  The trace distance (TD) possesses several of the good properties required for
a faithful distance measure in the quantum state space. Despite its importance
and ubiquitous use in quantum information science, one of its questionable
features, its possible non-monotonicity under taking tensor products of its
arguments (NMuTP), has been hitherto unexplored. In this article we advance
analytical and numerical investigations of this issue considering different
classes of states living in a discrete and finite dimensional Hilbert space.
Our results reveal that although this property of TD does not shows up for pure
states and for some particular classes of mixed states, it is present in a
non-negligible fraction of the regarded density operators. Hence, even though
the percentage of quartets of states leading to the NMuTP drawback of TD and
its strength decrease as the system's dimension grows, this property of TD must
be taken into account before using it as a figure of merit for distinguishing
mixed quantum states.
