bibkey: christiansen2025correlational authors: Martin Ravn Christiansen year: 2025 title: “A Correlational Bound for Eigenvalues of Fermionic 2-Body Operators” doi: null url: https://arxiv.org/abs/2505.21167v1 claim: “Conjecture 3 asks for a universal positive C bounding the two-body Rayleigh expectation by N(1-(N-2)/2 sum lambda_k^4+C(N lambda_max^2)^2) for even N with N lambda_max^2 <= 1.” strata_touched:

• D5/S3/Quantum/FermionicCorrelationalBound/SourceCorrelationalBound license: citation-only triage: anchor

A Correlational Bound for Eigenvalues of Fermionic 2-Body Operators

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Source: https://arxiv.org/abs/2505.21167v1

Section 1.1: Theorems 1 and 2 and Conjecture 3. Section 2: identity (2.4).

Conjecture 3

Section 1.1, p. 4:

Let $\Phi \in \mathfrak{h}\wedge \mathfrak{h}$ be normalized with canonical form $\Phi ={\sum }_{k=1}^{\infty }{\lambda }_k{u}_k\wedge v_k$. Then there exists a constant $C>0$, independent of $\Phi$, such that for every $N\in 2\mathbb{N}$ with $N{\lambda }_{\max }^2\le 1$

$$\underset{\Psi \in {\bigwedge }^N\mathfrak{h},\Vert \Psi \Vert =1}{\sup }⟨\Phi ,{\gamma }_2^{\Psi }\Phi ⟩\le N\left(1-\frac{N-2}{2}\sum _{k=1}^{\infty }{\lambda }_k^4+C(N{\lambda }_{\max }^2{)}^2\right).$$

Here $u\wedge v=(u\otimes v-v\otimes u)/\sqrt{2}$, the pair vectors are mutually orthonormal, the coefficients are nonnegative with square sum one, and ${\lambda }_{\max }={\max }_k{\lambda }_k$. The physical RDM has trace $N(N-1)$. A supplied canonical family can be finite or countable; all remaining one-particle modes are retained as spectators.

Theorem 1 requires $\Phi$ to be an eigenvector. Conjecture 3 and Theorem 2 concern its Rayleigh expectation without that requirement. Identity (2.4) reads $⟨\Phi ,{\gamma }_2^{\Psi }\Phi ⟩=2⟨\Psi ,B^{\ast }B\Psi ⟩$, where $B={\sum }_k{\lambda }_{k}c(v_k)c(u_k)$.

Two-body density matrix

Section 1, p. 1, equation (1.1):

Given a normalized state $\Psi \in {\bigwedge }^N\mathfrak{h}$ we define the 2-body operator associated to $\Psi$, ${\gamma }_2^{\Psi }:\mathfrak{h}\otimes \mathfrak{h}\to \mathfrak{h}\otimes \mathfrak{h}$, by

$$⟨{\phi }_1\otimes {\phi }_2,{\gamma }_2^{\Psi }({\psi }_1\otimes {\psi }_2)⟩=⟨\Psi ,c^{\ast }({\psi }_1)c^{\ast }({\psi }_2)c({\phi }_2)c({\phi }_1)\Psi ⟩$$

for any ${\phi }_1,{\phi }_2,{\psi }_1,{\psi }_2\in \mathfrak{h}$, where $c^{\ast }(\cdot )$ and $c(\cdot )$ denote the fermionic creation and annihilation operators.