|Numerical shadow and geometry of quantum states
|Year of Publication
|Dunkl CF, Gawron P, Holbrook J.A., Miszczak J, Puchała Z, Życzkowski K
|J. Phys. A: Math. Theor.
|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.