Hilbert–Schmidt Estimates for Fermionic 2-Body Operators

We prove that the 2-body operator \(\gamma _{2}^{\Psi }\) ${\gamma }_2^{\mathrm{\Psi }}$ of a fermionic N-particle state \(\Psi \) $\mathrm{\Psi }$ obeys \(\Vert \gamma _{2}^{\Psi }\Vert _{\textrm{HS}}\le \sqrt{5}N\) $\Vert {\gamma }_2^{\mathrm{\Psi }}{\Vert }_{\text{HS}}\le \sqrt{5}N$ , which complements the bound of Yang (Rev Mod Phys 34:694, 1962) that \(\Vert \gamma _{2}^{\Psi }\Vert _{\textrm{op}}\le N\) $\Vert {\gamma }_2^{\mathrm{\Psi }}{\Vert }_{\text{op}}\le N$ . 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 \(\gamma _{2}^{\Psi ,T}\) ${\gamma }_2^{\mathrm{\Psi },T}$ obeys the inequality \(\Vert \gamma _{2}^{\Psi ,T}\Vert _{\textrm{HS}}\le \sqrt{5N\,\textrm{tr}\,(\gamma _{1}^{\Psi }(1-\gamma _{1}^{\Psi }))}\) $\Vert {\gamma }_2^{\mathrm{\Psi },T}{\Vert }_{\text{HS}}\le \sqrt{5N\text{tr}({\gamma }_1^{\mathrm{\Psi }}(1-{\gamma }_1^{\mathrm{\Psi }}))}$ .