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Hilbert-Schmidt Estimates for Fermionic 2-Body Operators

Martin Ravn Christiansen

Abstract

We prove that the 2-body operator $γ_2^Ψ$ of a fermionic $N$-particle state $Ψ$ obeys $||γ_2^Ψ||_{HS} \leq \sqrt{5} N$, which complements the bound of Yang that $||γ_2^Ψ||_{op} \leq N$. This estimate furthermore resolves a conjecture of Carlen-Lieb-Reuvers (arXiv:1403.3816) concerning the entropy of the normalized 2-body operator. We also prove that the Hilbert-Schmidt norm of the truncated 2-body operator $γ_2^{Ψ,T}$ obeys the inequality $||γ_2^{Ψ,T}||_{HS} \leq \sqrt{5 N \, \mathrm{tr}(γ_1^Ψ(1 - γ_1^Ψ))}$.

Hilbert-Schmidt Estimates for Fermionic 2-Body Operators

Abstract

We prove that the 2-body operator of a fermionic -particle state obeys , which complements the bound of Yang that . This estimate furthermore resolves a conjecture of Carlen-Lieb-Reuvers (arXiv:1403.3816) concerning the entropy of the normalized 2-body operator. We also prove that the Hilbert-Schmidt norm of the truncated 2-body operator obeys the inequality .
Paper Structure (4 sections, 7 theorems, 40 equations)

This paper contains 4 sections, 7 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Acknowledgements
  4. Proof of the Theorems

Key Result

Theorem 1

For any $N\in\mathbb{N}$ and normalized $\Psi\in\bigwedge^{N}\mathfrak{h}$ it holds that

Theorems & Definitions (7)

  • Theorem 1
  • Theorem : Bach Bach-92
  • Theorem 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6