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arXiv · 2305.00834

Hilbert-Schmidt Estimates for Fermionic 2-Body Operators

Martin Ravn Christiansen
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Abstract

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

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Martin Ravn Christiansen. 2023-05-01. Hilbert-Schmidt Estimates for Fermionic 2-Body Operators. https://doi.org/10.1007/s00220-023-04914-3

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