Multipartite entanglement in fermionic systems via a geometric measure

TL;DR

This paper introduces a novel geometric entanglement measure for multipartite fermionic systems, applicable to systems with more than three parts, and demonstrates its invariance and utility in correlated electron systems.

Contribution

It proposes the first multipartite entanglement measure for fermionic systems beyond bipartite and tripartite cases, applicable to arbitrary partitions.

Findings

The measure is invariant under local unitaries.

It applies to systems with different subsystem dimensions.

Demonstrated in correlated electron systems.

Abstract

We study multipartite entanglement in a system consisting of indistinguishable fermions. Specifically, we have proposed a geometric entanglement measure for N spin-1/2 fermions distributed over 2L modes (single particle states). The measure is defined on the 2L qubit space isomorphic to the Fock space for 2L single particle states. This entanglement measure is defined for a given partition of 2L modes containing m >= 2 subsets. Thus this measure applies to m <= 2L partite fermionic system where L is any finite number, giving the number of sites. The Hilbert spaces associated with these subsets may have different dimensions. Further, we have defined the local quantum operations with respect to a given partition of modes. This definition is generic and unifies different ways of dividing a fermionic system into subsystems. We have shown, using a representative case, that the geometric measure is invariant under local unitaries corresponding to a given partition. We explicitly demonstrate the use of the measure to calculate multipartite entanglement in some correlated electron systems. To the best of our knowledge, there is no usable entanglement measure of m > 3 partite fermionic systems in the literature, so that this is the first measure of multipartite entanglement for fermionic systems going beyond the bipartite and tripartite cases.