  The Holevo bound is a bound on the mutual information for a given quantum
encoding. In 1996 Schumacher, Westmoreland and Wootters [Schumacher,
Westmoreland and Wootters, Phys. Rev. Lett. 76, 3452 (1996)] derived a bound
which reduces to the Holevo bound for complete measurements, but which is
tighter for incomplete measurements. The most general quantum operations may be
both incomplete and inefficient. Here we show that the bound derived by SWW can
be further extended to obtain one which is yet again tighter for inefficient
measurements. This allows us in addition to obtain a generalization of a bound
derived by Hall, and to show that the average reduction in the von Neumann
entropy during a quantum operation is concave in the initial state, for all
quantum operations. This is a quantum version of the concavity of the mutual
information. We also show that both this average entropy reduction, and the
mutual information for pure state ensembles, are Schur-concave for unitarily
covariant measurements; that is, for these measurements, information gain
increases with initial uncertainty.
