GHZ and W Entanglement Witnesses for the Non-interacting Fermi gas

TL;DR

This paper develops new entanglement witnesses to detect and differentiate types of tripartite entanglement in a noninteracting Fermi gas, revealing that only W-type entanglement exists in such systems.

Contribution

It introduces three classes of parameterized entanglement witnesses for tripartite entanglement detection and discrimination, specifically tailored for the noninteracting Fermi gas.

Findings

All detected tripartite entanglement is of W type.

States with GHZ extbackslash W entanglement do not occur.

Some PPT entangled states are identified.

Abstract

The existence and nature of tripartite entanglement of a noninteracting Fermi gas (NIFG) is investigated. Three new classes of parameterized 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\B and GHZ\W. By choosing appropriate EW operators, the problem of finding GHZ and W EWs is reduced to linear programming. Specifically, we devise new 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\B from GHZ\B 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\W 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\B genuine tripartite entanglement: three-party entanglement without any separable or biseparable admixture does not occur.