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arXiv · 1201.5333

Random pure quantum states via unitary Brownian motion

Ion Nechita·Clément Pellegrini
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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.

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Ion Nechita, Clément Pellegrini. 2012-01-25. Random pure quantum states via unitary Brownian motion. https://doi.org/10.1214/ecp.v18-2426

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