Local minimizers of the distances to the majorization flows

Abstract

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 μ 0 ∈   L ( σ ) and ν 0 ∈ U ( ρ ) be local minimizers of the distance functions Φ N ( μ ) = N ( ρ − μ ) , for μ ∈ L ( σ ) and Ψ N ( ν ) = N ( σ − ν ) , for ν ∈ U ( ρ ) . In this work we show that, for every unitarily invariant norm N ˜ ( ⋅ ) we have that N ˜ ρ − μ 0 ⩽ N ˜ ρ − μ , μ ∈ L σ and  N ˜ σ − ν 0 ⩽ N ˜ σ − ν ,   ν ∈ U ρ . That is, μ 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 μ 0 and ν 0 in terms of a simple finite step algorithm; we also describe the geometrical structure (eigenvectors) of μ 0 and ν 0 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.