Approximate Majorization

Abstract

Although an input distribution may not majorize a target distribution, it may majorize a distribution which is close to the target. Here we introduce a notion of approximate majorization. For any distribution, and given a distance $\delta$, we find the approximate distributions which majorize (are majorized by) all other distributions within the distance $\delta$. We call these the steepest and flattest approximation. This enables one to compute how close one can get to a given target distribution under a process governed by majorization. We show that the flattest and steepest approximations preserve ordering under majorization. Furthermore, we give a notion of majorization distance. This has applications ranging from thermodynamics, entanglement theory, and economics.

4 pages main text + 6 pages appendix. 2 figure

References in corpus (4)

• Operational one-to-one mapping between coherence and entanglement measures
• Quantum Relative Lorenz Curves
• Smoothed generalized free energies for thermodynamics
• Maximum and minimum entropy states yielding local continuity bounds

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• Entropic Uncertainty Relations via Direct-Sum Majorization Relation for Generalized Measurements
• Smooth entropy in axiomatic thermodynamics
• Extremal elements of a sublattice of the majorization lattice and approximate majorization
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• Single-shot entanglement manipulation of states and channels revisited
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