Separability Criteria and Entanglement Measures for Pure States of N Identical Fermions

TL;DR

This paper introduces simple, practical criteria for determining entanglement in pure states of N identical fermions, based on single-particle reduced density matrix properties, simplifying previous complex methods.

Contribution

It provides necessary and sufficient separability criteria for N-fermion pure states using a single identity involving purity or entropy, and proposes natural entanglement measures.

Findings

Separable states must satisfy a single identity involving purity or von Neumann entropy.

Derived inequalities lead to natural entanglement measures for N-fermion pure states.

Criteria are simpler and more practical than previous methods.

Abstract

The study of the entanglement properties of systems of N fermions has attracted considerable interest during the last few years. Various separability criteria for pure states of N identical fermions have been recently discussed but, excepting the case of two-fermions systems, these criteria are difficult to implement and of limited value from the practical point of view. Here we advance simple necessary and sufficient separability criteria for pure states of N identical fermions. We found that to be identified as separable a state has to comply with one single identity involving either the purity or the von Neumann entropy of the single-particle reduced density matrix. These criteria, based on the verification of only one identity, are drastically simpler than the criteria discussed in the recent literature. We also derive two inequalities verified respectively by the purity and the entropy of the single particle, reduced density matrix, that lead to natural entanglement measures for N-fermion pure states. Our present considerations are related to some classical results from the Hartree-Fock theory, which are here discussed from a different point of view in order to clarify some important points concerning the separability of fermionic pure states.