Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace

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 $\Vert P_{\nu }{\psi }_t{\Vert }^2$: 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 $t\mapsto \Vert P_{\nu }{\psi }_t{\Vert }^2$ 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 $t\in [0,\infty )$, $\Vert P_{\nu }{\psi }_t{\Vert }^2$ 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.

28 pages LaTeX, 3 figures

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• Typical Quantum States of the Universe are Observationally Indistinguishable
• Normal Typicality and Dynamical Typicality for a Random Block-Band Matrix Model