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Annales Henri Poincaré

, Volume 19, Issue 10, pp 2919–2953 | Cite as

The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have GaussianMaximizers

  • Giacomo De Palma
  • Dario Trevisan
  • Vittorio Giovannetti
Article

Abstract

We determine the \(p\rightarrow q\) norms of the Gaussian one-mode quantum-limited attenuator and amplifier and prove that they are achieved by Gaussian states, extending to noncommutative probability the seminal theorem “Gaussian kernels have only Gaussian maximizers” (Lieb in Invent Math 102(1):179–208, 1990). The quantum-limited attenuator and amplifier are the building blocks of quantum Gaussian channels, which play a key role in quantum communication theory since they model in the quantum regime the attenuation and the noise affecting any electromagnetic signal. Our result is crucial to prove the longstanding conjecture stating that Gaussian input states minimize the output entropy of one-mode phase-covariant quantum Gaussian channels for fixed input entropy. Our proof technique is based on a new noncommutative logarithmic Sobolev inequality, and it can be used to determine the \(p\rightarrow q\) norms of any quantum semigroup.

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References

  1. 1.
    Anderson, A., Halliwell, J.J.: Information-theoretic measure of uncertainty due to quantum and thermal fluctuations. Phys. Rev. D 48(6), 2753 (1993)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Audenaert, K.M.R.: A note on the p q norms of 2-positive maps. Linear Algeb. Its Appl. 430(4), 1436–1440 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barnett, S., Radmore, P.M.: Methods in Theoretical Quantum Optics. Oxford Series in Optical and Imaging Sciences. Clarendon Press, Oxford (2002)CrossRefzbMATHGoogle Scholar
  4. 4.
    Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization: Theory and Examples. CMS Books in Mathematics. Springer, New York (2013)Google Scholar
  5. 5.
    Braunstein, S.L., Van Loock, P.: Quantum information with continuous variables. Rev. Mod. Phys. 77(2), 513 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Carbone, R., Sasso, E.: Hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup. Probab. Theory Relat. Fields 140(3–4), 505–522 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Carlen, E.A., Maas, J.: An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker–Planck equation is gradient flow for the entropy. Commun. Math. Phys. 331(3), 887–926 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carlen, E.A., Maas, J.: Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance. J. Funct. Anal. 273(5), 1810–1869 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Caves, C.M., Drummond, P.D.: Quantum limits on bosonic communication rates. Rev. Mod. Phys. 66(2), 481 (1994)ADSCrossRefGoogle Scholar
  10. 10.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory. A Wiley-Interscience Publication. Wiley, Hoboken (2006)zbMATHGoogle Scholar
  11. 11.
    De Palma, G.: Gaussian Optimizers and Other Topics in Quantum Information. Ph.D. thesis, Scuola Normale Superiore, Pisa (Italy). Supervisor: Prof. Vittorio Giovannetti. arXiv:1710.09395 (2016)
  12. 12.
    De Palma, G., Mari, A., Giovannetti, V.: A generalization of the entropy power inequality to bosonic quantum systems. Nat. Photon. 8(12), 958–964 (2014)ADSCrossRefGoogle Scholar
  13. 13.
    De Palma, G., Mari, A., Lloyd, S., Giovannetti, V.: Multimode quantum entropy power inequality. Phys. Rev. A 91(3), 032320 (2015)ADSCrossRefGoogle Scholar
  14. 14.
    De Palma, G., Mari, A., Lloyd, S., Giovannetti, V.: Passive states as optimal inputs for single-jump lossy quantum channels. Phys. Rev. A 93(6), 062328 (2016)ADSCrossRefGoogle Scholar
  15. 15.
    De Palma, G., Trevisan, D., Giovannetti, V.: Passive states optimize the output of bosonic Gaussian quantum channels. IEEE Trans. Inf. Theory 62(5), 2895–2906 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    De Palma, G., Trevisan, D., Giovannetti, V.: Gaussian states minimize the output entropy of one-mode quantum Gaussian channels. Phys. Rev. Lett. 118, 160503 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    De Palma, G., Trevisan, D., Giovannetti, V.: Gaussian states minimize the output entropy of the one-mode quantum attenuator. IEEE Trans. Inf. Theory 63(1), 728–737 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    De Palma, G., Trevisan, D., Giovannetti, V., Ambrosio, L.: Gaussian optimizers for entropic inequalities in quantum information. arXiv:1803.02360 (2018)
  19. 19.
    Ferraro, A., Olivares, S., Paris, M.GA.: Gaussian states in continuous variable quantum information. arXiv:quant-ph/0503237 (2005)
  20. 20.
    Frank, R.L., Lieb, E.H.: Norms of quantum Gaussian multi-mode channels. J. Math. Phys. 58(6), 062204 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Garcia-Patron, R., Navarrete-Benlloch, C., Lloyd, S., Shapiro, J.H., Cerf, N.J.: Majorization theory approach to the Gaussian channel minimum entropy conjecture. Phys. Rev. Lett. 108(11), 110505 (2012)ADSCrossRefGoogle Scholar
  22. 22.
    Giovannetti, V., Semenovich Holevo, A., Mari, A.: Majorization and additivity for multimode bosonic Gaussian channels. Theor. Math. Phys. 182(2), 284–293 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Giovannetti, V., Holevo, A.S., García-Patrón, R.: A solution of Gaussian optimizer conjecture for quantum channels. Commun. Math. Phys. 334(3), 1553–1571 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Giovannetti, V., Holevo, A.S., Lloyd, S., Maccone, L.: Generalized minimal output entropy conjecture for one-mode Gaussian channels: definitions and some exact results. J. Phys. A Math. Theor. 43(41), 415305 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gorecki, J., Pusz, W.: Passive states for finite classical systems. Lett. Math. Phys. 4(6), 433–443 (1980)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Gross, L.: Logarithmic Sobolev inequalities and contractivity properties of semigroups. In: Dell’Antonio, G., Mosco, U. (eds.) Dirichlet Forms, pp. 54–88. Springer (1993)Google Scholar
  27. 27.
    Guha, S., Erkmen, B., Shapiro, J.H.: The entropy photon-number inequality and its consequences. In: Information Theory and Applications Workshop, 2008, pp. 128–130. IEEE (2008)Google Scholar
  28. 28.
    Guha, S., Shapiro, J.H.: Classical information capacity of the bosonic broadcast channel. In: IEEE International Symposium on Information Theory, 2007. ISIT 2007. pp. 1896–1900. IEEE (2007)Google Scholar
  29. 29.
    Guha, S., Shapiro, J.H., Erkmen, B.I.: Classical capacity of bosonic broadcast communication and a minimum output entropy conjecture. Phys. Rev. A 76(3), 032303 (2007)ADSCrossRefGoogle Scholar
  30. 30.
    Harremoës, P., Johnson, O., Kontoyiannis, I.: Thinning and the law of small numbers. In: IEEE International Symposium on Information Theory, 2007. ISIT 2007. pp. 1491–1495. IEEE (2007)Google Scholar
  31. 31.
    Harremoës, P., Johnson, O., Kontoyiannis, I.: Thinning, entropy, and the law of thin numbers. IEEE Trans. Inf. Theory 56(9), 4228–4244 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hartman, P.: Ordinary Differential Equations: Second Edition. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (2002)Google Scholar
  33. 33.
    Holevo, A.S.: Multiplicativity of p-norms of completely positive maps and the additivity problem in quantum information theory. Russ. Math. Surv. 61(2), 301 (2006)CrossRefzbMATHGoogle Scholar
  34. 34.
    Johnson, O., Yu, Y.: Monotonicity, thinning, and discrete versions of the entropy power inequality. IEEE Trans. Inf. Theory 56(11), 5387–5395 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    König, R., Smith, G.: The entropy power inequality for quantum systems. IEEE Trans. Inf. Theory 60(3), 1536–1548 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 481–492, University of California Press, Berkeley, CA (1951)Google Scholar
  37. 37.
    Lenard, A.: Thermodynamical proof of the Gibbs formula for elementary quantum systems. J. Stat. Phys. 19(6), 575–586 (1978)ADSCrossRefGoogle Scholar
  38. 38.
    Lieb, E.H.: Proof of an entropy conjecture of Wehrl. Commun. Math. Phys. 62(1), 35–41 (1978)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Lieb, E.H.: Gaussian kernels have only Gaussian maximizers. Invent. Math. 102(1), 179–208 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Lieb, E.H., Solovej, J.P.: Proof of an entropy conjecture for Bloch coherent spin states and its generalizations. Acta Math. 212(2), 379–398 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Mari, A., Giovannetti, V., Holevo, A.S.: Quantum state majorization at the output of bosonic Gaussian channels. Nat. Commun. 5, 3826 (2014)ADSCrossRefGoogle Scholar
  42. 42.
    Pusz, W., Woronowicz, S.L.: Passive states and KMS states for general quantum systems. Commun. Math. Phys. 58(3), 273–290 (1978)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    Qi, H., Wilde, M.M.: Capacities of quantum amplifier channels. Phys. Rev. A 95, 012339 (2017)ADSCrossRefGoogle Scholar
  44. 44.
    Qi, H., Wilde, M.M., Guha, S.: On the minimum output entropy of single-mode phase-insensitive Gaussian channels. arXiv:1607.05262 (2016)
  45. 45.
    Rényi, A.: A characterization of Poisson processes. Magyar Tud. Akad. Mat. Kutató Int. Közl 1, 519–527 (1956)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Schatten, R.: Norm Ideals of Completely Continuous Operators. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin (1960)CrossRefzbMATHGoogle Scholar
  47. 47.
    Semenovich Holevo, A.: Quantum Systems, Channels, Information: A Mathematical Introduction. De Gruyter Studies in Mathematical Physics. De Gruyter, Berlin (2013)Google Scholar
  48. 48.
    Semenovich Holevo, A.: Gaussian optimizers and the additivity problem in quantum information theory. Russ. Math. Surv. 70(2), 331 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Solomon Ivan, J., Kumar Sabapathy, K., Simon, R.: Operator-sum representation for bosonic Gaussian channels. Phys. Rev. A 84(4), 042311 (2011)ADSCrossRefGoogle Scholar
  50. 50.
    Serafini, A.: Quantum Continuous Variables: A Primer of Theoretical Methods. CRC Press, Boca Raton (2017)CrossRefzbMATHGoogle Scholar
  51. 51.
    Tomamichel, M.: Quantum Information Processing with Finite Resources: Mathematical Foundations. SpringerBriefs in Mathematical Physics. Springer, Berlin (2015)zbMATHGoogle Scholar
  52. 52.
    Weedbrook, C., Pirandola, S., Garcia-Patron, R., Cerf, N.J.: Timothy C Ralph, Jeffrey H Shapiro, and Seth Lloyd. Gaussian quantum information. Rev. Mod. Phys. 84(2), 621 (2012)ADSCrossRefGoogle Scholar
  53. 53.
    Wehrl, A.: On the relation between classical and quantum-mechanical entropy. Rep. Math. Phys. 16(3), 353–358 (1979)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Wilde, M.M.: Quantum Information Theory. Cambridge University Press, Cambridge (2017)CrossRefzbMATHGoogle Scholar
  55. 55.
    Yu, Y.: Monotonic convergence in an information-theoretic law of small numbers. IEEE Trans. Inf. Theory 55(12), 5412–5422 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Yu, Y., Johnson, O.: Concavity of entropy under thinning. In: IEEE International Symposium on Information Theory, 2009. ISIT 2009. pp. 144–148. IEEE (2009)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Giacomo De Palma
    • 1
    • 2
    • 3
  • Dario Trevisan
    • 4
  • Vittorio Giovannetti
    • 2
  1. 1.QMATH, Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark
  2. 2.NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNRPisaItaly
  3. 3.INFNPisaItaly
  4. 4.Università degli Studi di PisaPisaItaly

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