Communications in Mathematical Physics

, Volume 328, Issue 1, pp 285–301 | Cite as

Hypercontractivity for Semigroups of Unital Qubit Channels

  • Christopher KingEmail author


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 2 n -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.


Quantum Channel Logarithmic Sobolev Inequality Complete Positivity Unitary Invariance Noisy Quantum Channel 
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  1. 1.
    Audenaert K.M.R.: A note on the pq norms of completely positive maps. Lin. Alg. Appl 430, 1436–1440 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 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. 307–326Google Scholar
  3. 3.
    Beckner W.: Inequalities in Fourier analysis. Ann. Math. 102(1), 159–182 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 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. 477–486Google Scholar
  5. 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. 79–88Google Scholar
  6. 6.
    Biane P.: Free hypercontractivity. Commun. Math. Phys. 184, 457–474 (1997)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bonami A.: Etude des coefficients de Fourier des fonctions de L p(G). Ann. Inst. Fourier 20(2), 335–402 (1970)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 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)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Federbush P.: A parially alternate derivation of a result of Nelson. J. Math. Phys. 10, 50–52 (1969)ADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Glimm J.: Boson fields with nonlinear self-interaction in two dimensions. Commun. Math. Phys. 8, 12–25 (1968)ADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Gross L.: Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)CrossRefGoogle Scholar
  13. 13.
    Gross L.: Hypercontractivity and logarithmic Sobolev inequalities for the Clifford–Dirichlet form. Duke Math. J. 43, 383–396 (1975)CrossRefGoogle Scholar
  14. 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, 2006Google Scholar
  15. 15.
    Hoegh-Krohn R., Simon B.: Hypercontractive semigroups and two dimensional self-coupled Bose fields. J. Funct. Anal. 9, 121–180 (1972)MathSciNetCrossRefGoogle Scholar
  16. 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. 68–80Google Scholar
  17. 17.
    Kastoryano M.J., Temme K.: Quantum logarithmic Sobolev inequalities and rapid mixing. J. Math. Phys. 54, 052202 (2013)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    King C., Ruskai M.B.: Minimal entropy of states emerging from noisy quantum channels. IEEE Trans. Info. Theory 47, 1–19 (2001)MathSciNetGoogle Scholar
  19. 19.
    King C.: Inequalities for trace norms of 2 ×  2 block matrices. Commun. Math. Phys. 242, 531–545 (2003)ADSzbMATHCrossRefGoogle Scholar
  20. 20.
    King C.: Additivity for unital qubit channels. J. Math. Phys. 43(10), 4641–4653 (2002)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Montanaro A.: Some applications of hypercontractive inequalities in quantum information theory. J. Math. Phys. 53, 122206 (2012)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Montanaro, A., Osborne, T.: Quantum boolean functions. Chicago J. Theor. Comput. Sci. Article 1 (2010)Google Scholar
  23. 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. 69–73Google Scholar
  24. 24.
    Nelson E.: The free Markov field. J. Funct. Anal. 12, 211–227 (1973)zbMATHCrossRefGoogle Scholar
  25. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Stinespring, W.F.: Positive functions on C*-algebras. Proceedings of the American Mathematical Society, 1955, pp. 211–216Google Scholar
  27. 27.
    Watrous J.: Notes on super-operator norms induced by Schatten norms. Quantum Inf. Comput. 5, 57–67 (2005)zbMATHMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsNortheastern UniversityBostonUSA

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