K-separability And K-ME Concurrence Of N-partite Quantum States

1. Separability criteria for several classes of n-partite quantum statesWe mainly discuss the separability of n-partite quantum states from elements of densitymatrices. Practical separability criteria for different classes of n-qubit and n-qudit quantumstates are obtained. Some of them are also sufficient conditions for genuine entanglement ofn-partite quantum states. Moreover, one of the resulting criteria is also necessary and sufficientfor a class of n-partite states.2. Detection of genuinely entangled and non-separable n-partite quantum statesWe investigate the detection of entanglement in n-partite quantum states. We obtain prac-tical separability criteria to identify genuinely entangled and non-separable mixed quantumstates. No numerical optimization or eigenvalue evaluation is needed, and our criteria can beevaluated by simple computations involving components of the density matrix. We provide ex-amples in which our criteria perform better than all known separability criteria. Specifically, weare able to detect genuine n-partite entanglement which has previously not been identified. Inaddition, our criteria can be used in today’s experiment.3. k-separability criterion of mixed multipartite quantum statesWe investigate classification and detection of multipartite entanglement in a very generalsetting. The k-separability of mixed n-partite （n≥3） quantum states is mainly discussed. Weobtain practical k-separability criteria which identify k-nonseparable mixed multipartite statesin arbitrary dimensional quantum systems, characterize the different classes of multipartiteentanglement and detect multipartite entanglement （k-nonseparable n-partite quantum states）which have previously not been identified. Our criteria can be used to distinguish the differentclasses of multipartite entanglement and can detect many important multipartite entanglementstates such as GHZ state, W state, and anti W state efficiently. They can be used for detect-ing not only genuine n-partite entangled mixed states （k=2） but also k-nonseparable mixedmultipartite states （not k-separable states）（k=3,4,···, n）. No optimization or eigenvalueevaluation is needed, our criteria can be evaluated by simple computations involving compo-nents of the density matrix and can be implemented in today,s experiments. That is, our criteriacan be used for experimental detection of multipartite entanglement.4. Measure of multipartite entanglement with computable lower boundsWe present a measure of multipartite entanglement （k-nonseparable）, k-ME concurrenceC_k-ME（ρ）, that unambiguously detects all k-nonseparable states in arbitrary dimensions, where the special case2-ME concurrence C_2-ME（ρ） is a measure of genuine multipartite entangle-ment. The measure k-ME concurrence satisfies important characteristics of an entanglementmeasure, including the entanglement monotone, vanishing on k-separable states, convexity,subadditivity, and being strictly greater than zero for all k-nonseparable states. Two powerfullower bounds on this measure are given. These lower bounds are experimentally implementablewithout quantum state tomography and are easily computable as no optimization or eigenvalueevaluation is needed. We illustrate detailed examples in which the given bounds perform betterthan other known detection criteria.