Local minimizers of the distances to the majorization flows

María José Benac, Pedro Massey, Noelia Belén Rios, Mariano Ruiz
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$\\Phi_N(\\mu)=N(\\rho-\\mu)$, 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 
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\\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.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"14 5","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Physics A","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1088/1751-8121/ad07c6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract

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.
到主要流的距离的局部最小值
摘要:设$\mathcal{D}(d)$表示大小为$d$的密度矩阵的凸集,并设$\rho,\,\sigma\in\mathcal{D}(d)$使得$\rho\not\prec \sigma$。考虑多数化流程$\mathcal{L}(\sigma)=\{\mu \in\mathcal{D}(d) \ : \ \mu\prec \sigma\}$和$\mathcal{U}(\rho)=\{\nu\in\mathcal{D}(d) \ : \ \rho\prec \nu\}$,其中$\prec$表示多数化预购关系。我们赋予$\mathcal{L}(\sigma)$和$\mathcal{U}(\rho)$由谱范数引起的度规。设$N(\cdot)$为严格凸酉不变范数,设$\mu_0\in\ \mathcal{L}(\sigma)$和$\nu_0\in\mathcal{U}(\rho)$为距离函数的局部极小值;$\Phi_N(\mu)=N(\rho-\mu)$是$\mu\in\mathcal{L}(\sigma)$, $\Psi_N(\nu)=N(\sigma-\nu)$是$\nu\in\mathcal{U}(\rho)$。在这项工作中,我们证明,对于每一个酉不变范数$\tilde N(\cdot)$,我们有
$$
\tilde N(\rho-\mu_0)\leq \tilde N(\rho-\mu)\, , \, \mu\in\mathcal{L}(\sigma)\peso{and} 
\tilde N(\sigma-\nu_0)\leq \tilde N(\sigma-\nu)\, , \, \nu\in\mathcal{U}(\rho)\,.
$$也就是说,$\mu_0$和$\nu_0$是相对于每个酉不变范数,到相应的多数化流的距离的全局最小值。我们用一个简单的有限步算法描述了$\mu_0$和$\nu_0$的(唯一的)谱结构(特征值);我们还分别用$\sigma$和$\rho$的几何结构描述了$\mu_0$和$\nu_0$的几何结构(特征向量)。我们包括对我们的结果的物理和计算意义的讨论。我们还将我们的结果与最近在量子信息理论背景下的一些相关结果进行了比较。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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