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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 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 (xn yn 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
The increasing interest that quantum entanglement has received in the past decade is due, in
part, to its use as a resource in quantum information processing. We investigate the problem of
entanglement transformation: under which conditions can an entangled state | be transformed into
another entangled state | ? We restrict ourselves to LOCC protocols: Alice and Bob share |
and have at their disposal only local operations (such as unitaries UA IB for Alice) and classical

  

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

 

Collections: Mathematics