Communications in Mathematical Physics

, Volume 278, Issue 1, pp 133–144 | Cite as

Catalytic Majorization and \(\ell_p\) Norms

  • Guillaume Aubrun
  • Ion Nechita


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.


Entangle State Probability Vector Classical Communication Local Operation Large Deviation Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Université de Lyon, Université Lyon 1, CNRS, UMR 5208 Institut Camille Jordan, Batiment du Doyen Jean BraconnierVilleurbanne CedexFrance

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