Classification of phases for mixed states via fast dissipative evolution

Andrea Coser and David Pérez-García

Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain.
ICMAT, C/ Nicolás Cabrera, Campus de Cantoblanco, 28049 Madrid, Spain.

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Abstract

We propose the following definition of topological quantum phases valid for mixed states: two states are in the same phase if there exists a time independent, fast and local Lindbladian evolution driving one state into the other. The underlying idea, motivated by [1], is that it takes time to create new topological correlations, even with the use of dissipation.
We show that it is a good definition in the following sense: (1) It divides the set of states into equivalent classes and it establishes a partial order between those according to their level of ``topological complexity''. (2) It provides a path between any two states belonging to the same phase where observables behave smoothly.
We then focus on pure states to relate the new definition in this particular case with the usual definition for quantum phases of closed systems in terms of the existence of a gapped path of Hamiltonians connecting both states in the corresponding ground state path. We show first that if two pure states are in the same phase in the Hamiltonian sense, they are also in the same phase in the Lindbladian sense considered here.
We then turn to analyse the reverse implication, where we point out a very different behaviour in the case of symmetry protected topological (SPT) phases in 1D. Whereas at the Hamiltonian level, phases are known to be classified with the second cohomology group of the symmetry group, we show that symmetry cannot give any protection in 1D in the Lindbladian sense: there is only one SPT phase in 1D independently of the symmetry group.
We finish analysing the case of 2D topological quantum systems. There we expect that different topological phases in the Hamiltonian sense remain different in the Lindbladian sense. We show this formally only for the $\mathbb{Z}_n$ quantum double models $D(\mathbb{Z}_n)$. Concretely, we prove that, if $m$ is a divisor of $n$, there cannot exist any fast local Lindbladian connecting a ground state of $D(\mathbb{Z}_m)$ with one of $D(\mathbb{Z}_n)$, making rigorous the initial intuition that it takes long time to create those correlations present in the $\mathbb{Z}_n$ case that do not exist in the $\mathbb{Z}_m$ case and that, hence, the $\mathbb{Z}_n$ phase is strictly more complex in the Lindbladian case than the $\mathbb{Z}_m$ phase. We conjecture that such Lindbladian does exist in the opposite direction since Lindbladians can destroy correlations.

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► References

[1] Robert König and Fernando Pastawski ``Generating topological order: no speedup by dissipation'' Phys. Rev. B 90, 045101 (2013).
https:/​/​doi.org/​10.1103/​PhysRevB.90.045101
arXiv:1310.1037

[2] Jay D. Sau, Roman M. Lutchyn, Sumanta Tewari, and S. Das Sarma, ``A generic new platform for topological quantum computation using semiconductor heterostructures'' Phys. Rev. Lett. 104, 040502 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.104.040502
arXiv:0907.2239

[3] Di Xiao, Wenguang Zhu, Ying Ran, Naoto Nagaosa, and Satoshi Okamoto, ``Interface engineering of quantum Hall effects in digital transition metal oxide heterostructures'' Nat. Commun. 2, 596 (2011).
https:/​/​doi.org/​10.1038/​ncomms1602
arXiv:1106.4296

[4] N Goldman, J C Budich, and P Zoller, ``Topological quantum matter with ultracold gases in optical lattices'' Nat. Phys. 12, 639–645 (2016).
https:/​/​doi.org/​10.1038/​nphys3803
arXiv:1607.03902

[5] Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, ``Non-Abelian anyons and topological quantum computation'' Rev. Mod. Phys. 80, 1083–1159 (2008).
https:/​/​doi.org/​10.1103/​RevModPhys.80.1083
arXiv:0707.1889

[6] M. Z. Hasanand C. L. Kane ``Colloquium : Topological insulators'' Rev. Mod. Phys. 82, 3045–3067 (2010).
https:/​/​doi.org/​10.1103/​RevModPhys.82.3045
arXiv:1002.3895

[7] Xiao-Liang Qiand Shou-Cheng Zhang ``Topological insulators and superconductors'' Rev. Mod. Phys. 83, 1057–1110 (2011).
https:/​/​doi.org/​10.1103/​RevModPhys.83.1057
arXiv:1008.2026

[8] Xiao-Gang Wen ``Colloquium : Zoo of quantum-topological phases of matter'' Rev. Mod. Phys. 89, 041004 (2017).
https:/​/​doi.org/​10.1103/​RevModPhys.89.041004
arXiv:1610.03911

[9] Bruno Nachtergaele, Yoshiko Ogata, and Robert Sims, ``Propagation of Correlations in Quantum Lattice Systems'' J. Stat. Phys. 124, 1–13 (2006).
https:/​/​doi.org/​10.1007/​s10955-006-9143-6
arXiv:0603064

[10] Ola Bratteliand Derek William Robinson ``Operator Algebras and Quantum Statistical Mechanics: Volume 1: C*-and W*-Algebras. Symmetry Groups. Decomposition of States'' Springer Science & Business Media (2012).

[11] Claudio Castelnovoand Claudio Chamon ``Entanglement and topological entropy of the toric code at finite temperature'' Phys. Rev. B 76, 184442 (2007).
https:/​/​doi.org/​10.1103/​PhysRevB.76.184442
arXiv:0704.3616

[12] Claudio Castelnovoand Claudio Chamon ``Topological order in a three-dimensional toric code at finite temperature'' Phys. Rev. B 78, 155120 (2008).
https:/​/​doi.org/​10.1103/​PhysRevB.78.155120
arXiv:0804.3591

[13] S. Iblisdir, D. Pérez-García, M. Aguado, and J. Pachos, ``Scaling law for topologically ordered systems at finite temperature'' Phys. Rev. B 79, 134303 (2009).
https:/​/​doi.org/​10.1103/​PhysRevB.79.134303
arXiv:0806.1853

[14] Matthew B. Hastings ``Topological Order at Nonzero Temperature'' Phys. Rev. Lett. 107, 210501 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.107.210501
arXiv:1106.6026

[15] Sebastian Diehl, Enrique Rico, Mikhail A. Baranov, and Peter Zoller, ``Topology by dissipation in atomic quantum wires'' Nat. Phys. 7, 971–977 (2011).
https:/​/​doi.org/​10.1038/​nphys2106
arXiv:1105.5947

[16] C.-E. Bardyn, M. A. Baranov, E. Rico, A. İmamoğlu, P. Zoller, and S. Diehl, ``Majorana Modes in Driven-Dissipative Atomic Superfluids with a Zero Chern Number'' Phys. Rev. Lett. 109, 130402 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.109.130402
arXiv:1201.2112

[17] 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, 085001 (2013).
https:/​/​doi.org/​10.1088/​1367-2630/​15/​8/​085001
arXiv:1302.5135

[18] A. Rivas, O. Viyuela, and M. A. Martin-Delgado, ``Density-matrix Chern insulators: Finite-temperature generalization of topological insulators'' Phys. Rev. B 88, 155141 (2013).
https:/​/​doi.org/​10.1103/​PhysRevB.88.155141
arXiv:1301.4872v1

[19] O. Viyuela, A. Rivas, and M. A. Martin-Delgado, ``Uhlmann Phase as a Topological Measure for One-Dimensional Fermion Systems'' Phys. Rev. Lett. 112, 130401 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.112.130401
arXiv:1309.1174

[20] Fabian Grusdt ``Topological order of mixed states in correlated quantum many-body systems'' Phys. Rev. B 95, 075106 (2017).
https:/​/​doi.org/​10.1103/​PhysRevB.95.075106
arXiv:1609.02432

[21] Tobias J. Osborne ``Simulating adiabatic evolution of gapped spin systems'' Phys. Rev. A 75, 032321 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.75.032321
arXiv:0601019

[22] Yichen Huangand Xie Chen ``Quantum circuit complexity of one-dimensional topological phases'' Phys. Rev. B 91, 195143 (2015).
https:/​/​doi.org/​10.1103/​PhysRevB.91.195143
arXiv:1401.3820

[23] Tobias J. Osborne ``Efficient Approximation of the Dynamics of One-Dimensional Quantum Spin Systems'' Phys. Rev. Lett. 97, 157202 (2006).
https:/​/​doi.org/​10.1103/​PhysRevLett.97.157202
arXiv:0508031

[24] Jeongwan Haah, Matthew Hastings, Robin Kothari, and Guang Hao Low, ``Quantum Algorithm for Simulating Real Time Evolution of Lattice Hamiltonians'' 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) 350–360 (2018).
https:/​/​doi.org/​10.1109/​FOCS.2018.00041
arXiv:1801.03922

[25] Oleg Szehr, David Reeb, and Michael M. Wolf, ``Spectral Convergence Bounds for Classical and Quantum Markov Processes'' Commun. Math. Phys. 333, 565–595 (2015).
https:/​/​doi.org/​10.1007/​s00220-014-2188-5
arXiv:1301.4827

[26] David Aldousand Persi Diaconis ``The Asymmetric One-Dimensional Constrained Ising Model'' J. Stat. Phys. 107, 945–975 (2002).
https:/​/​doi.org/​10.1023/​A:1015170205728
arXiv:0110023

[27] Shirshendu Ganguly, Eyal Lubetzky, and Fabio Martinelli, ``Cutoff for the East Process'' Commun. Math. Phys. 335, 1287–1322 (2015).
https:/​/​doi.org/​10.1007/​s00220-015-2316-x
arXiv:1312.7863

[28] Michael J. Kastoryano, David Reeb, and Michael M. Wolf, ``A cutoff phenomenon for quantum Markov chains'' J. Phys. A 45, 075307 (2012).
https:/​/​doi.org/​10.1088/​1751-8113/​45/​7/​075307
arXiv:1111.2123

[29] M. J. Kastoryano, M. M. Wolf, and J. Eisert, ``Precisely Timing Dissipative Quantum Information Processing'' Phys. Rev. Lett. 110, 110501 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.110.110501
arXiv:1205.0985

[30] Toby S. Cubitt, Angelo Lucia, Spyridon Michalakis, and David Pérez-García, ``Stability of Local Quantum Dissipative Systems'' Commun. Math. Phys. 337, 1275–1315 (2015).
https:/​/​doi.org/​10.1007/​s00220-015-2355-3
arXiv:1303.4744

[31] Fernando G. S. L. Brandão, Toby S. Cubitt, Angelo Lucia, Spyridon Michalakis, and David Pérez-García, ``Area law for fixed points of rapidly mixing dissipative quantum systems'' J. Math. Phys. 56, 102202 (2015).
https:/​/​doi.org/​10.1063/​1.4932612
arXiv:1505.02776

[32] Sven Bachmann, Spyridon Michalakis, Bruno Nachtergaele, and Robert Sims, ``Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems'' Commun. Math. Phys. 309, 835–871 (2012).
https:/​/​doi.org/​10.1007/​s00220-011-1380-0
arXiv:1102.0842

[33] Bruno Nachtergaele, Anna Vershynina, and Valentin Zagrebnov, ``Lieb-Robinson bounds and existence of the thermodynamic limit for a class of irreversible quantum dynamics'' AMS Contemporary Mathematics 552, 161–175 (2011).
https:/​/​doi.org/​10.1090/​conm/​552/​10916
arXiv:1103.1122
http:/​/​www.ams.org/​conm/​552/​

[34] Angelo Lucia, Toby S. Cubitt, Spyridon Michalakis, and David Pérez-García, ``Rapid mixing and stability of quantum dissipative systems'' Phys. Rev. A 91, 040302 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.040302
arXiv:1409.7809

[35] O. Viyuela, A. Rivas, and M. A. Martin-Delgado, ``Generalized Toric Codes Coupled to Thermal Baths'' New J. Phys. 14, 033044 (2011).
https:/​/​doi.org/​10.1088/​1367-2630/​14/​3/​033044
arXiv:1112.1017

[36] O. Viyuela, A. Rivas, and M. A. Martin-Delgado, ``Thermal instability of protected end states in a one-dimensional topological insulator'' Phys. Rev. B 86, 155140 (2012).
https:/​/​doi.org/​10.1103/​PhysRevB.86.155140
arXiv:1207.2198

[37] Andreas P Schnyder, Shinsei Ryu, Akira Furusaki, and Andreas W W Ludwig, ``Classification of topological insulators and superconductors in three spatial dimensions'' Phys. Rev. B 78, 195125 (2008).
https:/​/​doi.org/​10.1103/​PhysRevB.78.195125
arXiv:0803.2786

[38] M. B. Hastings ``Lieb-Schultz-Mattis in higher dimensions'' Phys. Rev. B 69, 104431 (2004).
https:/​/​doi.org/​10.1103/​PhysRevB.69.104431
arXiv:0305505

[39] Markus Heyl ``Dynamical quantum phase transitions: a review'' Rep. Prog. Phys. 81, 054001 (2018).
https:/​/​doi.org/​10.1088/​1361-6633/​aaaf9a
arXiv:1709.07461

[40] Minh C. Tran, Andrew Y. Guo, Yuan Su, James R. Garrison, Zachary Eldredge, Michael Foss-Feig, Andrew M. Childs, and Alexey V. Gorshkov, ``Locality and Digital Quantum Simulation of Power-Law Interactions'' Phys. Rev. X 9, 031006 (2019).
https:/​/​doi.org/​10.1103/​PhysRevX.9.031006
arXiv:1808.05225

[41] Norbert Schuch, David Pérez-García, and Ignacio Cirac, ``Classifying quantum phases using matrix product states and projected entangled pair states'' Phys. Rev. B 84, 165139 (2011).
https:/​/​doi.org/​10.1103/​PhysRevB.84.165139
arXiv:1010.3732

[42] José Garre-Rubio, Sofyan Iblisdir, and David Pérez-García, ``Symmetry reduction induced by anyon condensation: A tensor network approach'' Phys. Rev. B 96, 155123 (2017).
https:/​/​doi.org/​10.1103/​PhysRevB.96.155123
arXiv:1702.08759

[43] D. Pérez-García, Frank Verstraete, Michael M Wolf, and J Ignacio Cirac, ``Matrix Product State Representations'' Quantum Inf. Comput. 7, 401–430 (2006).
arXiv:0608197
http:/​/​arxiv.org/​abs/​quant-ph/​0608197

[44] Norbert Schuch, Ignacio Cirac, and David Pérez-García, ``PEPS as ground states: Degeneracy and topology'' Ann. Phys. 325, 2153–2192 (2010).
https:/​/​doi.org/​10.1016/​j.aop.2010.05.008
arXiv:1001.3807

[45] S. B. Bravyiand A. Yu. Kitaev ``Quantum codes on a lattice with boundary'' arXiv:quant-ph/​9811052 (1998).
arXiv:9811052
http:/​/​arxiv.org/​abs/​quant-ph/​9811052

[46] Jeongwan Haah ``An Invariant of Topologically Ordered States Under Local Unitary Transformations'' Commun. Math. Phys. 342, 771–801 (2016).
https:/​/​doi.org/​10.1007/​s00220-016-2594-y
arXiv:1407.2926

[47] F. A. Baisand J. K. Slingerland ``Condensate-induced transitions between topologically ordered phases'' Phys. Rev. B 79, 045316 (2009).
https:/​/​doi.org/​10.1103/​PhysRevB.79.045316
arXiv:0808.0627

[48] Kasper Duivenvoorden, Mohsin Iqbal, Jutho Haegeman, Frank Verstraete, and Norbert Schuch, ``Entanglement phases as holographic duals of anyon condensates'' Phys. Rev. B 95, 235119 (2017).
https:/​/​doi.org/​10.1103/​PhysRevB.95.235119
arXiv:1702.08469

[49] Mohsin Iqbal, Kasper Duivenvoorden, and Norbert Schuch, ``Study of anyon condensation and topological phase transitions from a <math> <msub> <mi mathvariant="double-struck">Z</​mi> <mn>4</​mn> </​msub> </​math> topological phase using the projected entangled pair states approach'' Phys. Rev. B 97, 195124 (2018).
https:/​/​doi.org/​10.1103/​PhysRevB.97.195124
arXiv:1712.04021

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[2] Moshe Goldstein, "Dissipation-induced topological insulators: A no-go theorem and a recipe", SciPost Physics 7 5, 067 (2019).

[3] Max McGinley and Nigel R. Cooper, "Interacting symmetry-protected topological phases out of equilibrium", Physical Review Research 1 3, 033204 (2019).

[4] J. Ignacio Cirac, José Garre-Rubio, and David Pérez-García, "Mathematical open problems in Projected Entangled Pair States", arXiv:1903.09439.

[5] José Garre-Rubio, "Symmetries in topological tensor network states: classification, construction and detection", arXiv:1912.08597.

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