Constructing entanglement measures for fermions

TL;DR

This paper develops a method to construct polynomial SLOCC invariants for fermionic systems, enabling the quantification of entanglement and Bell-nonlocal correlations, with applications to specific fermionic states and Hamiltonians.

Contribution

It introduces a novel approach to find entanglement measures for fermions based on polynomial invariants under SLOCC, considering particle number conservation and system constraints.

Findings

Invariants are nonzero only if Bell-nonlocal correlations are observable.

Constructed invariants for two and three fermions reveal different entanglement types.

A high entanglement measure describes ground states of a specific Hamiltonian at phase transitions.

Abstract

In this paper we describe a method for finding polynomial invariants under Stochastic Local Operations and Classical Communication (SLOCC), for a system of delocalized fermions shared between different parties, with global particle number conservation as the only constraint. These invariants can be used to construct entanglement measures for different types of entanglement in such a system. It is shown that the invariants, and the measures constructed from them, take a nonzero value only if the state of the system allows for the observation of Bell-nonlocal correlations. Invariants of this kind are constructed for systems of two and three spin-1/2 fermions and examples of maximally entangled states are given that illustrate the different types of entanglement distinguished by the invariants. A general condition for the existence of SLOCC invariants and their associated measures is given as a relation between the number of fermions, their spin, and the number of spatial modes of the system. In addition, the effect of further constraints on the system, including the localization of a subset of the fermions, is discussed. Finally, a hybrid Ising-Hubbard Hamiltonian is constructed for which the groundstate of a three site chain exhibits a high degree of entanglement at the transition between a regime dominated by on-site interaction and a regime dominated by Ising-interaction. This entanglement is well described by a measure constructed by the introduced method.