  The time evolution of an initially uncorrelated system is governed by a
completely positive (CP) map. More generally, the system may contain initial
(quantum) correlations with an environment, in which case the system evolves
according to a not-completely positive (NCP) map. It is an interesting question
what the relative measure is for these two types of maps within the set of
positive maps. After indicating the scope of the full problem of computing the
true volume for generic maps acting on a qubit, we study the case of Pauli
channels in an abstract space whose elements represent an equivalence class of
maps that are identical up to a non-Pauli unitary. In this space, we show that
the volume of NCP maps is twice that of CP maps.
