The theory of majorization and its variants, including thermomajorization,
have been found to play a central role in the formulation of many physical
resource theories, ranging from entanglement theory to quantum thermodynamics.
Here we formulate the framework of quantum relative Lorenz curves, and show how
it is able to unify majorization, thermomajorization, and their noncommutative
analogues. In doing so, we define the family of Hilbert α-divergences
and show how it relates with other divergences used in quantum information
theory. We then apply these tools to the problem of deciding the existence of a
suitable transformation from an initial pair of quantum states to a final one,
focusing in particular on applications to the resource theory of athermality, a
precursor of quantum thermodynamics.Comment: ver2: 14 pages, 3 figures, published version; ver1: 5 pages + 2
figures + 15 more pages + 1 more figur
