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^S₂, where S₂ 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.