: 042332, 2015) presented a family of separable criteria to improve the computable cross-norm or realignment criterion Rudolph (Phys. Rev. A 67: 032312, 2003); Chen (Quantum Inf. Comput. 3: 193-202, 2003). In this paper, we first show that these criteria achieve their optimization when the parameterized matrix is chosen to be a constant matrix. It is then proved that the optimized criterion is equivalent to the corresponding criterion with one additional row and one additional column. This reduces the computation cost, since the combined realignment matrix possesses a lower dimension. Finally, the optimized criterion is further used to achieve the separable criterion for multipartite quantum states, which, by using a numerical example, is more efficient than the corresponding previous criteria based on linear contraction methods and sequential realignment methods.