Majorization, across the (Quantum) Universe

BELLOMO, G.; BOSYK, G. M.

Quantum Worlds. Perspectives on the Ontology of Quantum Mechanics

Cambridge University Press

Lugar: Cambridge; Año: 2019; p. 324 - 340

In how many ways can one represent a given quantum mixed state as a mixture ofpure states? Why (and in which sense) are separable states more disorderedglobally than locally? Is it possible to transform a given pure state into another by means of local operations and classical communication? How should anadequate formulation of the uncertainty principle be? All these questions, asdissimilar as they may seem, share one element in common: They can be answeredby 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).At present, it is clear that anyone who is interested in the field would find itappropriate to begin by the celebrated book by Marshall and Olkin (1979), whosesecond edition was coauthored by Arnold (2010). Recently, Arnold (2007) published an article entitled ?Majorization: Here, there and everywhere,? in which he presents a sampling of diverse areas in which majorization has been found useful in the last few years, such as geometry, probability, statistical mechanics, and graph theory. However, those contributions do not explore its quantum theoretical implications, which are much more extensively covered, for example, in Nielsen?s lecture notes (Nielsen 2002, see also Nielsen and Vidal 2001). In this chapter, we attempt to make a brief review of the subject and then to highlight the most important results of this research line in the quantum realm, in order to offer a kind of quantum counterpart of Arnold?s work.