  We study general ``normally'' distributed random unitary transformations.
These distributions can be defined in terms of a diffusive random walk in the
respective group manifold, formally underpinned by the concept of infinite
divisibility. On the one hand, a normal distribution induces a unital quantum
channel. On the other hand, the diffusive random walk defines a unital quantum
process, which can be generated by a Lindblad master equation. In the single
qubit case, we show that it is possible to find different distributions which
induce the same quantum channel.
  In the case of two qubits, the normal quantum channels, i.e. quantum channels
induced by normal distributions in ${\rm SU}(2)\otimes{\rm SU}(2)$ provide an
appropriate framework for modeling correlated quantum errors. In contrast to
correlated Pauli errors, for instance, they conserve their Markovianity, and
they lead to very different results in error correcting codes or entanglement
distillation. We expect our work to find applications in the tomography and
modeling of one- and two-qubit errors in current quantum computer platforms,
but also in the distillation of Bell pairs across imperfect communication
channels, where it is conceivable that subsequently transmitted qubits are
subject to correlated errors.
