We propose generalizations of concurrence for multi-partite quantum systems that can distinguish qualitatively distinct quantum correlations. All introduced quantities can be evaluated efficiently for arbitrary mixed sates.
The k-separability of mixed n-partite (n≥3) quantum states is mainly discussed in this paper. We obtain practical k-separability criteria to identify k-nonseparable mixed multipartite states in arbitrary dimensional quantum systems and detect genuine n-partite entanglement (k=2) and entanglement (k=3) which have previously not been identified. No optimization or eigenvalue evaluation is needed, our criteria can be evaluated by simple computations involving components of the density matrix and can be implemented in today's experiments.
The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement is explored for bi-partite and multi-partite pure and mixed states. It is determined analytically for arbitrary two-qubit mixed states, generalized Werner, and isotropic states, and is also applied to certain multi-partite mixed states, including two distinct multi-partite bound entangled states. Moreover, the ground-state entanglement of the XY model in a transverse field is calculated and shown to exhibit singular behavior near the quantum critical line. Along the way, connections are pointed out between the geometric measure of entanglement, the Hartree approximation, entanglement witnesses, correlation functions, and the relative entropy of entanglement.