Krylov Distribution and Universal Convergence of Quantum Fisher Information

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.