Majorization, across the (quantum) universe

Abstract

In how many ways can one represent a given quantum mixed state as a mixture of pure states? Why (and in which sense) are separable states more disordered globally than locally? Is it possible to transform a given pure state into another by means of local operations and classical communication? How should an adequate formulation of the uncertainty principle be? All these questions, as dissimilar as they may seem, share one element in common: They can be answered by appealing to the notion of majorization partial order. Majorization is nowadays a well-established and powerful mathematical tool with many and different applications in several disciplines, such as economics, biology, and physics, among others. Indeed, the seminal idea of this concept had already been glimpsed by Lorenz (1905) while studying the inequality of wealth distribution and developing the representation of the (nowadays called) Lorenz curves. Moreover, the famous Gini coefficient (Gini 1912), widely accepted as a legitimate quantifier of income distribution inequality, is merely a ratio between graphical areas defined by a Lorenz curve. Other key contributions to the subject were those by Muirhead (1903), Dalton (1920), Schur (1923), and Hardy, Littlewood, and Pólya (1929). The name “majorization,” though, appears first in the prominent book by Hardy, Littlewood, and Pólya (1934).