Geometry of degenerate quantum states

We describe all (functionally-)independent invariants associated with a collection of $m$-dimensional subspaces of a Hilbert space $mathbb{C}^n$iffalse of dimension $n$fi. Equivalently, we identify all independent geometric objects that can be constructed from a collection of points on the Grassmanian $Gr_{m,n}$ with respect to the action of $U(n)$. We characterize the invariants from the perspective of linear algebra as well as in terms of local invariant tensors on $Gr_{m,n}$. For two subspaces, the configuration is described by a set of $k$ principle angles that generalize the notion of quantum distance to the $k$-dimensional case. For more subspaces, there are additional invariants assoicated with triples of subspaces. These invariants fall into two classes: (1) a set $k^3$ of phases that generalize the Berry-Pancharatnam phase and (2) a collection of $3 k^2$ angles that have no analogues for 1-dimensional spaces. The local tensors needed for this construction are curvature of the tautological $U(n)$ bundle (the Wilczek-Zee curvature in physics) and a matrix-valued metric tensor. We present a procedure for calculating global invariants as integrals of invariants tensors over $Gr_{m,n}$ submanifolds constructed from geodesics. At the technical level, we find it convenient to represent the subspaces via the corresponding orthogonal projectors.