Solving a Nonlinear Eigenvalue Equation in Quantum Information Theory: A Hybrid Approach to Entanglement Quantification

TL;DR

This paper introduces a hybrid analytical and numerical method to evaluate the geometric measure of entanglement in quantum states, combining fixed point iteration with perturbative corrections, and demonstrating convergence and accuracy on standard benchmarks.

Contribution

It develops a novel hybrid solver that explicitly handles nonlinear eigenvalue problems in quantum entanglement quantification, with proven convergence properties and benchmark validation.

Findings

Reproduces exact entanglement measures for GHZ extsubscript{3} and W extsubscript{3} states.

Establishes a monotonic convergence and bounded overlap increase during iterations.

Provides a new framework combining fixed point iteration with perturbative corrections for quantum eigenproblems.

Abstract

Nonlinear eigenvalue equations arise naturally in quantum information theory, particularly in the variational quantification of entanglement. In this work, we present a hybrid analytical and numerical framework for evaluating the geometric measure of entanglement. The method combines a Gauss Seidel fixed point iteration with a controlled perturbative correction scheme. We make the coupled nonlinear eigenstructure explicit by proving the equal multiplier stationarity identity, which states that at the optimum all block Lagrange multipliers coincide with the squared fidelity between the target state and its closest separable approximation. A normalization-preserving linearization is then derived by projecting the dynamics onto the local tangent spaces, yielding a well-defined first order correction and an explicit scalar shift in the eigenvalue. Furthermore, we establish a monotonic block ascent property the squared overlap between the evolving product state and the target state increases at every iteration, remains bounded by unity, and converges to a stationary value. The resulting hybrid solver reproduces the exact optimum for standard three qubit benchmarks, obtaining squared-overlap values of one-half for the Greenberger Horne Zeilinger (GHZ\(_3\)) state and four-ninths for the W\(_3\) state, with smooth monotonic convergence.