Catalytic majorization and ℓₚ norms

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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 ⊗ z y ⊗ z for some z, where denotes the standard majorization relation. Our main result is that the closure of T(y) in the ℓ₁ norm can be fully described by inequalities on the ℓₚ norms: xₚ ≤ yₚ for all p ≥ 1. This is a first step towards a complete description of T(y) itself. It can also be seen as a ℓₚ-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⊗ ⁿ y⊗ ⁿ for given n). The main new tool is a variant of Cramér theorem on large deviations for sums of i.i.d. random variables.