Communications in Mathematical Physics

, Volume 333, Issue 2, pp 761–798 | Cite as

Exponential Decay of Correlations Implies Area Law

  • Fernando G. S. L. BrandãoEmail author
  • Michał Horodecki


We prove that a finite correlation length, i.e., exponential decay of correlations, implies an area law for the entanglement entropy of quantum states defined on a line. The entropy bound is exponential in the correlation length of the state, thus reproducing as a particular case Hastings’s proof of an area law for groundstates of 1D gapped Hamiltonians.

As a consequence, we show that 1D quantum states with exponential decay of correlations have an efficient classical approximate description as a matrix product state of polynomial bond dimension, thus giving an equivalence between injective matrix product states and states with a finite correlation length.

The result can be seen as a rigorous justification, in one dimension, of the intuition that states with exponential decay of correlations, usually associated with non-critical phases of matter, are simple to describe. It also has implications for quantum computing: it shows that unless a pure state quantum computation involves states with long-range correlations, decaying at most algebraically with the distance, it can be efficiently simulated classically.

The proof relies on several previous tools from quantum information theory—including entanglement distillation protocols achieving the hashing bound, properties of single-shot smooth entropies, and the quantum substate theorem—and also on some newly developed ones. In particular we derive a new bound on correlations established by local random measurements, and we give a generalization to the max-entropy of a result of Hastings concerning the saturation of mutual information in multiparticle systems. The proof can also be interpreted as providing a limitation on the phenomenon of data hiding in quantum states.


Quantum State Mutual Information Exponential Decay Entanglement Entropy Reduce Density Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Araki H., Hepp K., Ruelle D.: Asymptotic behaviour of Wightman functions. Helv. Phys. Acta 35, 164 (1962)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Fredenhagen K.: A remark on the cluster theorem. Commun. Math. Phys. 97, 461 (1985)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Nachtergaele B., Sims R.: Lieb-Robinson bounds and the exponential clustering theorem. Commun. Math. Phys. 265, 119 (2006)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Hastings M.B.: Lieb-Schultz-Mattis in higher dimensions. Phys. Rev. B 69, 104431 (2004)ADSCrossRefGoogle Scholar
  5. 5.
    Hastings M.B.: Locality in quantum and markov dynamics on lattices and networks. Phys. Rev. Lett. 93, 140402 (2004)ADSCrossRefGoogle Scholar
  6. 6.
    M.B. Hastings: Decay of correlations in fermi systems at non-zero temperature. Phys. Rev. Lett. 93, 126402 (2004)ADSCrossRefGoogle Scholar
  7. 7.
    Hastings M.B., Koma T.: Spectral gap and exponential decay of correlations. Commun. Math. Phys. 265, 781 (2006)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Aharonov D., Arad I., Landau Z., Vazirani U.: The detectability lemma and its applications to quantum Hamiltonian complexity. New J. Phys. 13, 113043 (2011)ADSCrossRefGoogle Scholar
  9. 9.
    Horodecki R., Horodecki P., Horodecki M., Horodecki K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Eisert J., Cramer M., Plenio M.B.: Colloquium: area laws for the entanglement entropy. Rev. Mod. Phys. 82, 277 (2010)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Bennett C.H., Bernstein H.J., Popescu S., Schumacher B.: Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046 (1996)ADSCrossRefGoogle Scholar
  12. 12.
    Bekenstein J.D.: Black holes and entropy. Phys. Rev. D 7, 233 (1973)ADSCrossRefMathSciNetGoogle Scholar
  13. 13.
    Vidal G., Latorre J.I., Rico E., Kitaev A.: Entanglement in quantum critical phenomena. Phys. Rev. Lett. 90, 227902 (2003)ADSCrossRefGoogle Scholar
  14. 14.
    Calabrese, P., Cardy, J.: Entanglement entropy and quantum field theory, J. Stat. Mech. P06002 (2004)Google Scholar
  15. 15.
    Audenaert K., Eisert J., Plenio M.B., Werner R.F.: Entanglement properties of the harmonic chain. Phys. Rev. A 66, 042327 (2002)ADSCrossRefGoogle Scholar
  16. 16.
    Plenio M.B., Eisert J., Dreissig J., Cramer M.: Entropy, entanglement, and area: analytical results for harmonic lattice systems. Phys. Rev. Lett. 94, 060503 (2005)ADSCrossRefMathSciNetGoogle Scholar
  17. 17.
    Wolf M.M.: Violation of the entropic area law for Fermions. Phys. Rev. Lett. 96, 010404 (2006)ADSCrossRefGoogle Scholar
  18. 18.
    Aharonov D., Gottesman D., Irani S., Kempe J.: The power of quantum systems on a line. Commun. Math. Phys. 287, 41 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Irani S.: Ground state entanglement in one dimensional translationally invariant quantum systems. J. Math. Phys. 51, 022101 (2010)ADSCrossRefMathSciNetGoogle Scholar
  20. 20.
    Gottesman, D., Irani, S.: The quantum and classical complexity of translationally invariant tiling and hamiltonian problems. FOCS ’ 09Google Scholar
  21. 21.
    Hastings, M.: An area law for one dimensional quantum systems. JSTAT, P08024 (2007)Google Scholar
  22. 22.
    Lieb E.H., Robinson D.W.: The nite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251 (1972)ADSCrossRefMathSciNetGoogle Scholar
  23. 23.
    Arad, I., Kitaev, A., Landau, Z., Vazirani, U.: In preparation, (2012)Google Scholar
  24. 24.
    Arad I., Landau Z., Vazirani U.: An improved 1D area law for frustration-free systems. Phys. Rev. B 85, 195145 (2012)ADSCrossRefGoogle Scholar
  25. 25.
    Gottesman D., Hastings M.B.: Entanglement vs. gap for one-dimensional spin systems. New J. Phys. 12, 025002 (2010)ADSCrossRefMathSciNetGoogle Scholar
  26. 26.
    Verstraete F., Cirac J.I.: Matrix product states represent ground states faithfully. Phys. Rev. B 73, 094423 (2006)ADSCrossRefGoogle Scholar
  27. 27.
    Hayden P., Leung D.W., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265, 95 (2006)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    DiVincenzo D.P., Leung D.W., Terhal B.M.: Quantum data hiding. IEEE Trans. Inf. Theory 48, 580 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Brandão, F.G.S.L., Christandl, M., Yard, J.: A quasipolynomial-time algorithm for the quantum separability problem. In: Proceedings of ACM symposium on theory of computation (STOC’11)Google Scholar
  30. 30.
    Brandão F.G.S.L., Christandl M., Yard J.: Faithful squashed entanglement. Commun. Math. Phys. 306, 805 (2011)ADSCrossRefzbMATHGoogle Scholar
  31. 31.
    Hastings M.B.: Random unitaries give quantum expanders. Phys. Rev. A 76, 032315 (2007)ADSCrossRefMathSciNetGoogle Scholar
  32. 32.
    Hastings M.B.: Entropy and entanglement in quantum ground states. Phys. Rev. B 76, 035114 (2007)ADSCrossRefMathSciNetGoogle Scholar
  33. 33.
    Wolf M.M., Verstraete F., Hastings M.B., Cirac J.I.: Area laws in quantum systems: mutual information and correlations. Phys. Rev. Lett. 100, 070502 (2008)ADSCrossRefMathSciNetGoogle Scholar
  34. 34.
    Masanes L.l.: An area law for the entropy of low-energy states. Phys. Rev. A 80, 052104 (2009)ADSCrossRefGoogle Scholar
  35. 35.
    Osborne T.: Hamiltonian complexity. Rep. Prog. Phys. 75, 022001 (2012)ADSCrossRefMathSciNetGoogle Scholar
  36. 36.
    Brandão F.G.S.L., Horodecki M: Entanglement area law from exponential decay of correlations. Nat. Phys. 9, 721–726 (2013)CrossRefGoogle Scholar
  37. 37.
    Tomamichel M., Colbeck R., Renner R.: Duality between smooth min- and max-entropies. IEEE Trans. Inf. Theory 56, 4674 (2010)CrossRefMathSciNetGoogle Scholar
  38. 38.
    Renner, R.: Ph.D. thesis ETH Zurich (2005)Google Scholar
  39. 39.
    Fannes M., Nachtergaele B., Werner R.F.: Finitely correlated states on quantum spin chains. Commun. Math. Phys. 144, 443 (1992)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Ostlund, Y.S., Rommer, S.: Phys. Rev. Lett. 75, 3537 (1995)Google Scholar
  41. 41.
    Vidal G.: Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91, 147902 (2003)ADSCrossRefGoogle Scholar
  42. 42.
    Perez-Garcia D., Verstraete F., Wolf M.M., Cirac J.I.: Matrix product state representations. Q. Inf. Comp. 7, 401 (2007)zbMATHMathSciNetGoogle Scholar
  43. 43.
    White S.R.: Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863 (1992)ADSCrossRefGoogle Scholar
  44. 44.
    Abeyesinghe A., Devetak I., Hayden P., Winter A.: The mother of all protocols: restructuring quantum information’s family tree. J. Proc. R. Soc. A 465, 2537 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Ben-Aroya A., Ta-Shma A.: Quantum expanders: motivation and construction. Theory Comput. 6, 47 (2010)CrossRefMathSciNetGoogle Scholar
  46. 46.
    Gottesman, D.: The Heisenberg representation of quantum computers. arXiv:quant-ph/9807006 (1998)
  47. 47.
    Jozsa R., Linden N.: On the role of entanglement in quantum computational speed-up. Proc. R. Soc. Lond. A 459(2036), 2011–2032 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    Valiant L.G.: Quantum circuits that can be simulated classically in polynomial time. SIAM J. Comput. 31(4), 1229 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  49. 49.
    DiVincenzo D., Terhal B.: Classical simulation of noninteracting-fermion quantum circuits. Phys. Rev. A 65, 032325 (2002)ADSCrossRefGoogle Scholar
  50. 50.
    Markov I., Shi Y.: Simulating quantum computation by contracting tensor networks. SIAM J. Comp. 38, 963 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    Vanden Nest M.: Simulating quantum computers with probabilistic methods. Quant. Inf. Comp. 11, 784 (2011)MathSciNetGoogle Scholar
  52. 52.
    Arad I., Landau Z.: Quantum computation and the evaluation of tensor networks. Siam J. Comput. 39(7), 3089–3121 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  53. 53.
    Hastings, M.B.: Notes on some questions in mathematical physics and quantum information. arXiv:1404.4327
  54. 54.
    Devetak I., Devetak I., Devetak I.: Distillation of secret key and entanglement from quantum states. Proc. R. Soc. Lond. A 461, 207 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  55. 55.
    Horodecki M., Oppenheim J., Winter A.: Partial quantum information. Nature 436, 673 (2005)ADSCrossRefGoogle Scholar
  56. 56.
    Horodecki M., Oppenheim J., Winter A.: Quantum state merging and negative information. Commun. Math. Phys. 269, 107 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  57. 57.
    Araki H., Lieb E.H. (1970) Entropy inequalities. Commun. Math. Phys. 18:160Google Scholar
  58. 58.
    Dupuis F., Berta M., Wullschleger J., Renner R.: One-shot decoupling. Commun. Math. Phys. 328, 251–284 (2014)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  59. 59.
    Datta N., Hsieh M.-H.: The apex of the family tree of protocols: optimal rates and resource inequalities. New J. Phys. 13, 093042 (2011)ADSCrossRefGoogle Scholar
  60. 60.
    Berta M., Christandl M., Renner R.: The quantum reverse shannon theorem based on one-shot information theory. Commun. Math. Phys. 306, 579 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  61. 61.
    Jain, R., Radhakrishnan, J., Sen, P.: A theorem about relative entropy of quantum states with an application to privacy in quantum communication. arXiv:0705.2437
  62. 62.
    Jain, R., Nayak A.: Short proofs of the quantum substate theorem. IEEE Trans. Inf. Theory 58, no.6 (2012)Google Scholar
  63. 63.
    Tomamichel M., Colbeck R., Renner R.: A fully quantum asymptotic equipartition property. IEEE Trans. Inf. Theory 55, 5840 (2009)CrossRefMathSciNetGoogle Scholar
  64. 64.
    Tomamichel, M.: A framework for non-asymptotic quantum information theory. PhD Thesis, ETH Zürich 2011. arXiv:1203.2142
  65. 65.
    Ledoux, M.: The concentration of measure phenomenon, volume 89 of Mathematical Surveys and Monographs. American Mathematical Society (2001)Google Scholar
  66. 66.
    Fannes, M.: Commun. Math. Phys. 31, 291 (1973)Google Scholar
  67. 67.
    Audenaert K.M.R.: A sharp fannes-type inequality for the von neumann entropy. J. Phys. A 40, 8127 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  68. 68.
    Datta N.: Min- and max- relative entropies and a new entanglement monotone. IEEE Trans. Inf. Theory 55, 2816 (2009)CrossRefGoogle Scholar
  69. 69.
    Berta, M., Furrer, F., Scholz, V.B.: The smooth entropy formalism on von neumann algebras. arXiv:1107.5460
  70. 70.
    Bennett C.H., Devetak I, Harrow A.W., Shor P.W., Winter A.: Quantum reverse shannon theorem. IEEE Trans. Inf. Theory 60(5), 2926–2959 (2014)CrossRefMathSciNetGoogle Scholar
  71. 71.
    Hayden P., Horodecki M., Yard J., Winter A.: A decoupling approach to the quantum capacity. Open Syst. Inf. Dyn. 15, 7 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  72. 72.
    del Rio L., Aberg J., Renner R., Dahlsten O., Vedral V.: The thermodynamic meaning of negative entropy. Nature 474, 61 (2011)CrossRefGoogle Scholar
  73. 73.
    Horodecki M., Oppenheim J.: Fundamental limitations for quantum and nano thermodynamics. Nat. Commun. 4, 2059 (2013)ADSCrossRefGoogle Scholar
  74. 74.
    Aharonov D., Arad I., Irani S.: An efficient algorithm for approximating 1d ground states. Phys. Rev. A 82, 012315 (2010)ADSCrossRefGoogle Scholar
  75. 75.
    Schuch N., Cirac J.I.: Matrix product state and mean field solutions for one-dimensional systems can be found efficiently. Phys. Rev. A 82, 012314 (2010)ADSCrossRefMathSciNetGoogle Scholar
  76. 76.
    Renner, R., Wolf, S.: Smooth renyi entropy and applications. In: Proceedings international symposium on information theory, ISIT 2004 (2004)Google Scholar
  77. 77.
    Brandão F.G.S.L., Horodecki M.: On Hastings’ counterexamples to the minimum output entropy additivity conjecture. Open Syst. Inf. Dyn. 17, 31 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  78. 78.
    Hastings, M.B.: Private communication (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Fernando G. S. L. Brandão
    • 1
    Email author
  • Michał Horodecki
    • 2
  1. 1.Institute for Theoretical PhysicsETH ZürichSwitzerland
  2. 2.Institute for Theoretical Physics and AstrophysicsUniversity of GdańskGdańskPoland

Personalised recommendations