Random pure quantum states via unitary Brownian motion

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.

References in corpus (5)

• Random quantum channels I: graphical calculus and the Bell state phenomenon
• Asymptotics of random density matrices
• Random graph states, maximal flow and Fuss-Catalan distributions
• Random quantum channels II: Entanglement of random subspaces, Renyi entropy estimates and additivity problems
• Central limit theorem for the heat kernel measure on the unitary group

Cited by in corpus (2)

• Random matrix techniques in quantum information theory
• Entanglement formation under random interactions