  Let S be the von Neumann entropy of a finite ensemble E of pure quantum
states. We show that S may be naturally viewed as a function of a set of
geometrical volumes in Hilbert space defined by the states and that S is
monotonically increasing in each of these variables. Since S is the Schumacher
compression limit of E, this monotonicity property suggests a geometrical
interpretation of the quantum redundancy involved in the compression process.
It provides clarification of previous work in which it was shown that S may be
increased while increasing the overlap of each pair of states in the ensemble.
As a byproduct, our mathematical techniques also provide a new interpretation
of the subentropy of E.
