Marches de Bernoulli quantiques. (Quantum Bernoulli walks). (French) Zbl 0701.60103

Séminaire de probabilités XXIV 1988/89, Lect. Notes Math. 1426, 329-344 (1990).
[For the entire collection see Zbl 0695.00024.]
We study a quantum analogue of the classical Bernoulli random walk: instead of one Bernoulli random variable, one considers three noncommuting random variables, which can be represented by Pauli matrices; then one adds independent copies of these: three quantum random variables, obtaining three non-commuting Bernoulli random walks. Using the representation theory of the Lie algebra SU(2), one derives several probabilistic results on these quantum stochastic processes. In particular, the “Euclidean norm” of this process is shown to be Markovian, and the transition probabilities are computed using Clebsch- Gordan formulae.
Reviewer: Ph.Biane


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks


Zbl 0695.00024
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