Numerical shadow and geometry of quantum states
Quantum Physics
2011-08-09 v1 Mathematical Physics
math.MP
Operator Algebras
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.
Cite
@article{arxiv.1104.2760,
title = {Numerical shadow and geometry of quantum states},
author = {Charles F. Dunkl and Piotr Gawron and John A. Holbrook and Jarosław A. Miszczak and Zbigniew Puchała and Karol Życzkowski},
journal= {arXiv preprint arXiv:1104.2760},
year = {2011}
}
Comments
19 pages, 5 figures