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Krylov Distribution and Universal Convergence of Quantum Fisher Information

Mohsen Alishahiha, Fatemeh Tarighi Tabesh, Mohammad Javad Vasli

Abstract

We develop a spectral-resolvent framework for computing the quantum Fisher information (QFI) using Krylov subspace methods, extending the notion of the Krylov distribution. By expressing the QFI as a resolvent moment of the superoperator $\mathcal{K}_ρ$ associated with a density matrix, the Krylov distribution quantifies how the QFI weight is distributed across Krylov levels in operator space and provides a natural measure for controlling the truncation error in Krylov approximations. Leveraging orthogonal polynomial theory, we identify two universal convergence regimes: exponential decay when the Liouville-space spectrum is gapped away from zero, and algebraic decay governed by hard-edge (Bessel) universality when small eigenvalues accumulate near zero. This framework establishes a direct connection between quantum metrology, spectral geometry, and Krylov dynamics, offering both conceptual insight and practical tools for efficient QFI computation in high-dimensional and many-body systems.

Krylov Distribution and Universal Convergence of Quantum Fisher Information

Abstract

We develop a spectral-resolvent framework for computing the quantum Fisher information (QFI) using Krylov subspace methods, extending the notion of the Krylov distribution. By expressing the QFI as a resolvent moment of the superoperator associated with a density matrix, the Krylov distribution quantifies how the QFI weight is distributed across Krylov levels in operator space and provides a natural measure for controlling the truncation error in Krylov approximations. Leveraging orthogonal polynomial theory, we identify two universal convergence regimes: exponential decay when the Liouville-space spectrum is gapped away from zero, and algebraic decay governed by hard-edge (Bessel) universality when small eigenvalues accumulate near zero. This framework establishes a direct connection between quantum metrology, spectral geometry, and Krylov dynamics, offering both conceptual insight and practical tools for efficient QFI computation in high-dimensional and many-body systems.
Paper Structure (9 sections, 103 equations, 2 figures)

This paper contains 9 sections, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Quantum Fisher Information and Krylov Distribution
  3. Spectral representation and convergence of the Krylov approximation
  4. Gapped spectrum: exponential convergence
  5. Hard edge at zero: algebraic convergence
  6. An explicit example: mixed-field Ising chain
  7. Conclusions
  8. Linear Systems, Krylov Subspaces, and Moment Representations
  9. State-Dependent Operator Geometry

Figures (2)

  • Figure 1: Lanczos coefficients $a_n$ (blue) and $b_n$ (brown) for the Krylov basis generated by the superoperator $\mathcal{K}_\rho$ for a random density matrix $\rho$ with $L=5$ for which $d_0=499$. The diagonal coefficients $a_n$ are typically larger than the off-diagonal coefficients $b_n$, indicating that the effective Krylov chain is dominated by on-site terms rather than hopping amplitudes.
  • Figure 2: Average relative truncation error $1 - {\cal F}^{(n)}/{\cal F}$ as a function of Krylov index $n$, computed for $20$ independent random density matrices with $L=5$ in the chaotic regime ($g=-1.05$, $h=0.5$). The decay is qualitatively consistent with the power-law behavior predicted for hard-edge spectral measures.