  W generalize the scheme for detection of qubit-environment entanglement to
qudit-environment systems. This is of relevance for many-qubit systems and the
quantification of the operation of quantum algorithms under the influence of
external noise, since only decoherence that is not entangling in its nature can
be effectively described by quantum channels and similar methods in more
complicated scenarios. The generalization involves an increase of the class of
entangled states which are not detected by the scheme, but the type of
entanglement which cannot be detected is also least likely to qualitatively
influence decoherence. We exemplify the operation of the scheme on a
realistically modelled NV-center spin qutrit interacting with an environment of
nuclear spins.
