  We introduce and apply Hilbert's projective metric in the context of quantum
information theory. The metric is induced by convex cones such as the sets of
positive, separable or PPT operators. It provides bounds on measures for
statistical distinguishability of quantum states and on the decrease of
entanglement under LOCC protocols or other cone-preserving operations. The
results are formulated in terms of general cones and base norms and lead to
contractivity bounds for quantum channels, for instance improving Ruskai's
trace-norm contraction inequality. A new duality between distinguishability
measures and base norms is provided. For two given pairs of quantum states we
show that the contraction of Hilbert's projective metric is necessary and
sufficient for the existence of a probabilistic quantum operation that maps one
pair onto the other. Inequalities between Hilbert's projective metric and the
Chernoff bound, the fidelity and various norms are proven.
