Publication: Rapid thermalization of spin chain commuting Hamiltonians
Loading...
Official URL
Full text at PDC
Publication Date
2022
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
We prove that spin chains weakly coupled to a large heat bath thermalize rapidly at any temperature for finite-range, translation-invariant commuting Hamiltonians, reaching equilibrium in a time which scales logarithmically with the system size. From a physical point of view, our result rigorously establishes the absence of dissipative phase transitions for Davies evolutions over translation-invariant spin chains. The result has also implications in the understanding of Symmetry Protected Topological phases for open quantum systems.
Description
UCM subjects
Unesco subjects
Keywords
Citation
[1] K. Temme, T. J. Osborne, K. G. Vollbrecht, D. Poulin,
and F. Verstraete. Quantum metropolis sampling. Na-
ture, 471:87–90, 2011.
[2] K. Huang. Statistical Mechanics. John Wiley & Sons,
1987.
[3] R. D. Somma, S. Boixo, H. Barnum, and K. Emanuel.
Quantum simulations of classical annealing processes.
Phys. Rev. Lett., 101:130504, 2008.
[4] F. G. S. L. Brandão and K. M. Svore. Quantum speedups
for solving semidefinite programs. In 2017 IEEE 58th
Annual Symposium on Foundations of Computer Science
(FOCS), pages 415–426. IEEE, 2017.
[5] M. Kieferová and N. Wiebe. Tomography and generative
training with quantum Boltzmann machines. Phys. Rev.
A, 96(6):062327, 2017.
[6] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost,
N. Wiebe, and S. Lloyd. Quantum machine learning.
Nature, 549:195–202, 2017.
[7] T. S. Cubitt, A. Lucia, S. Michalakis, and D. Pérez-
García. Stability of local quantum dissipative systems.
Commun. Math. Phys., 337:1275–1315, 2015.
[8] A. Lucia, T. S. Cubitt, S. Michalakis, and D. Pérez-
García. Rapid mixing and stability of quantum dissipative
systems. Phys. Rev. A, 91(4), April 2015.
[9] R. J. Glauber. Time-dependent statistics of the Ising
model. J. Math. Phys., 4(2):294–307, 1963.
[10] R. A. Holley and D.W. Stroock. Uniform and L2 convergence
in one dimensional stochastic Ising models. Com-
mun. Math. Phys., 123(1):85–93, 1989.
[11] R. Holley. Rapid convergence to equilibrium in one dimensional
stochastic Ising models. Ann. Prob., pages
72–89, 1985.
[12] B. Zegarlinski. Log-Sobolev inequalities for infinite onedimensional
lattice systems. Commun. Math. Phys, 133
(1):147–162, 1990.
[13] M. J. Kastoryano and F. G. S. L. Brandao. Quantum
Gibbs samplers: The commuting case. Commun. Math.
Phys., 344(3):915–957, 2016.
[14] R. Alicki, M. Fannes, and M. Horodecki. On thermalization
in Kitaev’s 2D model. J. Phys. A: Math. Theor., 42
(6):065303, 2009.
[15] A. Kómár, O. Landon-Cardinal, and K. Temme. Necessity
of an energy barrier for self-correction of abelian
quantum doubles. Phys. Rev. A, 93(5):052337, 2016.
[16] A. Lucia, D. Pérez-García, and A. Pérez-Hernández.
Thermalization in Kitaev’s quantum double models
via Tensor Network techniques. arXiv preprint
arXiv:2107.01628, 2021.
[17] L. Gross. Hypercontractivity and logarithmic sobolev
inequalities for the clifford-dirichlet form. Duke Math.
J., 42(3), September 1975.
[18] M. J. Kastoryano and K. Temme. Quantum logarithmic
Sobolev inequalities and rapid mixing. J. Math. Phys.,
54(5):052202, May 2013.
[19] K. Temme, F. Pastawski, and M. J. Kastoryano. Hypercontractivity
of quasi-free quantum semigroups. J. Phys.
A: Math. Theor., 47(40):405303, September 2014.
[20] I. Bardet, Á. Capel, A. Lucia, D. Pérez-García, and
C. Rouzé. On the modified logarithmic Sobolev inequality
for the heat-bath dynamics for 1D systems. J. Math.
Phys., 62(6):061901, 2021.
[21] Á Capel, C. Rouzé, and D. Stilck França. The modified
logarithmic Sobolev inequality for quantum spin systems:
classical and commuting nearest neighbour interactions.
arXiv preprint, arXiv:2009.11817, 2020.
[22] Á. Capel, A. Lucia, and D. Pérez-García. Quantum
conditional relative entropy and quasi-factorization of
the relative entropy. J. Phys. A: Math. Theor., 51(48):
484001, 2018.
[23] S. Beigi, N. Datta, and C. Rouzé. Quantum reverse
hypercontractivity: Its tensorization and application to
strong converses. Commun. Math. Phys., 376(2):753–794,
May 2020.
[24] P. Werner, K. Völker, M. Troyer, and S. Chakravarty.
Phase Diagram and Critical Exponents of a Dissipative
Ising Spin Chain in a Transverse Magnetic Field. Phys.
Rev. Lett., 94:047201, 2005.
[25] L. Capriotti, A. Cuccoli, A. Fubini, V. Tognetti, and
R. Vaia. Dissipation-Driven Phase Transition in Two-
Dimensional Josephson Arrays. Phys. Rev. Lett., 94:
157001, 2005.
[26] S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler,
and P. Zoller. Quantum states and phases in driven open
quantum systems with cold atoms. Nat. Phys., 4:878–
883, 2008.
[27] S. Morrison and A. S. Parkins. Dissipation-driven quantum
phase transitions in collective spin systems. J. Phys.
B: At. Mol. Opt. Phys., 41:195502, 2008.
[28] E.M. Kessler, G. Giedke, A. Imamoglu, S. F. Yelin, M. D.
Lukin, and J. I. Cirac. Dissipative phase transitions in a
central spin system. Phys. Rev. A, 86:012116, 2012.
[29] B. Horstmann, J. I. Cirac, and G. Giedke. Noise-driven
dynamics and phase transitions in fermionic systems.
Phys. Rev. A, 87:012108, 2013.
[30] F. Verstraete, M. M. Wolf, and J. I. Cirac. Quantum
computation and quantum-state engineering driven by
dissipation. Nat. Phys., 5:633–636, 2009.
[31] X.-G. Wen. Quantum orders and symmetric spin liquids.
Phys. Rev. B, 65(16):165113, 2002.
[32] Z.-C. Gu and X.-G. Wen. Tensor-entanglement-filtering
renormalization approach and symmetry-protected topological
order. Phys. Rev. B, 80(15):155131, 2009.
[33] X. Chen, Z.-C. Gu, and X.-G. Wen. Classification of
gapped symmetric phases in one-dimensional spin systems.
Phys. Rev. B, 83(3):035107, 2011.
[34] S. Diehl, E. Rico, M. A. Baranov, and P. Zoller. Topology
by dissipation in atomic quantum wires. Nat. Phys., 7
(12):971–977, October 2011.
[35] D. Rainis and D. Loss. Majorana qubit decoherence by
quasiparticle poisoning. Phys. Rev. B, 85(17):174533,
2012.
[36] O. Viyuela, A. Rivas, and M. A. Martín-Delgado.
Thermal instability of protected end states in a onedimensional
topological insulator. Phys. Rev. B, 86(15),
October 2012.
[37] C.-E. Bardyn, M. A. Baranov, C. V. Kraus, E. Rico,
A. İmamoğlu, P. Zoller, and S. Diehl. Topology by dissipation.
New J. Phys., 15(8):085001, August 2013.
[38] O. Viyuela, A. Rivas, and M. A. Martín-Delgado.
Symmetry-protected topological phases at finite temperature.
2D Mater., 2(3):034006, 2015.
[39] S. Roberts, B. Yoshida, A. Kubica, and S. D. Bartlett.
Symmetry-protected topological order at nonzero temperature.
Phys. Rev. A, 96(2), August 2017.
[40] M. McGinley and N. R. Cooper. Interacting symmetryprotected
topological phases out of equilibrium. Phys.
Rev. Res., 1(3):033204, 2019.
[41] A. Coser and D. Pérez-García. Classification of phases
for mixed states via fast dissipative evolution. Quantum,
3:174, 2019.
[42] M. McGinley and N. R. Cooper. Fragility of time-reversal
symmetry protected topological phases. Nat. Phys., 16
(12):1181–1183, 2020.
[43] C. de Groot, A. Turzillo, and N. Schuch. Symmetry protected
topological order in open quantum systems. arXiv
preprint, arXiv:2112.04483, 2021.
[44] A. Altland, M. Fleischhauer, and S. Diehl. Symmetry
Classes of Open Fermionic Quantum Matter. Phys. Rev.
X, 11(2), May 2021.
[45] O. Viyuela, A. Rivas, S. Gasparinetti, A. Wallraff, S. Filipp,
and M. A. Martín-Delgado. Observation of topological
uhlmann phases with superconducting qubits. NPJ
Quantum Inf., 4(1):1–6, 2018.
[46] H. J. Briegel and R. Raussendorf. Persistent entanglement
in arrays of interacting particles. Phys. Rev. Lett.,
86(5):910–913, January 2001.
[47] R. Raussendorf and H. J. Briegel. A One-Way Quantum
Computer. Phys. Rev. Lett., 86(22):5188–5191, May
2001.
[48] R. Raussendorf and H. J. Briegel. Computational model
underlying the one-way quantum computer. Quantum
Inf. Comput., 2(6):443–486, October 2002.
[49] W. Son, L. Amico, R. Fazio, A. Hamma, S. Pascazio, and
V. Vedral. Quantum phase transition between cluster and
antiferromagnetic states. EPL (Europhysics Letters), 95
(5):50001, August 2011.
[50] B. Buča and T. Prosen. A note on symmetry reductions
of the lindblad equation: transport in constrained open
spin chains. New J. Phys., 14(7):073007, 2012.
[51] V. V. Albert and L. Jiang. Symmetries and conserved
quantities in Lindblad master equations. Phys. Rev. A,
89(2), February 2014.
[52] C. Palazuelos and T. Vidick. Survey on nonlocal games
and operator space theory. J. Math. Phys., 57:015220,
2016.
[53] E. B. Davies. Markovian master equations. Commun.
Math. Phys., 39(2):91–110, June 1974.
[54] I. Bardet, Á. Capel, L. Gao, A. Lucia, D. Pérez-García,
and C. Rouzé. Entropy decay for Davies semigroups of 1D
quantum spin chains. arXiv preprint, arXiv:2112.00601,
2021.
[55] M. M. Wolf. Quantum channels & operations: Guided
tour. Lecture notes available at http://www-m5. ma. tum.
de/foswiki/pub M, 5, 2012.
[56] H. Araki. Gibbs states of a one dimensional quantum
lattice. Commun. Math. Phys., 14(2):120–157, 1969.
[57] A. Bluhm, Á. Capel, and A. Pérez-Hernández. Exponential
decay of mutual information for Gibbs states of local
hamiltonians. arXiv preprint, arXiv:2104.04419, 2021.
[58] L. Gao and C. Rouzé. Complete entropic inequalities
for quantum Markov chains. arXiv preprint
arXiv:2102.04146, 2021.
[59] S. Bravyi, M. B. Hastings, and S. Michalakis. Topological
quantum order: stability under local perturbations. J.
Math. Phys., 51(9):093512, 2010.
[60] F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa.
Entanglement spectrum of a topological phase in one dimension.
Phys. Rev. B, 81(6):064439, 2010.
[61] J. Haegeman, D. Pérez-García, I. Cirac, and N. Schuch.
Order parameter for symmetry-protected phases in one
dimension. Phys. Rev. Lett., 109(5):050402, 2012.
[62] N. Schuch, D. Pérez-García, and I. Cirac. Classifying
quantum phases using matrix product states and projected
entangled pair states. Phys. Rev. B, 84(16), October
2011.
[63] F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa.
Symmetry protection of topological phases in
one-dimensional quantum spin systems. Phys. Rev. B,
85(7), February 2012.
[64] L. Fidkowski and A. Kitaev. Topological phases of
fermions in one dimension. Phys. Rev. B, 83(7):075103,
2011.
[65] M. Sanz, M. M. Wolf, D. Pérez-García, and J. I. Cirac.
Matrix product states: Symmetries and two-body Hamiltonians.
Phys. Rev. A, 79(4), April 2009.
[66] J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete.
Matrix product density operators: Renormalization
fixed points and boundary theories. Ann. Phys.,
378:100–149, March 2017.
[67] G. De las Cuevas, J. I. Cirac, N. Schuch, and D. Pérez-
García. Irreducible forms of matrix product states: Theory
and applications. J. Math. Phys., 58(12):121901, December
2017.
[68] Y. Ogata. Classification of gapped ground state
phases in quantum spin systems. arXiv preprint,
arXiv:2110.04675, 2021.
[69] I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete.
Matrix product states and projected entangled
pair states: Concepts, symmetries, and theorems. arXiv
preprint arXiv:2011.12127, 2020.
[70] R. Carbone, E. Sasso, and V. Umanità. Decoherence
for Quantum Markov Semi-Groups on Matrix Algebras.
Ann. Henri Poincaré, 14(4):681–697, August 2012.
[71] F. Cesi. Quasi-factorization of the entropy and logarithmic
Sobolev inequalities for Gibbs random fields. Probab.
Theory Relat. Fields, 120(4):569–584, 2001.
[72] P. Dai Pra, A. M. Paganoni, and G. Posta. Entropy
inequalities for unbounded spin systems. Ann. Probab.,
30(4):1959–1976, 10 2002.
[73] Á. Capel. Quantum Logarithmic Sobolev Inequalities for
Quantum Many-Body Systems: An approach via Quasi-
Factorization of the Relative Entropy. Ph.D. thesis at
Universidad Autónoma de Madrid, 2019.
[74] I. Bardet, Á. Capel, and C. Rouzé. Approximate tensorization
of the relative entropy for noncommuting conditional
expectations. Ann. Henri Poincaré, 23:101–140,
2022.
[75] N. LaRacuente. Quasi-factorization and multiplicative
comparison of subalgebra-relative entropy. arXiv
preprint, arXiv:1912.00983, 2019.
[76] M. Junge and J. Parcet. Mixed-norm inequalities and
operator space Lp embedding theory. American Mathematical
Soc., 2010.
[77] D. Aharonov, I. Arad, Z. Landau, and U. Vazirani. The
detectability lemma and quantum gap amplification. In
Proceedings of the forty-first annual ACM symposium on
Theory of computing, pages 417–426, 2009.
[78] D. Aharonov, I. Arad, Z. Landau, and U. Vazirani.
Quantum Hamiltonian complexity and the detectability
lemma. arXiv preprint, arXiv:1011.3445, 2010.
[79] I. Bardet and C. Rouzé. Hypercontractivity and logarithmic
sobolev inequality for non-primitive quantum
markov semigroups and estimation of decoherence rates.
arXiv preprint, arXiv:1803.05379, 2018.
[80] L. Gao, M. Junge, and N. LaRacuente. Fisher information
and logarithmic sobolev inequality for matrix-valued
functions. In Ann. Henri Poincaré, volume 21, pages
3409–3478. Springer, 2020.
[81] G. Pisier. Non-commutative vector valued Lp-spaces and
completely p-summing maps. Société mathématique de
France, 1998.