  We study the statistical properties of the triplet
$(\sigma_x,\sigma_y,\sigma_z)$ of Pauli matrices going through a sequence of
noisy channels, modeled by the repetition of a general, trace-preserving,
completely positive map. We show a non-commutative central limit theorem for
the distribution of this triplet, which shows up a 3-dimensional Brownian
motion in the limit with a non-trivial covariance matrix. We also prove a large
deviation principle associated to this convergence, with an explicit rate
function depending on the stationary state of the noisy channel.
