Efficient detection for quantum states containing fewer than k unentangled particles in multipartite quantum systems

In this paper, we mainly investigate the detection of quantum states containing fewer than k unentangled particles in multipartite quantum systems. Based on inequalities of nonlinear operators, we derive two families of criteria for detecting N-partite quantum states containing fewer than k unentangled particles. By concrete examples, we point out that both families of criteria can identify some quantum states containing fewer than k unentangled particles that cannot be tested by known criteria. This demonstrates the effectiveness of our criteria.