Local minimizers of the distances to the majorization flows

Abstract

Let $\mathcal{D}(d)$ denote the convex set of density matrices of size $d$ and let $\rho,\,\sigma\in\mathcal{D}(d)$ be such that $\rho\not\prec \sigma$. Consider the majorization flows $\mathcal{L}(\sigma)=\{\mu \in\mathcal{D}(d) \ : \ \mu\prec \sigma\}$ and $\mathcal{U}(\rho)=\{\nu\in\mathcal{D}(d) \ : \ \rho\prec \nu\}$, where $\prec$ stands for the majorization pre-order relation. We endow $\mathcal{L}(\sigma)$ and $\mathcal{U}(\rho)$ with the metric induced by the spectral norm. Let $N(\cdot)$ be a strictly convex unitarily invariant norm and let $\mu_0\in\ \mathcal{L}(\sigma)$ and $\nu_0\in\mathcal{U}(\rho)$ be local minimizers of the distance functions 
$\Phi_N(\mu)=N(\rho-\mu)$, for $\mu\in\mathcal{L}(\sigma)$ and $\Psi_N(\nu)=N(\sigma-\nu)$, for $\nu\in\mathcal{U}(\rho)$. In this work we show that, for every unitarily invariant norm $\tilde N(\cdot)$ we have that 
$$
\tilde N(\rho-\mu_0)\leq \tilde N(\rho-\mu)\, , \, \mu\in\mathcal{L}(\sigma)\peso{and} 
\tilde N(\sigma-\nu_0)\leq \tilde N(\sigma-\nu)\, , \, \nu\in\mathcal{U}(\rho)\,.
$$ That is, $\mu_0$ and $\nu_0$ 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 $\nu_0$ in terms of a simple finite step algorithm; we also describe the geometrical structure (eigenvectors) of $\mu_0$ and $\nu_0$ in terms of the geometrical structure of $\sigma$ and $\rho$, 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.