Dynamical typicality of quantum expectation values

TL;DR

This paper demonstrates that in high-dimensional quantum systems, most pure states with the same initial expectation value evolve to produce similar expectation values over time, supporting ensemble-based descriptions of quantum dynamics.

Contribution

It introduces the concept of dynamical typicality in quantum expectation values and applies the Hilbert space average method to analyze their evolution.

Findings

Most pure states with the same initial expectation value have similar future expectation values.

Ensemble averages effectively describe individual quantum state dynamics.

Numerical simulations support the analytical results.

Abstract

We show that the vast majority of all pure states featuring a common expectation value of some generic observable at a given time will yield very similar expectation values of the same observable at any later time. This is meant to apply to Schroedinger type dynamics in high dimensional Hilbert spaces. As a consequence individual dynamics of expectation values are then typically well described by the ensemble average. Our approach is based on the Hilbert space average method. We support the analytical investigations with numerics obtained by exact diagonalization of the full time-dependent Schroedinger equation for some pertinent, abstract Hamiltonian model. Furthermore, we discuss the implications on the applicability of projection operator methods with respect to initial states, as well as on irreversibility in general.