Rapid mixing implies exponential decay of correlations

Journal of Mathematical Physics

Volumn 54, Issue 10, 2013, Pages

  Kastoryano, Michael J ^a   Eisert, Jens ^a  

^a FREIE UNIVERSITÄT BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84886862717     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.4822481     Document Type: Article

References (41)

1. Hastings M.B. Koma T. Spectral gap and exponential decay of correlations. Commun. Math. Phys. 2006, 265:781. 10.1007/s00220-006-0030-4.
2. Nachtergaele B. Sims R. Lieb-Robinson bounds and the exponential clustering theorem. Commun. Math. Phys. 2006, 265:119. 10.1007/s00220-006-1556-1.
3. Eisert J. Cramer M. Plenio M.B. Area laws for the entanglement entropy. Rev. Mod. Phys. 2010, 82:277. 10.1103/RevModPhys.82.277.
4. Hastings M.B. An area law for one-dimensional quantum systems. J. Stat. Mech.: Theory Exp. 2007:P08024. 10.1088/1742-5468/2007/08/P08024.
5. Arad I. Kitaev A. Landau Z. Vazirani U. An area law and sub-exponential algorithm for 1D systems. e-print arXiv:1301.1162.
6. Schuch N. Wolf M.M. Verstraete F. Cirac J.I. Entropy scaling and simulability by matrix product states. Phys. Rev. Lett. 2008, 100:030504. 10.1103/PhysRevLett.100.030504.
7. Brandao F.G. S. L. Horodecki M. An area law for entanglement from exponential decay of correlations. Nature Physics 2013, 1745.
8. Bernigau H. Kastoryano M.J. Eisert J. Area laws for thermal free fermions. e-print arXiv:1301.5646.
9. Bravyi S. Hastings M.B. Michalakis S. Topological quantum order: Stability under local perturbations. J. Math. Phys. 2010, 51:093512. 10.1063/1.3490195.
10. Michalakis S. Pytel J. Stability of frustration-free Hamiltonians. e-print arXiv:1109.1588.
11. Bachmann S. Michalakis S. Nachtergaele B. Sims R. Automorphic equivalence within gapped phases of quantum lattice systems. Commun. Math. Phys. 2012, 309:835. 10.1007/s00220-011-1380-0.
12. Cubitt T.S. Lucia A. Michalakis S. Perez-Garcia D. Stability of local quantum dissipative systems. e-print arXiv:1303.4744.
13. Davies E.B. Generators of dynamical semi-groups. J. Funct. Anal. 1979, 34:421. 10.1016/0022-1236(79)90085-5.
14. Spohn H. Entropy production for quantum dynamical semi-groups. J. Math. Phys. 1978, 19:1227. 10.1063/1.523789.
15. Martinelli F. Lectures on Glauber dynamics for discrete spin models. Lectures on Probability Theory and Statistics 1999, in (Springer).
16. Dyer M. Sinclair A. Vigoda E. Weitz D. Mixing in time and space for lattice spin systems: A combinatorial view. Random Struct. Algorithms 2004, 24:461. 10.1002/rsa.20004.
17. Guionnet A. Zegarlinski B. Lectures on logarithmic Sobolev inequalities. Lect. Notes Math. 2003, 1801:1. 10.1007/978-3-540-36107-7_1.
18. Yoshida N. The equivalence of the log-Sobolev inequality and a mixing condition for unbounded spin systems on the lattice. Ann. I.H.P. Probab. Stat. 2001, 37(2):223. 10.1016/S0246-0203(00)01066-9.
19. Kastoryano M.J. Temme K. Quantum logarithmic Sobolev inequalities and rapid mixing. J. Math. Phys. 2013, 54:052202. 10.1063/1.4804995.
20. Temme K. Kastoryano M.J. Ruskai M.B. Wolf M.M. Verstraete F. The χ2 divergence and mixing times of quantum Markov chains. J. Stat. Mech.: Theory Exp. 2010:122201.
21. Poulin D. Lieb-Robinson bound and locality for general Markovian quantum dynamics. Phys. Rev. Lett. 2010, 104:190401. 10.1103/PhysRevLett.104.190401.
22. Diehl S. Micheli A. Kantian A. Kraus B. Büchler H.P. Zoller P. Quantum states and phases in driven open quantum systems with cold atoms. Nat. Phys. 2008, 4:878. 10.1038/nphys1073.
23. Verstraete F. Wolf M.M. Cirac J.I. Quantum computation, quantum state engineering, and quantum phase transitions driven by dissipation. Nat. Phys. 2009, 5:633. 10.1038/nphys1342.
24. Eisert J. Prosen T. Noise-driven criticality. e-print arXiv:1012.5013.
25. Prosen T. Third quantization: A general method to solve master equations for quadratic open Fermi systems. New J. Phys. 2008, 10:043026. 10.1088/1367-2630/10/4/043026.
26. Bardyn C.-E. Baranov M.A. Rico E. Imamoglu A. Zoller P. Diehl S. Topology by dissipation. e-print arXiv:1201.2112.
27. Kliesch M. Barthel T. Gogolin C. Kastoryano M. Eisert J. A dissipative quantum Church-Turing theorem. Phys. Rev. Lett. 2011, 107:120501. 10.1103/PhysRevLett.107.120501.
28. Barthel T. Kliesch M. Quasi-locality and efficient simulation of Markovian quantum dynamics. Phys. Rev. Lett. 2012, 108:230504. 10.1103/PhysRevLett.108.230504.
29. Nachtergaele B. Vershynina A. Zagrebnov V.A. Lieb-Robinson bounds and existence of the thermodynamic limit for a class of irreversible quantum dynamics. AMS Contemp. Math. 2011, 552:161. 10.1090/conm/552.
30. Alicki R. Fannes M. Horodecki M. On thermalization in Kitaev's 2D model. J. Phys. A: Math. Theor. 2009, 42:065303. 10.1088/1751-8113/42/6/065303.
31. Kastoryano M.J. Wolf M.M. Eisert J. Precisely timing dissipative quantum information processing. Phys. Rev. Lett. 2013, 110:110501. 10.1103/PhysRevLett.110.110501.
32. Sanz M. Perez-Garcia D. Wolf M.M. Cirac J.I. A quantum version of Wielandts inequality. IEEE Trans. Inf. Theory 2010, 56:4668. 10.1109/TIT.2010.2054552.
33. Majewski W.A. Streater R.F. Detailed balance and quantum dynamical maps. J. Phys. A 1998, 31:7981. 10.1088/0305-4470/31/39/013.
34. Temme K. Osborne T.J. Vollbrecht K.G. Poulin D. Verstraete F. Quantum Metropolis sampling. Nature 2011, 471:87. 10.1038/nature09770.
35. Levin D.A. Peres Y. Wilmer E.L. Markov Chains and Mixing Times 2009, and (AMS, Providence, RI).
36. Bravyi S. Hastings M.B. Verstraete F. Lieb-Robinson bounds and the generation of correlations and topological quantum order. Phys. Rev. Lett. 2006, 97:050401. 10.1103/PhysRevLett.97.050401.
37. Hastings M.B. Topological order at non-zero temperature. Phys. Rev. Lett. 2011, 107:210501. 10.1103/PhysRevLett.107.210501.
38. Alicki R. Fannes M. Dilations of quantum dynamical semi-groups with classical Brownian motion. Commun. Math. Phys. 1987, 108:353. 10.1007/BF01212314.
39. Wolf M.M. Verstraete F. Hastings M.B. Cirac J.I. Area laws in quantum systems: mutual information and correlations. Phys. Rev. Lett. 2008, 100:070502. 10.1103/PhysRevLett.100.070502.
40. Verstraete F. Garcia-Ripoll J.J. Cirac J.I. Matrix product density operators: Simulation of finite-T and dissipative systems. Phys. Rev. Lett. 2004, 93:207204. 10.1103/PhysRevLett.93.207204.
41. Zwolak M. Vidal G. Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algorithm. Phys. Rev. Lett. 2004, 93:207205. 10.1103/PhysRevLett.93.207205.