Communications in Mathematical Physics

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

Catalytic Majorization and \(\ell_p\) Norms

Article

Abstract

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.

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References

  1. 1.
    Aubrun, G., Nechita, I.: Stochastic domination for iterated convolutions and catalytic majorization. Preprint, Available at arXiv:0707.0211Google Scholar
  2. 2.
    Bandyopadhyay S., Roychowdhury V. and Sen U. (2002). Classification of nonasymptotic bipartite pure-state entanglement transformations. Phys. Rev. A 65: 052315 CrossRefADSGoogle Scholar
  3. 3.
    Bhatia, R.: Matrix Analysis. Graduate Texts in Mathematics, Volume 169, New York: Springer-Verlag, 1997Google Scholar
  4. 4.
    Daftuar, S.K.: Eigenvalues Inequalities in Quantum Information Processing. Ph. D. Thesis, California Institute of technology, 2004. Available at http://resolver.caltech.edu/CaltechETD:etd-03312004-100014
  5. 5.
    Daftuar S.K. and Klimesh M. (2001). Mathematical structure of entanglement catalysis. Phys. Rev. A (3) 64(4): 042314 CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Second edition. Applications of Mathematics (New York), 38, New York: Springer-Verlag, 1998Google Scholar
  7. 7.
    Duan R., Feng Y., Li X. and Ying M. (2005). Multiple-copy entanglement transformation and entanglement catalysis. Phys. Rev. A 71: 042319 CrossRefADSGoogle Scholar
  8. 8.
    Duan R., Ji Z., Feng Y., Li X. and Ying M. (2006). Some issues in quantum information theory. J. Comput. Sci. & Tech. 21(5): 776–789 CrossRefMathSciNetGoogle Scholar
  9. 9.
    Jonathan D. and Plenio M.B. (1999). Entanglement-assisted local manipulation of pure quantum states. Phys. Rev. Lett. 83(17): 3566–3569 MATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Klimesh, M.: Entropy measures and catalysis of bipartite quantum state transformations. Extended abstract, ISIT 2004, Chicago, USAGoogle Scholar
  11. 11.
    Klimesh, M.: Inequalities that collectively completely characterize the catalytic majorization relation. Preprint, Available at arXiv:0709.3680Google Scholar
  12. 12.
    Kuperberg G. (2003). The capacity of hybrid quantum memory. IEEE Trans. Inform. Theory 49: 1465–1473 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Marshall, A., Olkin, I.: Inequalities: Theory of Majorization and its Applications. Mathematics in Science and Engineering, 143. New York-London: Academic Press Inc., 1979Google Scholar
  14. 14.
    Mitra, T., Ok, E.: Majorization by L p-Norms. Preprint, available at http://homepages.nyu.edu/~eo1/Papers-PDF/Major.pdf
  15. 15.
    Nielsen M. (1999). Conditions for a class of entanglement transformations. Phys. Rev. Lett. 83: 436 CrossRefADSGoogle Scholar
  16. 16.
    Nielsen, M.: An introduction to majorization and its applications to quantum mechanics. Preprint, available at http://www.qinfo.org/talks/2002/maj/book.ps
  17. 17.
    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2000Google Scholar
  18. 18.
    Open problems in Quantum Information Theory, available at http://www.imaph.tu-bs.de/qi/problems/ or http://arxiv.org/list/quant-ph/0504166, 2005
  19. 19.
    Pólya, G., Szegö, G.: Problems and Theorems in Analysis. Berlin-New York: Springer-Verlag, 1978Google Scholar
  20. 20.
    Turgut, S.: Necessary and sufficient conditions for the trumping relation. Preprint, Available at arXiv:0707.0444Google Scholar

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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