  Given some observable H of a finite-dimensional quantum system, we
investigate the typical properties of random quantum state vectors that have a
fixed expectation value with respect to H. Under some some conditions on the
spectrum, we prove that this manifold of quantum states shows a concentration
of measure phenomenon: any continuous function on this set is almost everywhere
close to its mean. We also give a method to estimate the corresponding
expectation values analytically, and we prove a formula for the typical reduced
density matrix in the case that H is a sum of local observables. We discuss the
implications of our results as new proof tools in quantum information theory
and to study phenomena in quantum statistical mechanics. As a by-product, we
derive a method to sample the resulting distribution numerically, which
generalizes the well-known Gaussian method to draw random states from the
sphere.
