Effective eigenvalue approximation from moments for self-adjoint trace-class operators

TL;DR

This work introduces a moment-based method to approximate the spectrum of self-adjoint, trace-class operators by constructing sets $\Lambda_n$ from the first $n$ moments, defined via roots of truncated Newton sums $g_n(x)$. The authors prove that $\Lambda_n$ converges to the nonzero spectrum in the Hausdorff metric and establish super-exponential convergence for extremal eigenvalues, along with monotone lower bounds $q_{n,c}$ for $\lambda_{\min}$ that depend only on moments and an upper bound on the Schatten--von Neumann $1$-norm. They develop a decomposition-based framework to bound the $1$-norm from above, enabling rigorous lower bounds for $\lambda_{\min}$ and upper bounds for negativity, with tight results for Gaussian and polynomial-Gaussian kernels. The approach is demonstrated through Gaussian and non-Gaussian kernels and is applicable to quantum-information quantities such as von Neumann entropy and logarithmic negativity, offering a diagonalization-free technique for infinite-dimensional problems. The method is computationally stable with high-precision moments and provides practical tools for entropy and entanglement assessments in quantum optics and open quantum systems.

Abstract

Spectral properties of bounded linear operators play a crucial role in several areas of mathematics and physics. For each self-adjoint, trace-class operator $O$ we define a set $Λ_n\subset \mathbb{R}$, and we show that it converges to the spectrum of $O$ in the Hausdorff metric under mild conditions. Our set $Λ_n$ only depends on the first $n$ moments of $O$. We show that it can be effectively calculated for physically relevant operators, and it approximates the spectrum well without diagonalization. We prove that using the above method we can converge to the minimal and maximal eigenvalues with super-exponential speed. We also construct monotone increasing lower bounds $q_n$ for the minimal eigenvalue (or decreasing upper bounds for the maximal eigenvalue). This sequence only depends on the moments of $O$ and a concrete upper estimate of its $1$-norm; we also demonstrate that $q_n$ can be effectively calculated for a large class of physically relevant operators. This rigorous lower bound $q_n$ tends to the minimal eigenvalue with super-exponential speed provided that $O$ is not positive semidefinite. As a by-product, we obtain computable upper bounds for the $1$-norm of $O$, too. Numerical examples demonstrate the relevance of our approximation in estimating entropy and negativity, which is useful, among others, in quantum optical and in open quantum system models. The results can be directly applicable to problems in quantum information, statistical mechanics, and quantum thermodynamics, where using traditional techniques based on diagonalization is impractical.

Figures (3)

• Figure 1: Upper bounds for the Schatten--von Neumann $1$-norm of the operator $\widehat{K}$ as a function of the polynomial parameter $C_{P}$. The Gaussian parameters are given as $A=\frac{3}{2}$, $C=1$, $B=D=E=0$, and the polynomial parameters are $A_{P}=-1$, $F_{P}=1$, and $B_{P}=D_{P}=E_{P}=0$. Note that the operator $\widehat{K}$ is easily seen to be positive semidefinite at $C_{P}=1$. Here $L_1^{(1)}=\sqrt{R(w_\text{min})}$ is the solid line, $L_1^{(2)}=\sqrt{R(w_\text{min}^G)}$ is the dashed curve. The dash-dotted curve $L_1^{(3)}$ shows an estimate obtained from numerical diagonalization of the operator $\widehat{K}$, for the details see Subsection \ref{['ss:num']}.
• Figure 2: Consider the positive semidefinite polynomial Gaussian operator $\widehat{K}$ of the form \ref{['eq:rho_Q']} with parameters $A=\frac{3}{2}$, $C=1$, $B=D=E=0$, $A_P=-1$, $C_P=1$, $F_P=1$, and $B_P=D_P=E_P=0$. By \ref{['eq:104']} the upper bound of $\|\widehat{K} \|_1$ can be chosen to be $c=1.04054$. We plotted $|q_n|=|q_{n,c}|$, $q_n^<=\frac{c}{n+1}$, and $q_n^>=\frac{ec}{n+1}$ as a function of $n$ calculated from \ref{['eq:q_n_definition']}.
• Figure 3: Consider the polynomial Gaussian operator $\widehat{K}$ of the form \ref{['eq:rho_Q']} with parameters $A=\frac{3}{2}$, $C=1$, $B=D=E=0$, $A_P=-1$, $C_P=40$, $F_P=1$, and $B_P=D_P=E_P=0$. We plotted $q_n^{(i)}=q_{n,c}$ as a function of $n$ calculated from \ref{['eq:q_n_definition']} and \ref{['eq:qn0']}. For $i=1$ we obtain $c=\sum_j |\lambda_j|=1.16445$ from diagonalization, for the details see Subsection \ref{['ss:num']}. For $i=2$ we take $c=0$, and for $i=3$ the value $c=1.4941$ was calculated from the Hölder's upper bound and optimization given in Subsection \ref{['ss:quad']}. The horizontal dashed line indicates the common limit $-0.082228$.

Theorems & Definitions (19)

• Definition 2.1
• Claim 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.4
• proof
• Definition 4.1
• proof
• Theorem 4.3
• proof