Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace

TL;DR

This paper proves that in macroscopic quantum systems, the evolution of macro state weights is nearly independent of initial states over finite times and tends to stable values, extending the concept of typicality.

Contribution

It establishes that macro state weights evolve predictably and stabilize over time for most initial states, generalizing and simplifying von Neumann's normal typicality.

Findings

Superposition weights are nearly independent of initial states over finite times.

Weights tend to stable, state- and time-independent values.

Results extend and simplify the concept of normal typicality.

Abstract

We consider a macroscopic quantum system with unitarily evolving pure state $\psi_t\in \mathcal{H}$ and take it for granted that different macro states correspond to mutually orthogonal, high-dimensional subspaces $\mathcal{H}_\nu$ (macro spaces) of $\mathcal{H}$. Let $P_\nu$ denote the projection to $\mathcal{H}_\nu$. We prove two facts about the evolution of the superposition weights $\|P_\nu\psi_t\|^2$: First, given any $T>0$, for most initial states $\psi_0$ from any particular macro space $\mathcal{H}_\mu$ (possibly far from thermal equilibrium), the curve $t\mapsto \|P_\nu \psi_t\|^2$ is approximately the same (i.e., nearly independent of $\psi_0$) on the time interval $[0,T]$. And second, for most $\psi_0$ from $\mathcal{H}_\mu$ and most $t\in[0,\infty)$, $\|P_\nu \psi_t\|^2$ is close to a value $M_{\mu\nu}$ that is independent of both $t$ and $\psi_0$. The first is an instance of the phenomenon of dynamical typicality observed by Bartsch, Gemmer, and Reimann, and the second modifies, extends, and in a way simplifies the concept, introduced by von Neumann, now known as normal typicality.