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An asymmetry lower bound on fermionic non-Gaussianity

Filiberto Ares, Michele Mazzoni, Sara Murciano, Dávid Szász-Schagrin, Pasquale Calabrese, Lorenzo Piroli

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.

An asymmetry lower bound on fermionic 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.
Paper Structure (14 sections, 85 equations, 1 figure)

This paper contains 14 sections, 85 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fermionic Gaussian states
  4. Non-Gaussianity, particle number Shannon entropy, and asymmetry
  5. Particle-number distribution of Gaussian states
  6. A bound on the particle-number Shannon entropy
  7. An elementary bound for the trace distance
  8. A lower bound on fermionic non-Gaussianity
  9. Preliminary considerations
  10. Superposition of kink states
  11. Derivation of the lower-bound
  12. Outlook and discussions
  13. Correlation matrices and lower bound for the superposition of kink states
  14. Proof of the concentration inequality for mixed states

Figures (1)

  • Figure 1: Non-Gaussianity of the superposition of kink states \ref{['eq:kink-family']} plotted against their asymmetry for different values of the parameter $R$ (red) for $N = 100$. The black dashed line denotes the lower bound \ref{['eq:tr-bound-kink']}.