  A quantum probability measure--or quantum measurement--is said to be clean if
it cannot be irreversibly connected to any other quantum probability measure
via a quantum channel. The notion of a clean quantum measure was introduced by
Buscemi et al (2005) for finite-dimensional Hilbert space, and was studied
subsequently by Kahn (2007) and Pellonp\"a\"a (2011). The present paper
provides new descriptions of clean quantum probability measures in the case of
finite-dimensional Hilbert space. For Hilbert spaces of infinite dimension, we
introduce the notion of `approximately clean quantum probability measures' and
characterise this property for measures whose range determines a
finite-dimensional operator system.
