# Dynamical typicality for initial states with a preset measurement statistics of several commuting observables

## TL;DR

This paper demonstrates that in quantum many-body systems, states with identical initial measurement statistics tend to evolve to exhibit similar expectation values, supporting the idea of dynamical typicality and simplifying experimental reproducibility.

## Contribution

The paper establishes a general analytical framework for dynamical typicality in quantum systems with fixed initial measurement statistics, applicable to pure and mixed states.

## Key findings

- Most states with the same initial measurement statistics lead to similar expectation values over time.
- The result provides a theoretical basis for the reproducibility of macroscopic measurement outcomes.
- The analytical framework can be used as a numerical tool for studying quantum dynamics.

## Abstract

We consider all pure or mixed states of a quantum many-body system which exhibit the same, arbitrary but fixed measurement outcome statistics for several commuting observables. Taking those states as initial conditions, which are then propagated by the pertinent Schr\"odinger or von Neumann equation up to some later time point, and invoking a few additional, fairly weak and realistic assumptions, we show that most of them still entail very similar expectation values for any given observable. This so-called dynamical typicality property thus corroborates the widespread observation that a few macroscopic features are sufficient to ensure the reproducibility of experimental measurements despite many unknown and uncontrollable microscopic details of the system. We also discuss and exemplify the usefulness of our general analytical result as a powerful numerical tool.