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

## TL;DR

This paper introduces a new criterion based on the rank of the realignment matrix to determine when bipartite and multipartite quantum states are fully product, providing a practical tool for quantum state analysis.

## Contribution

It establishes a necessary and sufficient product criterion using the realignment matrix rank and generalizes it to multipartite systems, introducing the concept of semiproduct.

## Key findings

- Criterion is necessary and sufficient for bipartite product states
- Generalization to multipartite systems via semiproduct concept
- Examples demonstrate the criterion's convenience and operational usefulness

## Abstract

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.