Physics 4999E Study Guide - Final Guide: Doubly Stochastic Matrix, Self-Adjoint Operator, Convex Set

• School: Western University
• Department: Physics
• Course: Physics 4999E
• Professor: john stockholm

Document Summary

Cs 766/qic 820 theory of quantum information (fall 2011) This lecture discusses the notion of majorization and some of its connections to quantum infor- mation. There are other interesting applications of the notion, however, and a few of them will be discussed in this lecture. Let be a nite, nonempty set, and for the sake of this discussion let us focus on the real vector space r . This condition is equivalent to requiring that aeb is a probability vector for each b , or equivalently that a maps probability vectors to probability vectors. An operator a l(cid:0)r (cid:1) is doubly stochastic if it is the case that both a and at (or, equivalently, Next, let us write sym( ) to denote the set of one-to-one and onto functions of the form. : (or, in other words, the permutations of ).