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

Abrar Ahmed Naqash, Fardeen Ahmad Sofi, Mohammad Haris Khan, Sundus Abdi

TL;DR

This work tackles the nonlinear eigenvalue problem that arises in computing the geometric measure of entanglement for pure quantum states by developing a hybrid analytical–numerical framework. The method combines perturbative expansions around a reference separable state with a normalization-preserving Gauss–Seidel fixed-point refinement, yielding a tangent-space linearization and a common eigenvalue shift. It provides monotone convergence guarantees for the squared overlap and reproduces exact optimal overlaps for canonical three-qubit states: $Λ_{max}^2=1/2$ for $GHZ_3$ and $4/9$ for $W_3$. The approach demonstrates stability across bipartite and multipartite benchmarks, offering a scalable toolkit for entanglement quantification that can be extended to related nonlinear optimization problems in quantum information science.

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.

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

TL;DR

This work tackles the nonlinear eigenvalue problem that arises in computing the geometric measure of entanglement for pure quantum states by developing a hybrid analytical–numerical framework. The method combines perturbative expansions around a reference separable state with a normalization-preserving Gauss–Seidel fixed-point refinement, yielding a tangent-space linearization and a common eigenvalue shift. It provides monotone convergence guarantees for the squared overlap and reproduces exact optimal overlaps for canonical three-qubit states: for and for . The approach demonstrates stability across bipartite and multipartite benchmarks, offering a scalable toolkit for entanglement quantification that can be extended to related nonlinear optimization problems in quantum information science.

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) state and four-ninths for the W state, with smooth monotonic convergence.

Paper Structure

This paper contains 11 sections, 1 theorem, 32 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Nonlinear Eigenvalue Equation
  3. Solution Method
  4. Perturbative formulation
  5. First-order linearization
  6. Convergence considerations.
  7. Bipartite Schmidt states
  8. Multipartite GHZ state and W state
  9. Numerical validation
  10. Results And Discussion
  11. Conclusion

Key Result

Lemma 3.1

. Let with $\|\phi_i\|=1$ for all $i$ and finite dimensional Hilbert spaces. Given an iterate $\{\ket{\phi_j^{(k)}}\}$ define the partial contraction and perform a cyclic (Gauss-Seidel) block update Then the objective values $f(\{\ket{\phi_i^{(k)}}\})$ form a non decreasing sequence bounded above by $1$, and hence converge. Moreover, every limit point of the sequence of iterates is a stationary

Figures (4)

  • Figure 1: Convergence of the iterative refinement scheme for a bipartite Schmidt state with coefficients $(0.8, 0.6, 0)$. The overlap estimate approaches the largest Schmidt coefficient within a few iterations.
  • Figure 2: Approximate maximum overlaps with product states for the three-qubit GHZ and W states. The values approach the theoretical predictions $\Lambda_{\max}^2 = 1/2$ (GHZ) and $\Lambda_{\max}^2 = 4/9$ (W).
  • Figure 3: Convergence of $\Lambda^2$ for $\mathrm{GHZ}_3$ across five random initializations (iteration index starts at 0). Curves rise monotonically and saturate at the theoretical value $1/2$ (dashed), typically within 2 to 3 iterations; the subsequent plateau reflects numerical tolerance.
  • Figure 4: Convergence of $\Lambda^2$ for $W_3$ across five random initializations (iteration index starts at 0). Curves rise monotonically and saturate at the theoretical value $4/9$ (dashed) within a few iterations; the long flat tail indicates convergence to tolerance.

Theorems & Definitions (2)

  • Lemma 3.1: Monotone block ascent and convergence to stationarity
  • proof