Random pure quantum states via unitary Brownian motion

TL;DR

This paper introduces a family of probability measures on pure quantum states, generated by unitary Brownian motion, which interpolate between deterministic and uniform measures, with unique invariance properties.

Contribution

It defines a new class of measures on quantum states using Brownian motion on the unitary group, revealing novel invariance features and enabling new average computations.

Findings

Measures interpolate between deterministic and uniform distributions

Measures exhibit U_{N-1} invariance, unlike the uniform measure

Laplace transform used for average calculations

Abstract

We introduce a new family of probability distributions on the set of pure states of a finite dimensional quantum system. Without any a priori assumptions, the most natural measure on the set of pure state is the uniform (or Haar) measure. Our family of measures is indexed by a time parameter $t$ and interpolates between a deterministic measure ($t=0$) and the uniform measure ($t=\infty$). The measures are constructed using a Brownian motion on the unitary group $\mathcal U_N$. Remarkably, these measures have a $\mathcal U_{N-1}$ invariance, whereas the usual uniform measure has a $\mathcal U_N$ invariance. We compute several averages with respect to these measures using as a tool the Laplace transform of the coordinates.