Numerical shadow and geometry of quantum states

TitleNumerical shadow and geometry of quantum states
Publication TypeJournal Article
Year of Publication2011
AuthorsDunkl CF, Gawron P, Holbrook J.A., Miszczak J.A., Puchała Z, Życzkowski K
JournalJ. Phys. A: Math. Theor.
AbstractThe 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.