# Optimal common resource in majorization-based resource theories

## TL;DR

This paper investigates the problem of identifying the optimal common resource in majorization-based quantum resource theories, providing a comprehensive solution using the properties of the majorization lattice and Lorenz curves.

## Contribution

It establishes the completeness of the majorization lattice and applies this to find optimal resources, including for continuous target sets, with implications for quantum thermodynamics and coherence.

## Key findings

- Proves the completeness of the majorization lattice.
- Provides explicit methods to find infimum and supremum of target sets.
- Links approximate majorization to lattice completeness.

## Abstract

We address the problem of finding the optimal common resource for an arbitrary family of target states in quantum resource theories based on majorization, that is, theories whose conversion law between resources is determined by a majorization relationship, such as it happens with entanglement, coherence or purity. We provide a conclusive answer to this problem by appealing to the completeness property of the majorization lattice. We give a proof of this property that relies heavily on the more geometric construction provided by the Lorenz curves, which allows to explicitly obtain the corresponding infimum and supremum. Our framework includes the case of possibly non-denumerable sets of target states (i.e. targets sets described by continuous parameters). In addition, we show that a notion of approximate majorization, which has recently found application in quantum thermodynamics, is in close relation with the completeness of this lattice. Finally, we provide some examples of optimal common resources within the resource theory of quantum coherence.