Majorization theory, majorization preserving maps and applications to quantum Information

Vector majorization is a preorder relation on the space of
n-dimensional real vectors. It has been generalized to matrices in a
number of different ways. The origins of Majorization Theory can be
traced back to Muirhead's results on symmetric polynomials, Schur's
work on Hadamard's inequality, and the notion of Lorenz curve.

The wide applicability of majorization to quantum mechanics arises as
a result of two fundamental theorems connecting majorization to
unitary matrices: Horn’s lemma and Uhlmann’s theorem, which
characterize vector and matrix majorization in terms of unitary
transformations. Notably, as means for measuring disorder,
majorization is more powerful than the classical notions of entropy.

We will discuss linear operators preserving majorization for
stochastic matrices and applications of this and similar problems to
quantum information theory.