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Random pure quantum states via unitary Brownian motion. (English) Zbl 1337.60204

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.

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)
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