Numerical shadow and geometry of quantum states

Details Numerical shadow and geometry of quantum states Numerical shadow and geometry of quantum states

2011-04-14

arxiv.org/abs/1104.2760v1

J. Phys. A: Math. Theor. 44, 335301 (2011)

Physics

Quantum Physics

19 pages, 5 figures

Scientific paper

10.1088/1751-8113/44/33/335301

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.

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