Separability criterion for n-particle states

Abstract: This research introduces the concept of the purity number, which represents the number of separable s-particle sub-states within an n-particle state ($s<n$ ). It establishes that, for any , achieving the maximum purity number is both a necessary and sufficient condition for the separability of n-particle pure states, and a necessary condition for the separability of n-particle mixed states. Subsequently, the study delves into the concept of entanglement rate in n-particle pure states. The entanglement rate of an n-particle pure state, in which all entangled sub-states are maximally entangled, can be considered as a measure of entanglement.