Entanglement negativity bounds for fermionic Gaussian states

TL;DR

This paper introduces efficiently computable upper and lower bounds for entanglement negativity in fermionic Gaussian states, addressing computational challenges and enabling applications in quantum many-body and topological systems.

Contribution

It develops rigorous bounds for fermionic negativity using semi-definite programming and Gaussian operator techniques, overcoming the non-Gaussian partial transpose issue.

Findings

Bounds are computationally efficient and rigorous.

Applicable to quantum many-body systems and topological properties.

Provides a practical tool for studying fermionic entanglement.

Abstract

The entanglement negativity is a versatile measure of entanglement that has numerous applications in quantum information and in condensed matter theory. It can not only efficiently be computed in the Hilbert space dimension, but for non-interacting bosonic systems, one can compute the negativity efficiently in the number of modes. However, such an efficient computation does not carry over to the fermionic realm, the ultimate reason for this being that the partial transpose of a fermionic Gaussian state is no longer Gaussian. To provide a remedy for this state of affairs, in this work we introduce efficiently computable and rigorous upper and lower bounds to the negativity, making use of techniques of semi-definite programming, building upon the Lagrangian formulation of fermionic linear optics, and exploiting suitable products of Gaussian operators. We discuss examples in quantum many-body theory and hint at applications in the study of topological properties at finite temperature.