Optimal common resource in majorization-based resource theories

G. M. Bosyk, G. Bellomo, F. Holik, H. Freytes, G. Sergioli

Published 2019-02-05Version 1

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 proving that the majorization lattice is complete. The proof relies heavily on the more geometric construction provided by the Lorenz curves. 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.