Nous étudions comment les convolutions itérées des mesures de probabilités se comparent pour la domination stochastique. Nous donnons des conditions nécessaires et suffisantes pour l'existence d'un entier n tel que μ*n soit stochastiquement dominée par ν*n, étant données deux mesures de probabilités μ et ν. 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.
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 μ*n is stochastically dominated by ν*n for two given probability measures μ and ν. 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.
Mots clés : stochastic domination, iterated convolutions, large deviations, majorization, catalysis
@article{AIHPB_2009__45_3_611_0, author = {Aubrun, Guillaume and Nechita, Ion}, title = {Stochastic domination for iterated convolutions and catalytic majorization}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {611--625}, publisher = {Gauthier-Villars}, volume = {45}, number = {3}, year = {2009}, doi = {10.1214/08-AIHP175}, mrnumber = {2548496}, zbl = {1179.60008}, language = {en}, url = {http://www.numdam.org/articles/10.1214/08-AIHP175/} }
TY - JOUR AU - Aubrun, Guillaume AU - Nechita, Ion TI - Stochastic domination for iterated convolutions and catalytic majorization JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2009 SP - 611 EP - 625 VL - 45 IS - 3 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/08-AIHP175/ DO - 10.1214/08-AIHP175 LA - en ID - AIHPB_2009__45_3_611_0 ER -
%0 Journal Article %A Aubrun, Guillaume %A Nechita, Ion %T Stochastic domination for iterated convolutions and catalytic majorization %J Annales de l'I.H.P. Probabilités et statistiques %D 2009 %P 611-625 %V 45 %N 3 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/08-AIHP175/ %R 10.1214/08-AIHP175 %G en %F AIHPB_2009__45_3_611_0
Aubrun, Guillaume; Nechita, Ion. Stochastic domination for iterated convolutions and catalytic majorization. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 611-625. doi : 10.1214/08-AIHP175. http://www.numdam.org/articles/10.1214/08-AIHP175/
[1] Catalytic majorization and ℓp norms. Comm. Math. Phys. 278 (2008) 133-144. | MR | Zbl
and .[2] Classification of nonasymptotic bipartite pure-state entanglement transformations. Phys. Rev. A 65 (2002) 052315.
, and .[3] Matrix Analysis. Springer, New York, 1997. | MR | Zbl
.[4] Mathematical structure of entanglement catalysis. Phys. Rev. A (3) 64 (2001) 042314. | MR
and .[5] Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998. | MR | Zbl
and .[6] Some issues in quantum information theory. J. Comput. Sci. and Technol. 21 (2006) 776-789. | MR
, , , and .[7] An Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York, 1966. | MR | Zbl
.[8] Relation between catalyst-assisted entanglement transformation and multiple-copy transformation. Phys. Rev. A (3) 74 (2006) 042312.
, and .[9] Probability and Random Processes, 3rd edition. Oxford University Press, New York, 2001. | MR | Zbl
and .[10] Entanglement-assisted local manipulation of pure quantum states. Phys. Rev. Lett. 83 (1999) 3566-3569. | MR | Zbl
and .[11] The capacity of hybrid quantum memory. IEEE Trans. Inform. Theory 49 (2003) 1465-1473. | MR | Zbl
.[12] Inequalities: Theory of Majorization and Its Applications. Academic Press Inc., New York, 1979. | MR | Zbl
and .[13] Conditions for a class of entanglement transformations. Phys. Rev. Lett. 83 436 (1999).
.[14] ε-convertibility of entangled states and extension of Schmidt rank in infinite-dimensional systems. Quantum Inf. Comput. 8 (2008) 0030-0052. | MR | Zbl
, , and .[15] Problems and Theorems in Analysis. Springer, Berlin, 1978. | MR | Zbl
and .[16] Comparison Metrods for Queues and Other Stochastic Models. Wiley, Chichester, 1983. | MR | Zbl
.[17] Catalytic Transformations for bipartite pure states. J. Phys. A 40 (2007) 12185-12212. | MR | Zbl
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