Towards a geometrical interpretation of quantum-information compression

Let $S$ be the von Neumann entropy of a finite ensemble $\mathcal{E}$ 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 $\mathcal{E}$, 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 by-product, our mathematical techniques also provide an interpretation of the subentropy of $\mathcal{E}$.