Rank deficiency and the Euclidean geometry of quantum states

Quantum state tomography requires characterizing a collection of parameters whose size grows rapidly with the size of the quantum system under consideration. In practice one hopes that prior information about the system can reduce the number of parameters in need of characterization---for example, one might expect to find high-quality quantum systems in states of low rank. Interest in tomographic schemes that return rank-deficient estimates leads us to explore some geometric properties of the space of quantum states that are analogous to solid angles in three-dimensional Euclidean geometry.