Relative submajorization and its use in quantum resource theories

Abstract

We introduce and study a generalization of majorization called relative submajorization and show that it has many applications to the resource theories of thermodynamics, bipartite entanglement, and quantum coherence. Relative majorization is an ordering on pairs of vectors induced by stochastic transformations: One pair of vectors relatively majorizes another when there exists a stochastic transformation taking the former to the latter. Relative submajorization is a weakened version in which a substochastic matrix need only generate a vector pair obeying certain positivity conditions relative to the input pair. In the context of resource theories, we show that relative submajorization characterizes both the probability and approximation error that can be obtained when transforming one resource to another, also when assisted by additional standard resources such as useful work or maximallyentangled states. These characterizations have a geometric formulation as the ratios or differences, respectively, between the Lorenz curves associated with the input and output resources, making them efficient to compute. We also find several interesting bounds on the reversibility of a given transformation in terms of the properties of the forward transformation. The main technical tool used to establish these results is linear programming duality, which is used to show that any instance of relative submajorization can be “dilated” to an instance of strict relative majorization.