  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 $||\gamma_2^{\Psi,T}||_{HS} \leq \sqrt{5 N \,
\mathrm{tr}(\gamma_1^\Psi (1 - \gamma_1^\Psi))}$.
