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STOCHASTIC DOMINATION FOR ITERATED CONVOLUTIONS AND CATALYTIC MAJORIZATION
 

Summary: STOCHASTIC DOMINATION FOR ITERATED CONVOLUTIONS AND
CATALYTIC MAJORIZATION
GUILLAUME AUBRUN AND ION NECHITA
Abstract. We study how iterated convolutions of probability measures compare under stochastic
domination. We give necessary and sufficient conditions for the existence of an integer n such that µn
is stochastically dominated by n for two given probability measures µ and . As a consequence we
obtain a similar theorem on the majorization order for vectors in Rd. In particular we prove results
about catalysis in quantum information theory.
Domination stochastique pour les convolutions itérées et catalyse quantique
Résumé. Nous étudions comment les convolutions itérées des mesures de probabilités se compar-
ent pour la domination stochastique. Nous donnons des conditions nécessaires et suffisantes pour
l'existence d'un entier n tel que µn soit stochastiquement dominée par n, étant données deux
mesures de probabilités µ et . Nous obtenons en corollaire un théorème similaire pour des vecteurs
de Rd et la relation de Schur-domination. Plus spécifiquement, nous démontrons des résultats sur la
catalyse en théorie quantique de l'information.
Introduction and notations
This work is a continuation of [1], where we study the phenomenon of catalytic majorization in
quantum information theory. A probabilistic approach to this question involves stochastic domination
which we introduce in Section 1 and its behavior with respect to the convolution of measures. We
give in Section 2 a condition on measures µ and for the existence of an integer n such that µn

  

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

 

Collections: Mathematics