  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
$p \geq 1$. 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 Ky Fan dominance
theorem about unitarily invariant norms. The proofs 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 Cram\'er$ theorem on large deviations for sums of i.i.d. random
variables.
