Catalytic Majorization and $\ell_p$ Norms

Abstract

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 $x \otimes z \prec y \otimes z$ for some z, where $\prec$ denotes the standard majorization relation.

Our main result is that the closure of $T(y)$ in the $\ell_1$ norm can be fully described by inequalities on the $\ell_p$ norms: $||x||_p \leq ||y||_p$ for all p ≥ 1. This is a first step towards a complete description of T(y) itself. It can also be seen as a $\ell_p$ -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 ($x^{\otimes n} \prec y^{\otimes n}$ for given n). The main new tool is a variant of Cramér’s theorem on large deviations for sums of i.i.d. random variables.