Typical Mixing and Rare-State Bottlenecks in Open Quantum Systems

TL;DR

This paper investigates the typical mixing behavior of open quantum systems, revealing that relaxation curves concentrate around a mean and identifying conditions where mixing times differ significantly from worst-case benchmarks.

Contribution

It introduces a framework for understanding concentration phenomena in quantum relaxation, extending typicality to non-observable dynamical quantities and analyzing rare-state bottlenecks.

Findings

Relaxation curves concentrate around a deterministic mean.

Vertical concentration leads to fixed-time concentration of trace distance.

Rare-state bottleneck law varies across different quantum system regimes.

Abstract

Mixing in open quantum systems is often summarized by a single worst-case time, even though that benchmark can be set by exponentially rare initial states. We show that for broad unstructured ensembles the nonlinear trace-distance relaxation curve itself concentrates around a deterministic mean. For Haar-random pure states this yields fixed-time concentration of the instantaneous trace distance to the steady state, which we term vertical concentration since typical relaxation curves bundle along the distance axis. Whenever the mean curve crosses the distance threshold with a finite slope, it converts this vertical concentration into a horizontal concentration of the mixing time, extending typicality from standard physical observables to a fundamentally non-observable dynamical quantity. This sharp concentration naturally raises the question of how the typical mixing timescale compares to the worst-case benchmark. We show that in a one-mode tail regime, this separation is controlled by the logarithmic ratio of extremal to typical initial-state overlaps for the slow left eigenoperator. This rare-state bottleneck law yields a hierarchy that is logarithmic in skin-effect settings, linear for boundary-supported many-body slow modes, and exponential in a protected-sector family where generic states mix rapidly while rare states stagnate. The framework also extends beyond Haar to exact and approximate unitary 2-designs and Hilbert-Schmidt/induced ensembles.