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.

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


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.

► BibTeX data

► References

[1] Robert König and Fernando Pastawski ``Generating topological order: no speedup by dissipation'' Phys. Rev. B 90, 045101 (2013).

[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).

[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).

[4] N Goldman, J C Budich, and P Zoller, ``Topological quantum matter with ultracold gases in optical lattices'' Nat. Phys. 12, 639–645 (2016).

[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).

[6] M. Z. Hasanand C. L. Kane ``Colloquium : Topological insulators'' Rev. Mod. Phys. 82, 3045–3067 (2010).

[7] Xiao-Liang Qiand Shou-Cheng Zhang ``Topological insulators and superconductors'' Rev. Mod. Phys. 83, 1057–1110 (2011).

[8] Xiao-Gang Wen ``Colloquium : Zoo of quantum-topological phases of matter'' Rev. Mod. Phys. 89, 041004 (2017).

[9] Bruno Nachtergaele, Yoshiko Ogata, and Robert Sims, ``Propagation of Correlations in Quantum Lattice Systems'' J. Stat. Phys. 124, 1–13 (2006).

[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).

[12] Claudio Castelnovoand Claudio Chamon ``Topological order in a three-dimensional toric code at finite temperature'' Phys. Rev. B 78, 155120 (2008).

[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).

[14] Matthew B. Hastings ``Topological Order at Nonzero Temperature'' Phys. Rev. Lett. 107, 210501 (2011).

[15] Sebastian Diehl, Enrique Rico, Mikhail A. Baranov, and Peter Zoller, ``Topology by dissipation in atomic quantum wires'' Nat. Phys. 7, 971–977 (2011).

[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).

[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).

[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).

[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).

[20] Fabian Grusdt ``Topological order of mixed states in correlated quantum many-body systems'' Phys. Rev. B 95, 075106 (2017).

[21] Tobias J. Osborne ``Simulating adiabatic evolution of gapped spin systems'' Phys. Rev. A 75, 032321 (2007).

[22] Yichen Huangand Xie Chen ``Quantum circuit complexity of one-dimensional topological phases'' Phys. Rev. B 91, 195143 (2015).

[23] Tobias J. Osborne ``Efficient Approximation of the Dynamics of One-Dimensional Quantum Spin Systems'' Phys. Rev. Lett. 97, 157202 (2006).

[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).

[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).

[26] David Aldousand Persi Diaconis ``The Asymmetric One-Dimensional Constrained Ising Model'' J. Stat. Phys. 107, 945–975 (2002).

[27] Shirshendu Ganguly, Eyal Lubetzky, and Fabio Martinelli, ``Cutoff for the East Process'' Commun. Math. Phys. 335, 1287–1322 (2015).

[28] Michael J. Kastoryano, David Reeb, and Michael M. Wolf, ``A cutoff phenomenon for quantum Markov chains'' J. Phys. A 45, 075307 (2012).

[29] M. J. Kastoryano, M. M. Wolf, and J. Eisert, ``Precisely Timing Dissipative Quantum Information Processing'' Phys. Rev. Lett. 110, 110501 (2013).

[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).

[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).

[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).

[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).

[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).

[35] O. Viyuela, A. Rivas, and M. A. Martin-Delgado, ``Generalized Toric Codes Coupled to Thermal Baths'' New J. Phys. 14, 033044 (2011).

[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).

[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).

[38] M. B. Hastings ``Lieb-Schultz-Mattis in higher dimensions'' Phys. Rev. B 69, 104431 (2004).

[39] Markus Heyl ``Dynamical quantum phase transitions: a review'' Rep. Prog. Phys. 81, 054001 (2018).

[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).

[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).

[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).

[43] D. Pérez-García, Frank Verstraete, Michael M Wolf, and J Ignacio Cirac, ``Matrix Product State Representations'' Quantum Inf. Comput. 7, 401–430 (2006).

[44] Norbert Schuch, Ignacio Cirac, and David Pérez-García, ``PEPS as ground states: Degeneracy and topology'' Ann. Phys. 325, 2153–2192 (2010).

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

[46] Jeongwan Haah ``An Invariant of Topologically Ordered States Under Local Unitary Transformations'' Commun. Math. Phys. 342, 771–801 (2016).

[47] F. A. Baisand J. K. Slingerland ``Condensate-induced transitions between topologically ordered phases'' Phys. Rev. B 79, 045316 (2009).

[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).

[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).

Cited by

[1] Ruochen Ma and Chong Wang, "Average Symmetry-Protected Topological Phases", Physical Review X 13 3, 031016 (2023).

[2] Luan M. Veríssimo, Marcelo L. Lyra, and Román Orús, "Dissipative symmetry-protected topological order", Physical Review B 107 24, L241104 (2023).

[3] Zongping Gong, Christoph Sünderhauf, Norbert Schuch, and J. Ignacio Cirac, "Classification of Matrix-Product Unitaries with Symmetries", Physical Review Letters 124 10, 100402 (2020).

[4] Alon Beck and Moshe Goldstein, "Disorder in dissipation-induced topological states: Evidence for a different type of localization transition", Physical Review B 103 24, L241401 (2021).

[5] Ivan Bardet, Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, and Cambyse Rouzé, "Entropy Decay for Davies Semigroups of a One Dimensional Quantum Lattice", Communications in Mathematical Physics 405 2, 42 (2024).

[6] Simon Lieu, Max McGinley, Oles Shtanko, Nigel R. Cooper, and Alexey V. Gorshkov, "Kramers' degeneracy for open systems in thermal equilibrium", Physical Review B 105 12, L121104 (2022).

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

[8] Paolo Molignini and Nigel R. Cooper, "Topological phase transitions at finite temperature", Physical Review Research 5 2, 023004 (2023).

[9] Lorenzo Piroli, Georgios Styliaris, and J. Ignacio Cirac, "Quantum Circuits Assisted by Local Operations and Classical Communication: Transformations and Phases of Matter", Physical Review Letters 127 22, 220503 (2021).

[10] Daniel Malz, Georgios Styliaris, Zhi-Yuan Wei, and J. Ignacio Cirac, "Preparation of Matrix Product States with Log-Depth Quantum Circuits", Physical Review Letters 132 4, 040404 (2024).

[11] Lorenzo Piroli, Alex Turzillo, Sujeet K Shukla, and J Ignacio Cirac, "Fermionic quantum cellular automata and generalized matrix-product unitaries", Journal of Statistical Mechanics: Theory and Experiment 2021 1, 013107 (2021).

[12] Kohei Kawabata, Ramanjit Sohal, and Shinsei Ryu, "Lieb-Schultz-Mattis Theorem in Open Quantum Systems", Physical Review Letters 132 7, 070402 (2024).

[13] Alberto Ruiz-de-Alarcón, José Garre-Rubio, András Molnár, and David Pérez-García, "Matrix product operator algebras II: phases of matter for 1D mixed states", Letters in Mathematical Physics 114 2, 43 (2024).

[14] Amit Jamadagni and Arpan Bhattacharyya, "Topological phase transitions induced by varying topology and boundaries in the toric code", New Journal of Physics 23 10, 103001 (2021).

[15] Ivan Bardet, Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, and Cambyse Rouzé, "Rapid Thermalization of Spin Chain Commuting Hamiltonians", Physical Review Letters 130 6, 060401 (2023).

[16] Caroline de Groot, Alex Turzillo, and Norbert Schuch, "Symmetry Protected Topological Order in Open Quantum Systems", Quantum 6, 856 (2022).

[17] Alexander Altland, Michael Fleischhauer, and Sebastian Diehl, "Symmetry Classes of Open Fermionic Quantum Matter", Physical Review X 11 2, 021037 (2021).

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

[19] J. Ignacio Cirac, David Pérez-García, Norbert Schuch, and Frank Verstraete, "Matrix product states and projected entangled pair states: Concepts, symmetries, theorems", Reviews of Modern Physics 93 4, 045003 (2021).

[20] Ignacio Cirac, David Perez-Garcia, Norbert Schuch, and Frank Verstraete, "Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems", arXiv:2011.12127, (2020).

[21] Ruochen Ma and Alex Turzillo, "Symmetry Protected Topological Phases of Mixed States in the Doubled Space", arXiv:2403.13280, (2024).

[22] J. Ignacio Cirac, José Garre-Rubio, and David Pérez-García, "Mathematical open problems in Projected Entangled Pair States", arXiv:1903.09439, (2019).

[23] José Garre-Rubio, "Symmetries in topological tensor network states: classification, construction and detection", arXiv:1912.08597, (2019).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-26 08:12:44) and SAO/NASA ADS (last updated successfully 2024-05-26 08:12:45). The list may be incomplete as not all publishers provide suitable and complete citation data.