Local minimizers of the distances to the majorization flows

BENAC, MARÍA JOSÉ; MASSEY, PEDRO; RIOS, NOELIA; RUIZ, MARIANO

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL

IOP PUBLISHING LTD

Año: 2023 vol. 56

1751-8113

Let $cD(d)$ denote the convex set of density matrices of size $d$ and let $ho,,sigmaincD(d)$ be such that $hootprec sigma$. Consider the majorization flows $cL(sigma)={mu incD(d) : muprec sigma}$ and $cU(ho)={uincD(d) : hoprec u}$, where $prec$ stands for the majorization pre-order relation. We endow $cL(sigma)$ and $cU(ho)$ with the metric induced by the spectral norm. Let $N(cdot)$ be a strictly convex unitarily invariant norm and let $mu_0in cL(sigma)$ and $u_0incU(ho)$ be local minimizers of the distance functions $Phi_N(mu)=N(ho-mu)$, for $muincL(sigma)$ and $Psi_N(u)=N(sigma-u)$, for $uincU(ho)$. In this work we show that, for every unitarily invariant norm $ilde N(cdot)$ we have that $$ilde N(ho-mu_0)leq ilde N(ho-mu)coma muincL(sigma)py ilde N(sigma-u_0)leq ilde N(sigma-u)coma uincU(ho),.$$ That is, $mu_0$ and $u_0$ 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 $mu_0$ and $u_0$ in terms of a simple finite step algorithm; we also describe the geometrical structure (eigenvectors) of $mu_0$ and $u_0$ in terms of the geometrical structure of $sigma$ and $ho$, 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.