Hilbert-Schmidt Estimates for Fermionic 2-Body Operators

We prove that the 2-body operator ${\gamma }_2^{\mathrm{\Psi }}$ of a fermionic $N$-particle state $\mathrm{\Psi }$ obeys $\mathrm{\mid }\mathrm{\mid }{\gamma }_2^{\mathrm{\Psi }}\mathrm{\mid }{\mathrm{\mid }}_{HS}\le \sqrt{5}N$, which complements the bound of Yang that $\mathrm{\mid }\mathrm{\mid }{\gamma }_2^{\mathrm{\Psi }}\mathrm{\mid }{\mathrm{\mid }}_{op}\le N$. 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 ${\gamma }_2^{\mathrm{\Psi },T}$ obeys the inequality $\mathrm{\mid }\mathrm{\mid }{\gamma }_2^{\mathrm{\Psi },T}\mathrm{\mid }{\mathrm{\mid }}_{HS}\le \sqrt{5N\mathrm{t}\mathrm{r}({\gamma }_1^{\mathrm{\Psi }}(1-{\gamma }_1^{\mathrm{\Psi }}))}$.