Fidelity Decay Saturation Level for Initial Eigenstates

TL;DR

This paper investigates how the fidelity decay of initial eigenstates under chaotic evolution saturates early, with a quadratic dependence on perturbation strength, supported by theory and numerical evidence.

Contribution

It introduces the concept of early saturation of fidelity decay for initial eigenstates and establishes its quadratic dependence on perturbation strength.

Findings

Fidelity decay saturates before the 1/N limit for initial eigenstates.

Saturation level depends quadratically on perturbation strength.

Numerical evidence supports the theoretical prediction.

Abstract

We show that the fidelity decay between an initial eigenstate state evolved under a unitary chaotic operator and the same eigenstate evolved under a perturbed operator saturates well before the 1/N limit, where $N$ is the size of the Hilbert space, expected for a generic initial state. We provide a theoretical argument and numerical evidence that, for intermediate perturbation strengths, the saturation level depends quadratically on the perturbation strength.