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There are many different classifications of entanglement for multipartitequantum systems, one of which is based on the number of unentangled particles.In this paper, we mainly study the quantum states containing at most $k-1$unentangled particles and provide several entanglement criteria based ondifferent forms of inequalities which can both identify quantum statescontaining at most $k-1$ unentangled particles. We show that these criteria aremore effective for some states by concrete examples.

Detection of the quantum states containing at most $k-1$ unentangled particles