Majorization is a type of a partial order that has been receiving growing attention in the quantum information community. Initially, it was adopted to entanglement theory, determining whether one pure entangled bipartite state can be converted into another by local means. In recent years, majorization found many applications in other areas of physics including quantum uncertainty relations, quantum thermodynamics, quantum coherence, the resource theory of asymmetry, and more. In this talk I will introduce a new type of majorization, called quantum majorization. It is a quantum generalization of ”matrix majorization” - a classical variant of majorization that was introduced by Dhal in 1999. I will discuss its properties and show that it provides an elegant generalization to many variants of majorization such as thermomajorization. Particularly, I will show that it can be viewed as a partial order within the space of quantum channels (or mixed bipartite states), determining if one quantum channel can be degraded to another. I will discuss its characterization in terms of the conditional min-entropy, and end with its applications to the resource theory of asymmetry and quantum thermodynamics.

This is joint work with David Jennings, Francesco Buscemi, Runyao Duan, and Iman Marvian. This work was supported by NSERC.