Complexity and geometry of quantum state manifolds

We show that the Hilbert space spanned by a continuously parametrized wavefunction family---i.e., a quantum state manifold---is dominated by a subspace, onto which all member states have close to unity projection weight. Its characteristic dimensionality $D_P$ is much smaller than the full Hilbert space dimension, and is equivalent to a statistical complexity measure $e^{S_2}$, where $S_2$ is the $2^{nd}$ Renyi entropy of the manifold. In the thermodynamic limit, $D_P$ closely approximates the quantum geometric volume of the manifold under the Fubini-Study metric, revealing an intriguing connection between information and geometry. This connection persists in compact manifolds such as a twisted boundary phase, where the corresponding geometric circumference is lower bounded by a term proportional to its topological index, reminiscent of entanglement entropy.