Existence Criterion of Genuine Tripartite Entanglement

TL;DR

This paper introduces a permutation-invariant mathematical criterion for detecting genuine tripartite entanglement in higher-dimensional quantum systems, extending previous residual entanglement measures to mixed states.

Contribution

It generalizes residual entanglement to higher dimensions with a permutation-invariant formulation, applicable to both pure and mixed tripartite states.

Findings

Provides an analytic approximation for weakly mixed states

Validates the criterion on specific weakly mixed tripartite states

Extends the residual entanglement concept to higher dimensions

Abstract

In this paper, an intuitive mathematical formulation is provided to generalize the residual entanglement for tripartite systems of qubits [Phys. Rev. A 61, 052306 (2000)] to the tripartite systems in higher dimension. The spirit lies in the tensor treatment of tripartite pure states [Phys. Rev. A 72, 022333 (2005)]. A distinct characteristic of the present generalization is that the formulation for higher dimensional systems is invariant under permutation of the subsystems, hence is employed as a criterion to test the existence of genuine tripartite entanglement. Furthermore, the formulation for pure states can be conveniently extended to the case of mixed states by utilizing the Kronecker product approximate technique. As applications, we give the analytic approximation of the criterion for weakly mixed tripartite quantum states and consider the existence of genuine tripartite entanglement of some weakly mixed states.