Title | Numerical shadow and geometry of quantum states |
Publication Type | Journal Article |
Year of Publication | 2011 |
Authors | Dunkl CF , Gawron P , Holbrook J.A. , Miszczak J , Puchała Z , Życzkowski K |
Journal | J. Phys. A: Math. Theor. |
Volume | 44 |
ISSN | 1751-8113 |
Abstract | 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. |