Nonlinear Eigenvalue Methods for Quantifying Quantum Entanglement

We present a hybrid analytical numerical method to evaluate the geometric measure of entanglement for pure multipartite states by formulating the closest separable state problem as a coupled nonlinear eigenvalue condition. We develop a hybrid analytical numerical framework in which a formal perturbative linearization around a reference product state is combined with an iterative fixed-point scheme. The approach combines a Gauss-Seidel block fixed-point iteration with a controlled first order linearization about a stationary reference product state. The perturbative analysis provides local structural insight and initialization guidance, while the iterative method yields accurate numerical estimates of the geometric measure of entanglement. We make explicit and prove an equal multiplier stationarity identity showing that, at an optimum, all block Lagrange multipliers coincide and are fixed by the optimal fidelity to the target state. A normalization preserving linearization is obtained by projecting onto local tangent spaces, which yields an explicit first order correction and a corresponding scalar shift in the effective eigenvalue. We further establish a monotonic block ascent property: the overlap with the target state increases at every update, remains bounded, and converges to a stationary value. For standard three qubit benchmarks, the hybrid solver converges smoothly and reproduces the known exact optima for the GHZ and W states.

کلیدواژه ها:

Quantum information theory ، Geometric measure of entanglement ، Nonlinear eigenvalue problem ، Fixed-point iteration ، Perturbation theory ، Gauss–Seidel method