Local minimizers of the distances to the majorization flows

Abstract

Let denote the convex set of density matrices of size d and let be such that . Consider the majorization flows and , where stands for the majorization pre-order relation. We endow and with the metric induced by the spectral norm. Let be a strictly convex unitarily invariant norm and let and be local minimizers of the distance functions , for and , for . In this work we show that, for every unitarily invariant norm we have that N˜ρ−μ0⩽N˜ρ−μ,μ∈Lσand N˜σ−ν0⩽N˜σ−ν, ν∈Uρ. That is, 0 and ν 0 are global minimizers of the distances to the corresponding majorization flows, with respect to every unitarily invariant norm. We describe the (unique) spectral structure (eigenvalues) of 0 and ν 0 in terms of a simple finite step algorithm; we also describe the geometrical structure (eigenvectors) of 0 and ν 0 in terms of the geometrical structure of and ρ, respectively. We include a discussion of the physical and computational implications of our results. We also compare our results to some recent related results in the context of quantum information theory.