Entanglement negativity of fermions: monotonicity, separability criterion, and classification of few-mode states

TL;DR

This paper investigates the properties of fermionic entanglement negativity, establishing it as a valid entanglement measure, comparing it with bosonic cases, and classifying fermionic states based on their entanglement characteristics.

Contribution

It introduces fermionic entanglement negativity as a monotone, compares fermionic and bosonic partial transpose, and classifies multi-fermion states based on entanglement criteria.

Findings

Fermionic negativity is an entanglement monotone under local fermion-number parity-preserving operations.

Vanishing negativity is necessary and sufficient for separability of multi-fermion modes.

Bosonic partial transpose fails to distinguish fermionic separable states.

Abstract

We study quantum information aspects of the fermionic entanglement negativity recently introduced in [Phys. Rev. B 95, 165101 (2017)] based on the fermionic partial transpose. In particular, we show that it is an entanglement monotone under the action of local quantum operations and classical communications--which preserves the local fermion-number parity-- and satisfies other common properties expected for an entanglement measure of mixed states. We present fermionic analogs of tripartite entangled states such as W and Greenberger-Horne-Zeilinger states and compare the results of bosonic and fermionic partial transpose in various fermionic states, where we explain why the bosonic partial transpose fails in distinguishing separable states of fermions. Finally, we explore a set of entanglement quantities which distinguish different classes of entangled states of a system with two and three fermionic modes. In doing so, we prove that vanishing entanglement negativity is a necessary and sufficient condition for separability of $N\geq 2$ fermionic modes with respect to the bipartition into one mode and the rest. We further conjecture that the entanglement negativity of inseparable states which mix local fermion-number parity is always non-vanishing.