  Changes of some unitarily invariant norms and anti-norms under the operation
of partial trace are examined. The norms considered form a two-parametric
family, including both the Ky Fan and Schatten norms as particular cases. The
obtained results concern operators acting on the tensor product of two
finite-dimensional Hilbert spaces. For any such operator, we obtain upper
bounds on norms of its partial trace in terms of the corresponding
dimensionality and norms of this operator. Similar inequalities, but in the
opposite direction, are obtained for certain anti-norms of positive matrices.
Through the Stinespring representation, the results are put in the context of
trace-preserving completely positive maps. We also derive inequalities between
the unified entropies of a composite quantum system and one of its subsystems,
where traced-out dimensionality is involved as well.
