Separability criterion for n-particle states

TL;DR

This paper introduces the purity number as a new criterion for assessing the separability of n-particle quantum states, providing a necessary and sufficient condition for pure states and a necessary condition for mixed states.

Contribution

It defines the purity number and establishes its role as a separability criterion, along with analyzing the entanglement rate in n-particle pure states.

Findings

Maximum purity number indicates separability of pure states.

Purity number serves as a necessary condition for mixed state separability.

Entanglement rate measures the degree of entanglement in pure 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.