An asymmetry lower bound on fermionic non-Gaussianity

TL;DR

This paper establishes a practical lower bound on fermionic non-Gaussianity using the Shannon entropy of particle-number distribution, linking it to particle-number asymmetry and enabling easier estimation in many-body quantum systems.

Contribution

It introduces a novel lower bound on fermionic non-Gaussianity based on Shannon entropy, connecting non-Gaussianity measures with particle-number asymmetry and demonstrating its tightness numerically.

Findings

The lower bound is non-trivial for large asymmetry values.

The bound is related to the exponential of the Shannon entropy.

The approach is practical for experimental measurement of non-Gaussianity.

Abstract

Fermionic Gaussian states are a fundamental tool in many-body physics, faithfully representing non-interacting quantum systems and allowing for efficient numerical simulations. Given a many-body wave function, it is therefore interesting to ask how much it differs from that of a Gaussian state, as quantified by the notion of non-Gaussianity. In this work, we relate measures of non-Gaussianity with the Shannon entropy of the particle-number distribution, coinciding with the particle-number asymmetry for pure states. We derive a lower bound on the relative entropy of non-Gaussianity in terms of the exponential of the Shannon entropy, and study numerically its tightness for large system sizes. Our bound is non-trivial for large values of the asymmetry and relies on the concentration of the particle-number distribution of (mixed) fermionic Gaussian states. Since the Shannon entropy of the particle-number distribution is often efficient to compute or experimentally measure, our results can be viewed as a practical way to lower bound non-Gaussianity, highlighting a non-trivial interplay with particle-number asymmetry.