Local minimizers of the distances to the majorization flows

Abstract

Let D(d) denote the convex set of density matrices of size d and let ρ, σ ∈ D(d) be such that ρ ̸≺ σ. Consider the majorization flows L(σ) = {µ ∈ D(d) : µ ≺ σ} and U(ρ) = {ν ∈ D(d) : ρ ≺ ν}, where ≺ stands for the majorization pre-order relation. We endow L(σ) and U(ρ) with the metric induced by the spectral norm. Let N(·) be a strictly convex unitarily invariant norm and let µ0 ∈ L(σ) and ν0 ∈ U(ρ) be local minimizers of the distance functions ΦN (µ) = N(ρ − µ), for µ ∈ L(σ) and ΨN (ν) = N(σ − ν), for ν ∈ U(ρ). In this work we show that, for every unitarily invariant norm Ñ(·) we have that Ñ(ρ − µ0) ≤ Ñ(ρ − µ), µ ∈ L(σ) and Ñ(σ − ν0) ≤ Ñ(σ − ν), ν ∈ U(ρ). That is, µ0 and ν0 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 µ0 and ν0 in terms of a simple finite step algorithm; we also describe the geometrical structure (eigenvectors) of µ0 and ν0 in terms of the geometrical structure of σ 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.