Statistics and Complexity of Wavefunction Spreading in Quantum Dynamical Systems

TL;DR

This paper analyzes the statistical properties of wavefunction spreading in quantum systems, linking measurement distributions to spread complexities, and explores their behavior in different Hamiltonian models including random matrices.

Contribution

It introduces a framework connecting measurement statistics to generalized spread complexities and analyzes their dynamics in Lie algebra and random matrix Hamiltonian models.

Findings

Distribution of measurement results derived and characterized.

Generalized spread complexities peak at certain times in random matrix models.

An upper bound on complexity change related to Hamiltonian operator norm.

Abstract

We consider the statistics of the results of a measurement of the spreading operator in the Krylov basis generated by the Hamiltonian of a quantum system starting from a specified initial pure state. We first obtain the probability distribution of the results of measurements of this spreading operator at a certain instant of time, and compute the characteristic function of this distribution. We show that the moments of this characteristic function are related to the so-called generalised spread complexities, and obtain expressions for them in several cases when the Hamiltonian is an element of a Lie algebra. Furthermore, by considering a continuum limit of the Krylov basis, we show that the generalised spread complexities of higher orders have a peak in the time evolution for a random matrix Hamiltonian belonging to the Gaussian unitary ensemble. We also obtain an upper bound on the change in generalised spread complexity at an arbitrary time in terms of the operator norm of the Hamiltonian and discuss the significance of these results.