Hilbert-Schmidt Estimates for Fermionic 2-Body Operators

We prove that the 2-body operator $\gamma_2^\Psi$ of a fermionic $N$-particle state $\Psi$ obeys $||\gamma_2^\Psi||_{HS} \leq \sqrt{5} N$, which complements the bound of Yang that $||\gamma_2^\Psi||_{op} \leq N$. 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 $\gamma_2^{\Psi,T}$ obeys the inequality $||\gamma_2^{\Psi,T}||_{HS} \leq \sqrt{5 N \, \mathrm{tr}(\gamma_1^\Psi (1 - \gamma_1^\Psi))}$.