Necessary and sufficient product criteria for quantum states via the rank of realignment matrix of density matrix

We present a necessary and sufficient product criterion for bipartite quantum states based on the rank of realignment matrix of density matrix. Then, this approach is generalized to multipartite systems. We first introduce the concept of semiproduct in a similar manner to the semiseparable and prove that semiproduct is equivalent to fully product. Therefore, a quantum state is bipartite product with respect to all possible partitions implies fully product which is different from the case of separability. For pure states, it can easily be seen that several necessary and sufficient separability criteria for multipartite systems are derived as a special case of our results. Several specific examples illustrate that our criteria are convenient and operational.