Numerical shadow and geometry of quantum states

TL;DR

This paper explores the geometric structure of quantum states using numerical shadows and projections, providing new tools for analyzing quantum state sets and their dynamics.

Contribution

It introduces the concept of numerical shadows for quantum states and generalizes them to mixed states, linking geometry with quantum state analysis.

Findings

Numerical shadows relate to the numerical ranges of matrices.

Projections of quantum state sets resemble numerical ranges.

Generalized shadows help analyze quantum state structures.

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.