Numerical shadow and geometry of quantum states
  The totality of normalised density matrices of order N forms a convex set Q_N in R^(N^2-1). 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 order N. For a matrix A of a order 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 CP^(N-1). We define generalized, mixed-states shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein. 
Charles F. Dunkl and Piotr Gawron and John A. Holbrook and Jaros{\l}aw   A. Miszczak and Zbigniew Pucha{\l}a and Karol \.Zyc...
quant-ph math-ph math.MP math.OA