Greenberger-Horne-Zeilinger and W entanglement witnesses for the noninteracting Fermi gas

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, W\textbackslashB and GHZ\textbackslashW (the convex set of B states is comprised of mixed states of product and biseparable states; that of W states is comprised of mixed states of B states and W-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 W EWs is reduced to linear programming. Specifically, we devise W 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 W\textbackslashB from GHZ\textbackslashB entanglement, which is not possible with W EWs. Implementing these classes of EWs, it is found that all states containing genuine tripartite entanglement are of W type, and hence states containing GHZ\textbackslashW 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'' W\textbackslashB genuine tripartite entanglement: three-party entanglement without any separable or biseparable admixture does not occur.