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 calculations about operators, we derive two practical criteria for judging \(N\)-partite quantum states owning fewer than \(k\) unentangled particles. In addition, we demonstrate the effectiveness of our frameworks through some concrete examples, and specifically point out the quantum states having fewer than \(k\) unentangled particles that our methods can detect, while other criteria cannot recognize.