## Electronic Communications in Probability

### 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.

#### Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 27, 13 pp.

Dates
Accepted: 15 April 2013
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465315566

Digital Object Identifier
doi:10.1214/ECP.v18-2426

Mathematical Reviews number (MathSciNet)
MR3056064

Zentralblatt MATH identifier
1337.60204

Rights

#### Citation

Nechita, Ion; Pellegrini, Clément. Random pure quantum states via unitary Brownian motion. Electron. Commun. Probab. 18 (2013), paper no. 27, 13 pp. doi:10.1214/ECP.v18-2426. https://projecteuclid.org/euclid.ecp/1465315566

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