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CATALYTIC MAJORIZATION AND # p NORMS GUILLAUME AUBRUN AND ION NECHITA
 

Summary: CATALYTIC MAJORIZATION AND # p NORMS
GUILLAUME AUBRUN AND ION NECHITA
Abstract. An important problem in quantum information theory is the mathematical character­
ization 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 clas­
sical 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 # 1 norm can be fully described by inequalities
on the #p norms: #x# p # #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 #p­norm analogue of Ky Fan dominance theorem about unitarily
invariant norms. The proofs exploits links with another quantum phenomenon: the possibiliy of
multiple­copy transformations
(x# n
#
y# 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.
1. Introduction

  

Source: Aubrun, Guillaume - Institut Camille Jordan, Université Claude Bernard Lyon-I

 

Collections: Mathematics