Approximate Reversal of Quantum Gaussian Dynamics

Authors Ludovico Lami, Siddhartha Das, Mark M. Wilde

Thumbnail PDF


  • Filesize: 0.5 MB
  • 18 pages

Document Identifiers

Author Details

Ludovico Lami
Siddhartha Das
Mark M. Wilde

Cite AsGet BibTex

Ludovico Lami, Siddhartha Das, and Mark M. Wilde. Approximate Reversal of Quantum Gaussian Dynamics. In 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 73, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Recently, there has been focus on determining the conditions under which the data processing inequality for quantum relative entropy is satisfied with approximate equality. The solution of the exact equality case is due to Petz, who showed that the quantum relative entropy between two quantum states stays the same after the action of a quantum channel if and only if there is a reversal channel that recovers the original states after the channel acts. Furthermore, this reversal channel can be constructed explicitly and is now called the Petz recovery map. Recent developments have shown that a variation of the Petz recovery map works well for recovery in the case of approximate equality of the data processing inequality. Our main contribution here is a proof that bosonic Gaussian states and channels possess a particular closure property, namely, that the Petz recovery map associated to a bosonic Gaussian state \sigma and a bosonic Gaussian channel N is itself a bosonic Gaussian channel. We furthermore give an explicit construction of the Petz recovery map in this case, in terms of the mean vector and covariance matrix of the state \sigma and the Gaussian specification of the channel N.
  • Gaussian dynamics
  • Petz recovery map


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Gerardo Adesso, Sammy Ragy, and Antony R. Lee. Continuous variable quantum information: Gaussian states and beyond. Open Systems and Information Dynamics, 21(01-02):1440001, June 2014. arXiv:1401.4679. Google Scholar
  2. Alvaro M. Alhambra, Stephanie Wehner, Mark M. Wilde, and Mischa P. Woods. Work and reversibility in quantum thermodynamics, June 2015. arXiv:1506.08145. Google Scholar
  3. Alvaro M. Alhambra and Mischa P. Woods. Dynamical maps, quantum detailed balance and Petz recovery map, September 2016. arXiv:1609.07496. Google Scholar
  4. Hans-A. Bachor and Timothy C. Ralph. A Guide to Experiments in Quantum Optics. Wiley, second edition, March 2004. Google Scholar
  5. Leonardo Banchi, Samuel L. Braunstein, and Stefano Pirandola. Quantum fidelity for arbitrary Gaussian states. Physical Review Letters, 115(26):260501, December 2015. arXiv:1507.01941. URL:
  6. J. Bergh and Jorgen Löfström. Interpolation Spaces. Springer-Verlag Berlin Heidelberg, 1976. Google Scholar
  7. Mario Berta, Kaushik Seshadreesan, and Mark M. Wilde. Rényi generalizations of the conditional quantum mutual information. Journal of Mathematical Physics, 56(2):022205, February 2015. arXiv:1403.6102. Google Scholar
  8. Mario Berta, Kaushik P. Seshadreesan, and Mark M. Wilde. Rényi generalizations of quantum information measures. Physical Review A, 91(2):022333, February 2015. arXiv:1502.07977. URL:
  9. Fernando G. S. L. Brandao, Matthias Christandl, and Jon Yard. Faithful squashed entanglement. Communications in Mathematical Physics, 306(3):805-830, September 2011. arXiv:1010.1750. URL:
  10. Fernando G. S. L. Brandao, Aram W. Harrow, Jonathan Oppenheim, and Sergii Strelchuk. Quantum conditional mutual information, reconstructed states, and state redistribution. Physical Review Letters, 115(5):050501, July 2014. arXiv:1411.4921. Google Scholar
  11. Francesco Buscemi, Siddhartha Das, and Mark M. Wilde. Approximate reversibility in the context of entropy gain, information gain, and complete positivity. Physical Review A, 93(6):062314, June 2016. arXiv:1601.01207. URL:
  12. Filippo Caruso, Jens Eisert, Vittorio Giovannetti, and Alexander S. Holevo. Multi-mode bosonic Gaussian channels. New Journal of Physics, 10:083030, August 2008. arXiv:0804.0511. Google Scholar
  13. Xiao-yu Chen. Gaussian relative entropy of entanglement. Physical Review A, 71(6):062320, June 2005. arXiv:quant-ph/0402109. URL:
  14. Matthias Christandl and Andreas Winter. "Squashed entanglement" - an additive entanglement measure. Journal of Mathematical Physics, 45(3):829-840, March 2004. arXiv:quant-ph/0308088. Google Scholar
  15. Igor Devetak and Jon Yard. Exact cost of redistributing multipartite quantum states. Physical Review Letters, 100(23):230501, June 2008. URL:
  16. Frederic Dupuis and Mark M. Wilde. Swiveled Rényi entropies. Quantum Information Processing, 15(3):1309-1345, March 2016. arXiv:1506.00981. Google Scholar
  17. Edward G. Effros. A matrix convexity approach to some celebrated quantum inequalities. Proceedings of the National Academy of Sciences of the United States of America, 106(4):1006-1008, January 2009. arXiv:0802.1234. Google Scholar
  18. Omar Fawzi and Renato Renner. Quantum conditional mutual information and approximate Markov chains. Communications in Mathematical Physics, 340(2):575-611, December 2015. arXiv:1410.0664. Google Scholar
  19. Marco G. Genoni, Ludovico Lami, and Alessio Serafini. Conditional and unconditional Gaussian quantum dynamics. Contemporary Physics, 57(3):331-349, January 2016. arXiv:1607.02619. URL:
  20. Christopher Gerry and Peter Knight. Introductory Quantum Optics. Cambridge University Press, November 2004. Google Scholar
  21. Vittorio Giovannetti, Raul Garcia-Patron, Nicolas J. Cerf, and Alexander S. Holevo. Ultimate classical communication rates of quantum optical channels. Nature Photonics, 8:796-800, September 2014. arXiv:1312.6225. Google Scholar
  22. Vittorio Giovannetti, Saikat Guha, Seth Lloyd, Lorenzo Maccone, Jeffrey H. Shapiro, and Horace P. Yuen. Classical capacity of the lossy bosonic channel: The exact solution. Physical Review Letters, 92(2):027902, January 2004. arXiv:quant-ph/0308012. URL:
  23. Vittorio Giovannetti, Alexander S. Holevo, and Raul Garcia-Patron. A solution of Gaussian optimizer conjecture for quantum channels. Communications in Mathematical Physics, 334(3):1553-1571, March 2015. arXiv:1312.2251. Google Scholar
  24. Patrick Hayden, Richard Jozsa, Denes Petz, and Andreas Winter. Structure of states which satisfy strong subadditivity of quantum entropy with equality. Communications in Mathematical Physics, 246(2):359-374, April 2003. arXiv:quant-ph/0304007. Google Scholar
  25. Isidore Isaac Hirschman. A convexity theorem for certain groups of transformations. Journal d'Analyse Mathématique, 2(2):209-218, December 1952. Google Scholar
  26. Oscar Lanford III and Derek W. Robinson. Mean entropy of states in quantum-statistical mechanics. Journal of Mathematical Physics, 9(7):1120-1125, July 1968. Google Scholar
  27. Marius Junge, Renato Renner, David Sutter, Mark M. Wilde, and Andreas Winter. Universal recovery from a decrease of quantum relative entropy, September 2015. arXiv:1509.07127. Google Scholar
  28. Isaac H. Kim. Operator extension of strong subadditivity of entropy. Journal of Mathematical Physics, 53(12):122204, December 2012. arXiv:1210.5190. Google Scholar
  29. Isaac H. Kim. Application of conditional independence to gapped quantum many-body systems, 2013. Google Scholar
  30. Ludovico Lami, Siddhartha Das, and Mark M. Wilde. Approximate reversal of quantum Gaussian dynamics, 2017. arXiv:1702.04737. Google Scholar
  31. Matthew S. Leifer and Robert W. Spekkens. Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference. Physical Review A, 88(5):052130, November 2013. arXiv:1107.5849. URL:
  32. Marius Lemm and Mark M. Wilde. Information-theoretic limitations on approximate quantum cloning and broadcasting, August 2016. arXiv:1608.07569. Google Scholar
  33. Ke Li and Andreas Winter. Squashed entanglement, k-extendibility, quantum Markov chains, and recovery maps, October 2014. arXiv:1410.4184. Google Scholar
  34. Elliott H. Lieb. Convex trace functions and the Wigner-Yanase-Dyson conjecture. Advances in Mathematics, 11(3):267-288, December 1973. Google Scholar
  35. Elliott H. Lieb and Mary Beth Ruskai. A fundamental property of quantum-mechanical entropy. Physical Review Letters, 30(10):434-436, March 1973. Google Scholar
  36. Elliott H. Lieb and Mary Beth Ruskai. Proof of the strong subadditivity of quantum mechanical entropy. Journal of Mathematical Physics, 14(12):1938-1941, December 1973. Google Scholar
  37. Göran Lindblad. Expectations and entropy inequalities for finite quantum systems. Communications in Mathematical Physics, 39(2):111-119, June 1974. Google Scholar
  38. Göran Lindblad. Completely positive maps and entropy inequalities. Communications in Mathematical Physics, 40(2):147-151, June 1975. URL:
  39. Iman Marvian and Seth Lloyd. From clocks to cloners: Catalytic transformations under covariant operations and recoverability, August 2016. arXiv:1608.07325. Google Scholar
  40. William Matthews, Stephanie Wehner, and Andreas Winter. Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding. Communications in Mathematical Physics, 291(3):813-843, November 2009. arXiv:0810.2327. Google Scholar
  41. Milán Mosonyi. Entropy, Information and Structure of Composite Quantum States. PhD thesis, Katholieke Universiteit Leuven, 2005. Available at 41 thesisbook9.pdf. Google Scholar
  42. Milán Mosonyi and Dénes Petz. Structure of sufficient quantum coarse-grainings. Letters in Mathematical Physics, 68(1):19-30, April 2004. arXiv:quant-ph/0312221. URL:
  43. Michael A. Nielsen and Denés Petz. A simple proof of the strong subadditivity inequality. Quantum Information and Computation, 5(6):507-513, September 2005. arXiv:quant-ph/0408130. Google Scholar
  44. Masanori Ohya and Denes Petz. Quantum Entropy and Its Use. Springer-Verlag, 1993. Google Scholar
  45. Giacomo De Palma, Andrea Mari, Vittorio Giovannetti, and Alexander S. Holevo. Normal form decomposition for Gaussian-to-Gaussian superoperators. Journal of Mathematical Physics, 56(5):052202, May 2015. arXiv:1502.01870. Google Scholar
  46. Giacomo De Palma, Dario Trevisan, and Vittorio Giovannetti. Gaussian states minimize the output entropy of one-mode quantum Gaussian channels, October 2016. arXiv:1610.09970. Google Scholar
  47. Giacomo De Palma, Dario Trevisan, and Vittorio Giovannetti. One-mode quantum-limited Gaussian channels have Gaussian maximizers, October 2016. arXiv:1610.09967. Google Scholar
  48. Giacomo De Palma, Dario Trevisan, and Vittorio Giovannetti. Gaussian states minimize the output entropy of the one-mode quantum attenuator. IEEE Transactions on Information Theory, 63(1):728-737, January 2017. arXiv:1605.00441. Google Scholar
  49. Gh.-S. Paraoanu and Horia Scutaru. Fidelity for multimode thermal squeezed states. Physical Review A, 61(2):022306, January 2000. arXiv:quant-ph/9907068. URL:
  50. Denes Petz. Sufficient subalgebras and the relative entropy of states of a von Neumann algebra. Communications in Mathematical Physics, 105(1):123-131, 1986. Google Scholar
  51. Denes Petz. Sufficiency of channels over von Neumann algebras. Quarterly Journal of Mathematics, 39(1):97-108, 1988. Google Scholar
  52. Denes Petz. Monotonicity of quantum relative entropy revisited. Reviews in Mathematical Physics, 15(01):79-91, March 2003. arXiv:quant-ph/0209053. Google Scholar
  53. Stefano Pirandola, Riccardo Laurenza, Carlo Ottaviani, and Leonardo Banchi. Fundamental limits of repeaterless quantum communications, September 2016. arXiv:1510.08863v6. Google Scholar
  54. Haoyu Qi and Mark M. Wilde. Capacities of quantum amplifier channels. Physical Review A, 95(1):012339, January 2017. arXiv:1605.04922. Google Scholar
  55. Derek W. Robinson and David Ruelle. Mean entropy of states in classical statistical mechanics. Communications in Mathematical Physics, 5(4):288-300, August 1967. Google Scholar
  56. Mary Beth Ruskai. Inequalities for quantum entropy: a review with conditions for equality. Journal of Mathematical Physics, 43:4358-4375, 2002. erratum 46, 019901 (2005); arXiv:quant-ph/0205064. Google Scholar
  57. Stefan Scheel and Dirk-Gunnar Welsch. Entanglement generation and degradation by passive optical devices. Physical Review A, 64(6):063811, November 2001. arXiv:quant-ph/0103167. URL:
  58. Alessio Serafini. Quantum Continuous Variables. CRC Press, 2017. Google Scholar
  59. Kaushik P. Seshadreesan, Mario Berta, and Mark M. Wilde. Rényi squashed entanglement, discord, and relative entropy differences. Journal of Physics A: Mathematical and Theoretical, 48(39):395303, September 2015. arXiv:1410.1443. Google Scholar
  60. R. Simon, N. Mukunda, and Biswadeb Dutta. Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms. Physical Review A, 49(3):1567-1583, March 1994. URL:
  61. David Sutter, Mario Berta, and Marco Tomamichel. Multivariate trace inequalities. Communications in Mathematical Physics, 352(1):37-58, May 2017. arXiv:1604.03023. Google Scholar
  62. David Sutter, Omar Fawzi, and Renato Renner. Universal recovery map for approximate Markov chains. Proceedings of the Royal Society A, 472(2186), February 2016. arXiv:1504.07251. Google Scholar
  63. David Sutter, Marco Tomamichel, and Aram W. Harrow. Strengthened monotonicity of relative entropy via pinched Petz recovery map. IEEE Transactions on Information Theory, 62(5):2907-2913, May 2016. arXiv:1507.00303. Google Scholar
  64. Robert R. Tucci. Quantum entanglement and conditional information transmission, September 1999. arXiv:quant-ph/9909041. Google Scholar
  65. Robert R. Tucci. Entanglement of distillation and conditional mutual information, February 2002. arXiv:quant-ph/0202144. Google Scholar
  66. Armin Uhlmann. Endlich dimensionale dichtmatrizen, ii. Wiss. Z. Karl-Marx-University Leipzig, 22(Jg. H. 2.):139, 1973. Google Scholar
  67. Armin Uhlmann. The "transition probability" in the state space of a *-algebra. Reports on Mathematical Physics, 9(2):273-279, 1976. Google Scholar
  68. Hisaharu Umegaki. Conditional expectations in an operator algebra IV (entropy and information). Kodai Mathematical Seminar Reports, 14(2):59-85, 1962. Google Scholar
  69. Xiang-Bin Wang, Tohya Hiroshima, Akihisa Tomita, and Masahito Hayashi. Quantum information with Gaussian states. Physics Reports, 448(1–-4):1-111, August 2007. arXiv:0801.4604. URL:
  70. Xiang-Bin Wang, L. C. Kwek, and C. H. Oh. Bures fidelity for diagonalizable quadratic Hamiltonians in multi-mode systems. Journal of Physics A: Mathematical and General, 33(27):4925, July 2000. URL:
  71. Mark M. Wilde. Recoverability in quantum information theory. Proceedings of the Royal Society A, 471(2182):20150338, October 2015. arXiv:1505.04661. Google Scholar
  72. Mark M. Wilde. Monotonicity of p-norms of multiple operators via unitary swivels, October 2016. arXiv:1610.01262. Google Scholar
  73. Mark M. Wilde, Patrick Hayden, and Saikat Guha. Information trade-offs for optical quantum communication. Physical Review Letters, 108(14):140501, April 2012. arXiv:1105.0119. Google Scholar
  74. Mark M. Wilde and Haoyu Qi. Energy-constrained private and quantum capacities of quantum channels, September 2016. arXiv:1609.01997. Google Scholar
  75. Andreas Winter and Ke Li. A stronger subadditivity relation? with applications to squashed entanglement, sharability and separability. notes available online at∼csajw/stronger_subadditivity.pdf , see also, 2012. Google Scholar
  76. Jon Yard and Igor Devetak. Optimal quantum source coding with quantum side information at the encoder and decoder. IEEE Transactions on Information Theory, 55(11):5339-5351, November 2009. arXiv:0706.2907. Google Scholar