Nechita, Ion; Pellegrini, Clément Random pure quantum states via unitary Brownian motion. (English) Zbl 1337.60204 Electron. Commun. Probab. 18, Paper No. 27, 13 p. (2013). Summary: 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. Cited in 1 ReviewCited in 3 Documents MSC: 60J65 Brownian motion 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60F05 Central limit and other weak theorems 81P16 Quantum state spaces, operational and probabilistic concepts 81P45 Quantum information, communication, networks (quantum-theoretic aspects) Keywords:quantum states; unitary Brownian motion; stochastic differential equations; quantum information theory PDFBibTeX XMLCite \textit{I. Nechita} and \textit{C. Pellegrini}, Electron. Commun. Probab. 18, Paper No. 27, 13 p. (2013; Zbl 1337.60204) Full Text: DOI arXiv