Noncommutative majorization

Abstract

We introduce a theory of noncommutative majorization that extends the classical majorization theory introduced by Hardy, Littlewood and Pólya to tuples of self-adjoint matrices that do not necessarily commute. We define and characterize a noncommutative majorization order that extends the classical majorization order. The definition is in terms of convex noncommutative functions, and we utilize the noncommutative convexity theory and noncommutative Choquet theory recently introduced by Davidson and the first author. As an application, we obtain a new necessary and sufficient condition for the existence of a trace-preserving completely positive map, i.e. a quantum channel, that interpolates between two finite sets of matrices. We give examples demonstrating that it is not always possible for this map to be chosen mixed unitary, even locally. We also address the computational difficulty of verifying the noncommutative majorization order. Our results further apply beyond the tracial case, and we obtain more general results characterizing the existence of unital completely positive maps that preserve an arbitrary faithful state or even unital completely positive map.