Numerical shadow and geometry of quantum states

The totality of normalized density matrices of dimension N forms a convex set Q(N) in R-N2 (1). Working with the flat geometry induced by the Hilbert-Schmidt distance, we consider images of orthogonal projections of Q(N) onto a two-plane and show that they are similar to the numerical ranges of matrices of dimension N. For a matrix A of dimension N, one defines its numerical shadow as a probability distribution supported on its numerical range W(A), induced by the unitarily invariant Fubini-Study measure on the complex projective manifold CPN-1. We define generalized, mixed-state shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein.