Catalytic majorization and $\ell_p$ norms

The possible transformations of jointly held quantum states under local operations and classical communication (LOCC) are described using majorization, which is a fairly well-known relation. This LOCC protocol can be generalized in two ways: using an auxiliary entangled state as a catalyst (Entanglement-assisted LOCC) or performing simultaneously several identical transformations (Multiple-copy LOCC). These protocols are encoded by relations on probability vectors that generalize the majorization relation. We study the geometry of the set of vectors dominated by a fixed vector for these two relations. We show that both closures in the $\ell_1$ norm coincide and characterize it using inequalities involving the $\ell_p$ norms, $p \geq 1$. Our proof uses a variant of Cram\'{e}r's theorem on large deviations