Hilbert–Schmidt Estimates for Fermionic 2-Body Operators

Abstract

AbstractWe prove that the 2-body operator $$\gamma _{2}^{\Psi }$$ γ 2 Ψ of a fermionic N-particle state $$\Psi $$ Ψ obeys $$\Vert \gamma _{2}^{\Psi }\Vert _{\textrm{HS}}\le \sqrt{5}N$$ ‖ γ 2 Ψ ‖ HS ≤ 5 N , which complements the bound of Yang (Rev Mod Phys 34:694, 1962) that $$\Vert \gamma _{2}^{\Psi }\Vert _{\textrm{op}}\le N$$ ‖ γ 2 Ψ ‖ op ≤ N . This estimate furthermore resolves a conjecture of Carlen–Lieb–Reuvers (Commun Math Phys 344:655–671, 2016) concerning the entropy of the normalized 2-body operator. We also prove that the Hilbert–Schmidt norm of the truncated 2-body operator $$\gamma _{2}^{\Psi ,T}$$ γ 2 Ψ , T obeys the inequality $$\Vert \gamma _{2}^{\Psi ,T}\Vert _{\textrm{HS}}\le \sqrt{5N\,\textrm{tr}\,(\gamma _{1}^{\Psi }(1-\gamma _{1}^{\Psi }))}$$ ‖ γ 2 Ψ , T ‖ HS ≤ 5 N tr ( γ 1 Ψ ( 1 - γ 1 Ψ ) ) .