Advertisement

Communications in Mathematical Physics

, Volume 346, Issue 1, pp 35–73 | Cite as

Entanglement Rates and the Stability of the Area Law for the Entanglement Entropy

  • Michaël Mariën
  • Koenraad M. R. Audenaert
  • Karel Van Acoleyen
  • Frank Verstraete
Article

Abstract

We prove a conjecture by Bravyi on an upper bound on entanglement rates of local Hamiltonians. We then use this bound to prove the stability of the area law for the entanglement entropy of quantum spin systems under adiabatic and quasi-adiabatic evolutions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambainis, A.: Quantum lower bounds by quantum arguments. In: Proceedings of the Thirty-second Annual ACM Symposium on Theory of Computing, ACM, pp. 636–643 (2000)Google Scholar
  2. 2.
    Audenaert K.M.: A sharp continuity estimate for the von Neumann entropy. J. Phys. A Math. Theor. 40(28), 8127 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Audenaert K.M.: Quantum skew divergence. J. Math. Phys. 55, 112202 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Audenaert, K.M., Kittaneh, F.: Problems and conjectures in matrix and operator inequalities (2012). preprint. arXiv:1201.5232
  5. 5.
    Avron J., Seiler R., Yaffe L.: Adiabatic theorems and applications to the quantum Hall effect. Commun. Math. Phys. 110(1), 33–49 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bañuls M.-C., Cirac J.I., Wolf M.M.: Entanglement in fermionic systems. Phys. Rev. A 76, 022311 (2007)ADSCrossRefGoogle Scholar
  7. 7.
    Bachmann S., Michalakis S., Nachtergaele B., Sims R.: Automorphic equivalence within gapped phases of quantum lattice systems. Commun. Math. Phys. 309(3), 835–871 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bennett C.H., Harrow A.W., Leung D.W., Smolin J.A.: On the capacities of bipartite Hamiltonians and unitary gates. Inf. Theory IEEE Trans. 49(8), 1895–1911 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics Vol. 2: Equilibrium States, Models in Quantum Statistical Mechanics. Springer, (1981)Google Scholar
  10. 10.
    Bravyi S.: Upper bounds on entangling rates of bipartite Hamiltonians. Phys. Rev. A 76, 052319 (2007)ADSCrossRefGoogle Scholar
  11. 11.
    Bravyi S., Hastings M.B., Michalakis S.: Topological quantum order: stability under local perturbations. J. Math. Phys. 51, 093512 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bravyi S., Hastings M.B., Verstraete F.: Lieb–Robinson bounds and the generation of correlations and topological quantum order. Phys. Rev. Lett. 97, 050401 (2006)ADSCrossRefGoogle Scholar
  13. 13.
    Chen X., Gu Z.-C., Wen X.-G.: Complete classification of one-dimensional gapped quantum phases in interacting spin systems. Phys. Rev. B 84, 235128 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    Childs A.M., Leung D.W., Verstraete F., Vidal G.: Asymptotic entanglement capacity of the Ising and anisotropic Heisenberg interactions. Quantum Inf. Comput. 01(3), 97–105 (2003)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Childs A.M., Leung D.W., Vidal G.: Reversible simulation of bipartite product Hamiltonians. Inf. Theory IEEE Trans. 50(6), 1189–1197 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cho, J.: Entanglement area law in thermodynamically gapped spin systems (2014). preprint. arXiv:1404.7616
  17. 17.
    Cubitt T.S., Verstraete F., Cirac J.I.: Entanglement flow in multipartite systems. Phys. Rev. A 71, 052308 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dennis E., Kitaev A., Landahl A., Preskill J.: Topological quantum memory. J. Math. Phys. 43(9), 4452–4505 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    DiVincenzo D.P., Horodecki M., Leung D.W., Smolin J.A., Terhal B.M.: Locking classical correlations in quantum states. Phys. Rev. Lett. 92, 067902 (2004)ADSCrossRefGoogle Scholar
  20. 20.
    Dür W., Vidal G., Cirac J.I., Linden N., Popescu S.: Entanglement capabilities of nonlocal Hamiltonians. Phys. Rev. Lett. 87, 137901 (2001)ADSCrossRefGoogle Scholar
  21. 21.
    Dziubanski J., Hernández E.: Band-limited wavelets with subexponential decay. Can. Math. Bull. 41(4), 398–403 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Eisert J., Cramer M., Plenio M.B.: Colloquium: Area laws for the entanglement entropy. Rev. Mod. Phys. 82, 277–306 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Evenbly G., Vidal G.: Scaling of entanglement entropy in the (branching) multiscale entanglement renormalization ansatz. Phys. Rev. B 89, 235113 (2014)ADSCrossRefGoogle Scholar
  24. 24.
    Fannes M.: A continuity property of the entropy density for spin lattice systems. Commun. Math. Phys. 31(4), 291–294 (1973)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Foong S.K., Kanno S.: Proof of Page’s conjecture on the average entropy of a subsystem. Phys. Rev. Lett. 72, 1148–1151 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Graham S., Vaaler J.D.: A class of extremal functions for the fourier transform. Trans. Am. Math. Soc. 265(1), 283–302 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Haah J.: Local stabilizer codes in three dimensions without string logical operators. Phys. Rev. A 83, 042330 (2011)ADSCrossRefGoogle Scholar
  28. 28.
    Haah J.: Bifurcation in entanglement renormalization group flow of a gapped spin model. Phys. Rev. B 89, 075119 (2014)ADSCrossRefGoogle Scholar
  29. 29.
    Hastings M.: Quasi-adiabatic continuation in gapped spin and fermion systems: Goldstone’s theorem and flux periodicity. J. Stat. Mech. Theory Exp. 2007(05), P05010 (2007)CrossRefGoogle Scholar
  30. 30.
    Hastings M.: Locality in quantum systems. Quantum Theory Small Large Scales 95, 171–212 (2010)zbMATHGoogle Scholar
  31. 31.
    Hastings, M.: Quasi-adiabatic continuation for disordered systems: applications to correlations, Lieb–Schultz–Mattis, and Hall conductance (2010). preprint. arXiv:1001.5280
  32. 32.
    Hastings, M., Michalakis, S.: Quantization of Hall conductance for interacting electrons on a torus. Commun. Math. Phys., 1–39 (2014)Google Scholar
  33. 33.
    Hastings M.B.: Lieb–Schultz–Mattis in higher dimensions. Phys. Rev. B 69, 104431 (2004)ADSCrossRefGoogle Scholar
  34. 34.
    Hastings M.B.: An area law for one-dimensional quantum systems. J. Stat. Mech. Theory Exp. 2007(08), P08024 (2007)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Hastings M.B., Koma T.: Spectral gap and exponential decay of correlations. Commun. Math. Phys. 265(3), 781–804 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Hastings M.B., Wen X.-G.: Quasiadiabatic continuation of quantum states: the stability of topological ground-state degeneracy and emergent gauge invariance. Phys. Rev. B 72, 045141 (2005)ADSCrossRefGoogle Scholar
  37. 37.
    Hayden P., Leung D.W., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265(1), 95–117 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Hutter A., Wehner S.: Almost all quantum states have low entropy rates for any coupling to the environment. Phys. Rev. Lett. 108, 070501 (2012)ADSCrossRefGoogle Scholar
  39. 39.
    Ingham A.: A note on Fourier transforms. J. Lond. Math. Soc. 1(1), 29–32 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Jordan, P., Wigner, E.: Über das Paulische Äquivalenzverbot. Z. Phys. 47, 631–651 (1928)ADSCrossRefzbMATHGoogle Scholar
  41. 41.
    Kitaev, A.: Anyons in an exactly solved model and beyond. Ann. Phys. 321(1), 2–111 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Kitaev A., Kong L.: Models for gapped boundaries and domain walls. Commun. Math. Phys. 313(2), 351–373 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Kittaneh F.: Inequalities for commutators of positive operators. J. Funct. Anal. 250(1), 132–143 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Kittaneh F.: Norm inequalities for commutators of self-adjoint operators. Integral Equ. Oper. Theory 62(1), 129–135 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    König R., Reichardt B.W., Vidal G.: Exact entanglement renormalization for string-net models. Phys. Rev. B 79, 195123 (2009)ADSCrossRefGoogle Scholar
  46. 46.
    Landau, Z., Vazirani, U., Vidick, T.: An efficient algorithm for finding the ground state of 1d gapped local Hamiltonians. In: Proceedings of the 5th Conference on Innovations in Theoretical Computer Science, ITCS ’14, pp. 301–302, New York, NY, USA, ACM (2014)Google Scholar
  47. 47.
    Levin M.: Protected edge modes without symmetry. Phys. Rev. X 3, 021009 (2013)Google Scholar
  48. 48.
    Lieb E.H., Robinson D.W.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28(3), 251–257 (1972)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Lieb E.H., Vershynina A.: Upper bounds on mixing rates. Quantum Inf. Comput. 13(11-12), 0986–0994 (2013)MathSciNetGoogle Scholar
  50. 50.
    Lin C.-H., Levin M.: Generalizations and limitations of string-net models. Phys. Rev. B 89, 195130 (2014)ADSCrossRefGoogle Scholar
  51. 51.
    Linden N., Popescu S., Smolin J.A.: Entanglement of superpositions. Phys. Rev. Lett. 97, 100502 (2006)ADSCrossRefzbMATHGoogle Scholar
  52. 52.
    Michalakis, S.: Stability of the area law for the entropy of entanglement (2012). preprint arXiv:1206.6900
  53. 53.
    Nachtergaele B., Ogata Y., Sims R.: Propagation of correlations in quantum lattice systems. J. Stat. Phys. 124(1), 1–13 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Nachtergaele B., Raz H., Schlein B., Sims R.: Lieb–Robinson bounds for harmonic and anharmonic lattice systems. Commun. Math. Phys. 286(3), 1073–1098 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Nachtergaele B., Schlein B., Sims R., Starr S., Zagrebnov V.: On the existence of the dynamics for anharmonic quantum oscillator systems. Rev. Math. Phys. 22(02), 207–231 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Nachtergaele B., Sims R.: Lieb-Robinson bounds and the exponential clustering theorem. Commun. Math. Phys. 265(1), 119–130 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Osborne T.J.: Simulating adiabatic evolution of gapped spin systems. Phys. Rev. A 75, 032321 (2007)ADSMathSciNetCrossRefGoogle Scholar
  58. 58.
    Page D.N.: Average entropy of a subsystem. Phys. Rev. Lett. 71, 1291–1294 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Schlosshauer M.: Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Mod. Phys. 76, 1267–1305 (2005)ADSCrossRefGoogle Scholar
  60. 60.
    Schuch N., Pérez-García D., Cirac I.: Classifying quantum phases using matrix product states and projected entangled pair states. Phys. Rev. B 84, 165139 (2011)ADSCrossRefGoogle Scholar
  61. 61.
    Shor P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493–R2496 (1995)ADSCrossRefGoogle Scholar
  62. 62.
    Swingle B., McGreevy J.: Renormalization group constructions of topological quantum liquids and beyond. Phys. Rev. B 93, 045127 (2016)ADSCrossRefGoogle Scholar
  63. 63.
    Vaaler J.D.: Some extremal functions in fourier analysis. Bull. Am. Math. Soc. 12(2), 183–216 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Van Acoleyen K., Mariën M., Verstraete F.: Entanglement rates and area laws. Phys. Rev. Lett. 111, 170501 (2013)ADSCrossRefGoogle Scholar
  65. 65.
    Verstraete F., Cirac J.I.: Matrix product states represent ground states faithfully. Phys. Rev. B 73, 094423 (2006)ADSCrossRefGoogle Scholar
  66. 66.
    Verstraete F., Murg V., Cirac J.I.: Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems. Adv. Phys. 57(2), 143–224 (2008)ADSCrossRefGoogle Scholar
  67. 67.
    Wang X., Sanders B.C.: Entanglement capability of a self-inverse Hamiltonian evolution. Phys. Rev. A 68, 014301 (2003)ADSCrossRefGoogle Scholar
  68. 68.
    Wen X.-G.: Topological orders and edge excitations in fractional quantum Hall states. Adv. Phys. 44(5), 405–473 (1995)ADSCrossRefGoogle Scholar
  69. 69.
    Zeng, B., Wen, X.-G.: Stochastic local transformations, emergence of unitarity, long-range entanglement, gapped quantum liquids, and topological order (2014). preprint. arXiv:1406.5090

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Michaël Mariën
    • 1
  • Koenraad M. R. Audenaert
    • 1
    • 2
  • Karel Van Acoleyen
    • 1
  • Frank Verstraete
    • 1
    • 3
  1. 1.Department of Physics and AstronomyGhent UniversityGhentBelgium
  2. 2.Mathematics DepartmentRoyal Holloway University of LondonEghamUnited Kingdom
  3. 3.Fakultät für PhysikUniversität WienWienAustria

Personalised recommendations