The $d$-Majorization Polytope

Abstract

We investigate geometric and topological properties of $d$-majorization -- a generalization of classical majorization to positive weight vectors $d\in {\mathbb{R}}^n$. In particular, we derive a new, simplified characterization of $d$-majorization which allows us to work out a halfspace description of the corresponding $d$-majorization polytopes. That is, we write the set of all vectors which are $d$-majorized by some given vector $y\in {\mathbb{R}}^n$ as an intersection of finitely many half spaces, i.e. as solutions to an inequality of the type $Mx\le b$. Here $b$ depends on $y$ while $M$ can be chosen independently of $y$. This description lets us prove continuity of the $d$-majorization polytope (jointly with respect to $d$ and $y$) and, furthermore, lets us fully characterize its extreme points. Interestingly, for $y\ge 0$ one of these extreme points classically majorizes every other element of the $d$-majorization polytope. Moreover, we show that the induced preorder structure on ${\mathbb{R}}^n$ admits minimal and maximal elements. While the former are always unique the latter are unique if and only if they correspond to the unique minimal entry of the $d$-vector.

35 pages, 2 figures, revised version. Re-submitted to Linear Algebra and its Applications

References in corpus (8)

• Description of quantum coherence in thermodynamic processes requires constraints beyond free energy
• Quantum coherence, time-translation symmetry and thermodynamics
• Gibbs-Preserving Maps outperform Thermal Operations in the quantum regime
• The $d$-Majorization Polytope
• Two constructive proofs on d-majorization and thermo-majorization
• Exploring the gap between thermal operations and enhanced thermal operations
• Strict Positivity and $D$-Majorization
• Reachability in Controlled Markovian Quantum Systems: An Operator-Theoretic Approach

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