Catalytic majorization and lp norms

An important problem in quantum information theory is the mathematical characterization of the phenomenon of quantum catalysis: when can the surrounding entanglement be used to perform transformations of a jointly held quantum state under LOCC (local operations and classical communication)? Mathematically, the question amounts to describe, for a fixed vector y, the set T (y) of vectors x such that we have x circle times z < y circle times z for some z, where < denotes the standard majorization relation. Our main result is that the closure of T (y) in the l(1) norm can be fully described by inequalities on the l(p) norms: parallel to x parallel to(p) <= parallel to y parallel to(p) for all p >= 1. This is a first step towards a complete description of T (y) itself. It can also be seen as a l(p)-norm analogue of the Ky Fan dominance theorem about unitarily invariant norms. The proof exploits links with another quantum phenomenon: the possibiliy of multiple-copy transformations (x(circle times n) < y(circle times n) for given n). The main new tool is a variant of Cramer's theorem on large deviations for sums of i.i.d. random variables.