Krylov Distribution and Universal Convergence of Quantum Fisher Information

TL;DR

This paper introduces a spectral-resolvent framework using Krylov methods to compute quantum Fisher information, revealing universal convergence behaviors and connecting spectral geometry with quantum metrology.

Contribution

It extends Krylov subspace techniques to quantum Fisher information, identifying universal convergence regimes and providing practical tools for high-dimensional quantum systems.

Findings

Krylov distribution quantifies QFI weight distribution in operator space.

Exponential convergence when Liouville spectrum is gapped.

Algebraic convergence governed by Bessel universality near zero eigenvalues.

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 $\mathcal{K}_\rho$ 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.