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One-and-a-Half Quantum de Finetti Theorems

Abstract

When nk systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound on the trace distance of this approximation is given by \({2\frac{kd^2}{n}}\) , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation \({U_{\mu +\nu} \subset U_\mu \otimes U_\nu}\) of the unitary group U(d), for highest weights μ, ν and μ + ν. Let ξμ be the state obtained by tracing out U ν. Then ξμ is close to a convex combination of the coherent states \({U_\mu(g)|{v_\mu\rangle}}\) , where \({g\in U(d)}\) and \({|v_\mu\rangle}\) is the highest weight vector in U μ.

For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second de Finetti-style theorem (our “half” theorem). It arises from a combinatorial formula for the distance of certain special symmetric Werner states to states of fixed spectrum, making a connection to the recently defined shifted Schur functions [1]. This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike their classical counterparts) must depend on the dimension d. The last part of this paper analyses the structure of the set of symmetric Werner states and shows that the product states in this set do not form a polytope in general.

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References

  1. Okounkov, A., Olshanski, G.: Alg. i Anal. 9, no. 2, 13–146 (1997) (Russian); Eng. in st. Petersburg Math. J 9, no. 2 (1998)

  2. de Finetti B. (1937). Ann. Inst. H. Poincaré 7: 1

    MATH  MathSciNet  Google Scholar 

  3. Størmer E. (1969). J. Funct. Anal. 3: 48

    MATH  Article  Google Scholar 

  4. Hudson R.L., Moody G.R. and Wahrschein Z. (1976). verw. Geb. 33: 343

    MATH  Article  Google Scholar 

  5. Petz D. (1990). Prob. Th. Rel. Fields. 85: 1–11

    Article  MathSciNet  Google Scholar 

  6. Caves C.M., Fuchs C.A. and Schack R. (2002). J. Math. Phys. 43: 4537

    MATH  Article  ADS  MathSciNet  Google Scholar 

  7. Fuchs CA, Schack R: In: Quantum Estimation Theory, M.G.A. Paris, J. Rehaeck (eds), Berlin: Springer, 2004

  8. Fuchs C.A., Schack R. and Scudo P.F. (2004). Phys. Rev. A 69: 062305

    Article  ADS  Google Scholar 

  9. König R. and Renner R. (2005). J. Math. Phys. 46: 122108

    Article  MathSciNet  Google Scholar 

  10. Renner, R.: Security of Quantum Key Distribution. PhD thesis, ETH Zurich, 2005, available at http://axiv.org/list/quant-ph/0512258, 2005

  11. Werner R.F. (1989). Phys. Rev. A 40: 4277

    Article  ADS  Google Scholar 

  12. Carter, R., Segal, G., MacDonald, I.: Lectures on Lie Groups and Lie Algebras. London Mathematical Society Student Texts vol. 32, 1st ed, Cambridge: Cambridge Univ. Press, 1995

  13. Perelomov, A.: Generalized coherent states and their application. Texts and Monographs in Physics, Berlin: Springer-Verlag, 1986

  14. Weyl H. (1950). The Theory of Groups and Quantum Mechanics. Dover Publications, Inc, New York

    Google Scholar 

  15. Diaconis P. and Freedman D. (1980). The Annals of Probability 8: 745

    MATH  MathSciNet  Google Scholar 

  16. Fannes M., Lewis J.T. and Verbeure A. (1988). Lett. Math. Phys. 15: 255

    MATH  Article  MathSciNet  Google Scholar 

  17. Raggio G.A. and Werner R.F. (1989). Helv. Phys. Acta 62: 980

    MathSciNet  Google Scholar 

  18. Ioannou, L.M.: Deterministic computational complexity of the quantum separability problem. http://arxiv.org/list/quant-ph/0603199; 2006, to appear in QIP, 2006

  19. Doherty, A.: Personal communication, 2006

  20. Audenaert, K.: Available at http://qols.ph.ic.ac.uk/ kauden/QITNotes_files/irreps.pdf, 2004

  21. Fulton, W.F.: Young Tableaux. Cambridge: Cambridge University Press, 1997

  22. Eggeling, T., Werner, R.F.: Phys. Rev. A 63, 04211 (2001)

    Google Scholar 

  23. Macdonald, I.G.: Symmetric functions and Hall polynomials. Oxford: Clarendon Press, 1979

  24. Klyachko, A.: http:/laxiv.org/list/quant-ph/0206012, 2002

  25. Keyl M. and Werner R.F. (2001). Phys. Rev. A 64: 052311

    Article  ADS  MathSciNet  Google Scholar 

  26. Hudson R.L. (1981). Found. Phys. 11: 805

    Article  MathSciNet  Google Scholar 

  27. Fannes M., Spohn H. and Verbeure A. (1980). J. Math. Phys. 21: 355

    MATH  Article  ADS  MathSciNet  Google Scholar 

  28. Brun T.A., Caves C.M. and Schack R. (2001). Phys. Rev. A 63: 042309

    Article  ADS  MathSciNet  Google Scholar 

  29. Doherty A.C., Parillo P.A. and Spedalieri F.M. (2004). Phys. Rev. A 69: 022308

    Article  ADS  Google Scholar 

  30. Audenaert KMR.: In: Proceedings of MTNS2004 (2004), available at http://arxiv.org/list/quant-ph/ 0402076, 2004

  31. Terhal B.M., Doherty A.C. and Schwab D. (2003). Phys. Rev. Lett 90: 157903

    Article  ADS  MathSciNet  Google Scholar 

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Correspondence to Robert König.

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Communicated by M.B. Ruskai

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Christandl, M., König, R., Mitchison, G. et al. One-and-a-Half Quantum de Finetti Theorems. Commun. Math. Phys. 273, 473–498 (2007). https://doi.org/10.1007/s00220-007-0189-3

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Keywords

  • Irreducible Representation
  • Coherent State
  • Unitary Group
  • Symmetric State
  • High Weight Vector