Multimode entanglement for fermions

TL;DR

This paper explores the structure and properties of multipartite entanglement among fermions, focusing on piecewise intrication and affine Slater determinants, with implications for understanding fermionic quantum states.

Contribution

It introduces a novel analysis of fermionic entanglement using affine determinants and examines their properties, extending the understanding of fermionic quantum states beyond traditional GHZ or W states.

Findings

Representation of fermionic entanglement via affine determinants

Analysis of properties of affine Slater determinants

Insights into the structure of multipartite fermionic states

Abstract

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 ${\bf C}^2$, 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 ${\bf C}^d$ 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.