Hermitian Tensor Product Approximation of Complex Matrices and Separability

The quantum entangled states have become one of the key resources in quantum information processing. The study of quantum teleportation, quantum cryptography, quantum dense coding, quantum error correction and parallel computation [1, 2, 3] has spurred a flurry of activities in the investigation of quantum entanglements. Despite the potential applications of quantum entangled states, the theory of quantum entanglement itself is far from being satisfied. The separability for bipartite and multipartite quantum mixed states is one of the important problems in quantum entanglement. Let H1 (resp. H2) be an m (resp. n)-dimensional complex Hilbert space, with |i〉, i = 1, ..., m (resp. |j〉, j = 1, ..., n), as an orthonormal basis. A bipartite mixed state is said to be separable if the density matrix can be written as