A Correlational Bound for Eigenvalues of Fermionic 2-Body Operators
Martin Ravn Christiansen
TL;DR
This work derives a correlational bound for the eigenvalues of the fermionic 2-body operator $\gamma_{2}^{\Psi}$ associated with an $N$-particle state. By exploiting the canonical form $Φ=\sum_k \lambda_k u_k \wedge v_k$ of eigenvectors and introducing the operator $B=\sum_k \lambda_k c_{k,\downarrow} c_{k,\uparrow}$, the authors relate eigenvalues $\Lambda$ to the localization of the $\lambda_k$ via $\Lambda \le \frac{N}{1+\frac{N-2}{2}\sum_k \lambda_k^4}$, strengthening the classic Yang bound $\Lambda \le N$ and revealing that large $\Lambda$ implies highly correlated eigenvectors. They also provide a near-converse bound in the highly correlated regime and present a conjecture about sharpness, supported by constructive lower bounds obtained from trial states $\Psi_M=(B^{\ast})^{M}\Omega$. The results connect spectral constraints to structural correlation measures, with implications for understanding entanglement and correlation in fermionic quantum systems, and they highlight subtleties in achieving equality across $N$ and correlation regimes.
Abstract
We prove that the eigenvalues of a 2-body operator $γ_{2}^Ψ$ associated to a fermionic $N$-particle state $Ψ$ are highly constrained by the structure of the corresponding eigenvectors: If $Φ=\sum_{k=1}^{\infty}λ_{k}u_{k}\wedge v_{k}$ is the canonical form of an eigenvector $Φ$ with eigenvalue $Λ$, then $Λ\leq(1+\frac{N-2}{2}\sum_{k=1}^{\infty}λ_{k}^{4})^{-1}N$. We also prove a lower bound on $\sup_{\Vert Ψ\Vert =1}\langle Φ,γ_{2}^ΨΦ\rangle$ for fixed $Φ$, and state a conjecture motivated by these results.