Letters in Mathematical Physics

, Volume 107, Issue 6, pp 1131–1155 | Cite as

The minimum Rényi entropy output of a quantum channel is locally additive

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Abstract

We show that the minimum Rényi entropy output of a quantum channel is locally additive for Rényi parameter \(\alpha >1\). While our work extends the results of Gour and Friedland (IEEE Trans. Inf. Theory 59(1):603, 2012) (in which local additivity was proven for \(\alpha =1\)), it is based on several new techniques that incorporate the multiplicative nature of \(\ell _p\)-norms, in contrast to the additivity property of the von-Neumann entropy. Our results demonstrate that the counterexamples to the Rényi additivity conjectures exhibit purely global effects of quantum channels. Interestingly, the approach presented here cannot be extended to Rényi entropies with parameter \(\alpha <1\).

Keywords

Quantum information Entropy output Rényi entroy Additivity conjecture 

Mathematics Subject Classification

81P45 94A17 

References

  1. 1.
    Amosov, G.G., Holevo, A.S., Werner, R.F.: On the additivity conjecture in quantum information theory. Probl. Inf. Transm. C/C Probl. Peredachi Informatsii 36(4), 305–313 (2000)MATHGoogle Scholar
  2. 2.
    Audenaert, K.M., Braunstein, S.L.: On strong superadditivity of the entanglement of formation. Commun. Math. Phys. 246(3), 443–452 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Audenaert, K.M., Calsamiglia, J., Munoz-Tapia, R., Bagan, E., Masanes, L., Acin, A., Verstraete, F.: Discriminating states: the quantum Chernoff bound. Physi. Rev. Lett. 98(16), 160501 (2007)ADSCrossRefGoogle Scholar
  4. 4.
    Bennett, C.H., DiVincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, B.M.: Unextendible product bases and bound entanglement. Phys. Rev. Lett. 82(26), 5385 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brandao, F.G., Eisert, J., Horodecki, M., Yang, D.: Entangled inputs cannot make imperfect quantum channels perfect. Phys. Rev. Lett. 106(23), 230502 (2011)ADSCrossRefGoogle Scholar
  6. 6.
    Brandao, F.G., Horodecki, M.: On Hastings’ counterexamples to the minimum output entropy additivity conjecture. Open Syst. Inf. Dyn. 17(01), 31–52 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brandao, F., Horodecki, M., Ng, N., Oppenheim, J., Wehner, S.: The second laws of quantum thermodynamics. Proc. Nat. Acad. Sci. 112(11), 3275–3279 (2015)ADSCrossRefGoogle Scholar
  8. 8.
    Derksen, H., Friedland, S., Gour, G., Gross, D., Gurvits, L., Roy, A., Yard, J.: On minimum entropy output and the additivity conjecture. In: Notes of Quantum Information Group, American Mathematical Institute workshop on Geometry and representation theory of tensors for computer science, statistics and other areas (2008)Google Scholar
  9. 9.
    Fannes, M., Haegeman, B., Mosonyi, M., Vanpeteghem, D.: Additivity of minimal entropy output for a class of covariant channels. preprint, arXiv:quant-ph/0410195 (2004)
  10. 10.
    Friedland, S.: Matrices: Algebra, Analysis and Applications. World Scientific (2015)Google Scholar
  11. 11.
    Friedland, S., Gour, G., Roy, A.: Local extrema of entropy functions under tensor products. Quantum Inf. Comput. 11, 1028 (2011). eprint [arXiv:1105.5380] [math-ph]
  12. 12.
    Fukuda, M., King, C., Moser, D.K.: Comments on Hastings’ additivity counterexamples. Commun. Math. Phys. 296(1), 111–143 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gour, G., Friedland, S.: The minimum entropy output of a quantum channel is locally additive. IEEE Trans. Inf. Theory 59(1), 603 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gour, G., Wallach, N.R.: Entanglement of subspaces and error-correcting codes. Phys. Rev. A 76(4), 042309 (2007)ADSCrossRefGoogle Scholar
  15. 15.
    Hastings, M.B.: Superadditivity of communication capacity using entangled inputs. Nat. Phys. 5(4), 255–257 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hayden, P., Winter, A.: Counterexamples to the maximal \(p\)-norm multiplicativity conjecture for all \(p>1\). Commun. Math. Phys. 284(1), 263–280 (2008)ADSCrossRefMATHGoogle Scholar
  17. 17.
    Holevo, A.S.: The additivity problem in quantum information theory. In: Proceedings of the International Congress of Mathematicians (Madrid, 2006), vol. 3, pp. 999–1018 (2006)Google Scholar
  18. 18.
    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81(2), 865 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kato, T.: Perturbation theory for linear operators, vol. 132, Springer Science & Business Media (2013)Google Scholar
  20. 20.
    Kemp, T., Nourdin, I., Peccati, G., Speicher, R.: Wigner chaos and the fourth moment. Ann. Probab. 1577–1635 (2012)Google Scholar
  21. 21.
    King, C.: Additivity for unital qubit channels. J. Math. Phys. 43(10), 4641–4653 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    King, C.: Maximal \(p\)-norms of entanglement breaking channels. Quantum Inf. Comput. 3, 186–190 (2003)MathSciNetMATHGoogle Scholar
  23. 23.
    King, C., Ruskai, M.B.: Minimal entropy of states emerging from noisy quantum channels. IEEE Trans. Inf. Theory 47(1), 192–209 (2001)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Renner, R.: Security of quantum key distribution. Int. J. Quantum Inf. 6(01), 1–127 (2008)CrossRefMATHGoogle Scholar
  25. 25.
    Shirokov, M.E.: On the structure of optimal sets for a quantum channel. Probl. Inf. Transm. 42(4), 282–297 (2006)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Shor, P.W.: Additivity of the classical capacity of entanglement-breaking quantum channels. J. Math. Phys. 43(9), 4334–4340 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Shor, P.W.: Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246(3), 453–472 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Smith, G., Yard, J.: Quantum communication with zero-capacity channels. Science 321(5897), 1812–1815 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Turgut, S.: Catalytic transformations for bipartite pure states. J. Phys. A Math. Theor. 40(40), 12185 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Institute for Quantum Science and TechnologyUniversity of CalgaryCalgaryCanada
  2. 2.Department of MathematicsUniversity of CaliforniaSan DiegoUSA

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