Multimode entanglement for fermions

We are motivated by tripartite entanglement for fermions. While GHZ or W states involve 3-fold intrication, we consider here piecewise intrication of 3 fermions in ${\mathbf{C}}^2$, namely of type $ab+bc+ca$. Before interaction with Stern-Gerlach apparatus, qu-bits are distinguishable; at the output however they turn into un-distinguishable particles, whose anti-symmetric wave function is of the form $\det (b-a,c-a)$ (affine determinant). More generally, $d+1$ intricated fermions in ${\mathbf{C}}^d$ can be represented by the anti-symmetric wave function $\det (a_1-a_0,a_2-a_0,\cdots ,a_d-a_0)$. We investigate also properties of affine Slater determinants, as expectation values or reduced density matrices.