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

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