Skip to main content
SpringerLink
Log in

Hypercontractivity for Semigroups of Unital Qubit Channels

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Hypercontractivity is proved for products of qubit channels that belong to self-adjoint semigroups. The hypercontractive bound gives necessary and sufficient conditions for a product of the form \({e^{-t_1 H_1}\otimes \cdots \otimes e^{- t_n H_n}}\) to be a contraction from L p to L q, where L p is the algebra of 2n-dimensional matrices equipped with the normalized Schatten norm, and each generator H j is a self-adjoint positive semidefinite operator on the algebra of 2-dimensional matrices. As a particular case the result establishes the hypercontractive bound for a product of qubit depolarizing channels.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Audenaert K.M.R.: A note on the p → q norms of completely positive maps. Lin. Alg. Appl 430, 1436–1440 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Barak, B., Brando, F.G.S.L., Harrow, A.W., Kelner, J.A., Steurer, D., Zhou, Y. Hypercontractivity, Sum-of-Squares Proofs, and their Applications. Proceedings of the STOC 2012, pp. 307–326

  3. Beckner W.: Inequalities in Fourier analysis. Ann. Math. 102(1), 159–182 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ben-Aroya, A., Regev, O., de Wolf, R.: A hypercontractive inequality for matrix-valued functions with applications to quantum computing and LDCs. Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 477–486

  5. Bennett, C.H., Fuchs, C.A., Smolin, J.A.: Entanglement-enhanced classical communication on a noisy quantum channel. In: Hirota, O., Holevo, A.S., Caves, C.M. (eds.) Quantum Communication, Computing and Measurement, NY: Plenum Press, 1997, pp. 79–88

  6. Biane P.: Free hypercontractivity. Commun. Math. Phys. 184, 457–474 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Bonami A.: Etude des coefficients de Fourier des fonctions de L p (G). Ann. Inst. Fourier 20(2), 335–402 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  8. Carlen E.: Some integral identities and inequalities for entire functions and their application to the coherent state transform. J. Funct. Anal. 97, 231–249 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  9. Carlen E.A., Lieb E.H.: Optimal hypercontractivity for Fermi fields and related non-commutative integration inequalities. Commun. Math. Phys. 155, 27–46 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Federbush P.: A parially alternate derivation of a result of Nelson. J. Math. Phys. 10, 50–52 (1969)

    Article  ADS  MATH  Google Scholar 

  11. Glimm J.: Boson fields with nonlinear self-interaction in two dimensions. Commun. Math. Phys. 8, 12–25 (1968)

    Article  ADS  MATH  Google Scholar 

  12. Gross L.: Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)

    Article  Google Scholar 

  13. Gross L.: Hypercontractivity and logarithmic Sobolev inequalities for the Clifford–Dirichlet form. Duke Math. J. 43, 383–396 (1975)

    Article  Google Scholar 

  14. Gross, L.: Hypercontractivity, logarithmic Sobolev inequalities and applications: a survey of surveys. In: Faris, W.G.(ed.) Diffusion, Quantum Theory, and Radically Elementary Mathematics, Princeton, NJ: Princeton University Press, 2006

  15. Hoegh-Krohn R., Simon B.: Hypercontractive semigroups and two dimensional self-coupled Bose fields. J. Funct. Anal. 9, 121–180 (1972)

    Article  MathSciNet  Google Scholar 

  16. Kahn, J., Kalai, G., Linial, N.: The influence of variables on Boolean functions. In: Proceedings of the 29th Annual Symposium Foundations of Computer Science, 1988, pp. 68–80

  17. Kastoryano M.J., Temme K.: Quantum logarithmic Sobolev inequalities and rapid mixing. J. Math. Phys. 54, 052202 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  18. King C., Ruskai M.B.: Minimal entropy of states emerging from noisy quantum channels. IEEE Trans. Info. Theory 47, 1–19 (2001)

    MathSciNet  Google Scholar 

  19. King C.: Inequalities for trace norms of 2 ×  2 block matrices. Commun. Math. Phys. 242, 531–545 (2003)

    Article  ADS  MATH  Google Scholar 

  20. King C.: Additivity for unital qubit channels. J. Math. Phys. 43(10), 4641–4653 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Montanaro A.: Some applications of hypercontractive inequalities in quantum information theory. J. Math. Phys. 53, 122206 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  22. Montanaro, A., Osborne, T.: Quantum boolean functions. Chicago J. Theor. Comput. Sci. Article 1 (2010)

  23. Nelson, E.: A quartic interaction in two dimensions. In: Goodman R., Segal I.E. (eds.) Mathematical Theory of Elementary Particles (Dedham, Massachusetts, 1965), Cambridge MA: MIT Press, 1966, pp. 69–73

  24. Nelson E.: The free Markov field. J. Funct. Anal. 12, 211–227 (1973)

    Article  MATH  Google Scholar 

  25. Ruskai M.B., Szarek S., Werner W.: An analysis of completely-positive trace-preserving maps on 2 ×  2 matrices. Lin. Alg. Appl. 347, 159–187 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  26. Stinespring, W.F.: Positive functions on C*-algebras. Proceedings of the American Mathematical Society, 1955, pp. 211–216

  27. Watrous J.: Notes on super-operator norms induced by Schatten norms. Quantum Inf. Comput. 5, 57–67 (2005)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Northeastern University, Boston, MA, 02115, USA

    Christopher King

Authors
  1. Christopher King
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christopher King.

Additional information

Communicated by A. Winter

Rights and permissions

Reprints and permissions

About this article

Cite this article

King, C. Hypercontractivity for Semigroups of Unital Qubit Channels. Commun. Math. Phys. 328, 285–301 (2014). https://doi.org/10.1007/s00220-014-1982-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-1982-4

Keywords

Search

Navigation