Abstract
The existence and nature of tripartite entanglement of a noninteracting Fermi gas (NIFG) is investigated. Three classes of parametrized entanglement witnesses (EWs) are introduced with the aim of detecting genuine tripartite entanglement in the three-body reduced density matrix and discriminating between the presence of the two types of genuine tripartite entanglement, and (the convex set of states is comprised of mixed states of product and biseparable states; that of states is comprised of mixed states of states and -type pure entangled states; and the GHZ (Greenberger-Horne-Zeilinger) set contains generic mixtures of any kind for a tripartite system). By choosing appropriate EW operators, the problem of finding GHZ and EWs is reduced to linear programming. Specifically, we devise EWs based on a spin-chain model with periodic boundary conditions, and we construct a class of parametrized GHZ EWs by linearly combining projection operators corresponding to all the different state-vector types arising for a three-fermion system. A third class of EWs is provided by a GHZ stabilizer operator capable of distinguishing from entanglement, which is not possible with EWs. Implementing these classes of EWs, it is found that all states containing genuine tripartite entanglement are of type, and hence states containing genuine tripartite entanglement do not arise. Some genuine tripartite entangled states that have a positive partial transpose (PPT) with respect to some bipartition are detected. Finally, it is demonstrated that a NIFG does not exhibit “pure” genuine tripartite entanglement: three-party entanglement without any separable or biseparable admixture does not occur.
- Received 2 May 2009
DOI:https://doi.org/10.1103/PhysRevA.81.032302
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