Classification of phases for mixed states via fast dissipative evolution

Andrea Coser; David Pérez-García

doi:10.22331/q-2019-08-12-174

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.

Published:	2019-08-12, volume 3, page 174
Eprint:	arXiv:1810.05092v3
Doi:	https://doi.org/10.22331/q-2019-08-12-174
Citation:	Quantum 3, 174 (2019).

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

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.

► BibTeX data

@article{Coser2019classificationof,
  doi = {10.22331/q-2019-08-12-174},
  url = {https://doi.org/10.22331/q-2019-08-12-174},
  title = {Classification of phases for mixed states via fast dissipative evolution},
  author = {Coser, Andrea and P{\'{e}}rez-Garc{\'{i}}a, David},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {174},
  month = aug,
  year = {2019}
}

► 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. Ä°mamoÄlu, 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 Ä°mamoÄlu, 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

[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

[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

Cited by

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

[2] Ramanjit Sohal and Abhinav Prem, "Noisy Approach to Intrinsically Mixed-State Topological Order", PRX Quantum 6 1, 010313 (2025).

[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] 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).

[5] Carolyn Zhang, Yichen Xu, Jian-Hao Zhang, Cenke Xu, Zhen Bi, and Zhu-Xi Luo, "Strong-to-weak spontaneous breaking of 1-form symmetry and intrinsically mixed topological order", Physical Review B 111 11, 115137 (2025).

[6] Andreas Karch and Merna Youssef, "Dissipation in open holography", Journal of High Energy Physics 2025 12, 157 (2025).

[7] Seunghun Lee and Eun-Gook Moon, "Mixed-State Topological Order under Coherent Noise", PRX Quantum 6 3, 030355 (2025).

[8] Adam Gammon-Smith, Michael Knap, and Frank Pollmann, "Simulating topological order on quantum processors", Nature Reviews Physics 8 3, 160 (2026).

[9] Seunghun Lee and Eun-Gook Moon, "Robust mixed-state cluster states and spurious topological entanglement negativity", Physical Review B 112 16, L161121 (2025).

[10] 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).

[11] Natalie Parham, Proceedings of the 58th Annual ACM Symposium on Theory of Computing 734 (2026) ISBN:9798400725364.

[12] 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).

[13] Claudia Artiaco, Thomas Klein Kvorning, David Aceituno Chávez, Loïc Herviou, and Jens H. Bardarson, "Universal Characterization of Quantum Many-Body States through Local Information", Physical Review Letters 134 19, 190401 (2025).

[14] Yimu Bao, Ruihua Fan, Ashvin Vishwanath, and Ehud Altman, "Mixed-State Topological Order and the Errorfield Double Formulation of Decoherence-Induced Transitions", Physical Review Letters 136 22, 220402 (2026).

[15] Yifan F. Zhang and Sarang Gopalakrishnan, "Approximate Markovianity in high-temperature hidden Markov networks", Physical Review B 113 11, 115101 (2026).

[16] Pablo Sala, Sarang Gopalakrishnan, Masaki Oshikawa, and Yizhi You, "Spontaneous strong symmetry breaking in open systems: Purification perspective", Physical Review B 110 15, 155150 (2024).

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

[18] Yizhi You and Masaki Oshikawa, "Intrinsic symmetry-protected topological mixed state from modulated symmetries and hierarchical structure of boundary anomaly", Physical Review B 110 16, 165160 (2024).

[19] 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).

[20] Jeet Shah, Christopher Fechisin, Yu-Xin Wang, Joseph T. Iosue, James D. Watson, Yan-Qi Wang, Brayden Ware, Alexey V. Gorshkov, and Cheng-Ju Lin, "Instability of steady-state mixed-state symmetry-protected topological order to strong-to-weak spontaneous symmetry breaking", Quantum 9, 1912 (2025).

[21] Ruochen Ma and Alex Turzillo, "Symmetry-Protected Topological Phases of Mixed States in the Doubled Space", PRX Quantum 6 1, 010348 (2025).

[22] Ting-Tung Wang, Menghan Song, Zi Yang Meng, and Tarun Grover, "Analog of Topological Entanglement Entropy for Mixed States", PRX Quantum 6 1, 010358 (2025).

[23] Pablo Sala, Jason Alicea, and Ruben Verresen, "Decoherence and Wave-Function Deformation of D4 Non-Abelian Topological Order", Physical Review X 15 3, 031002 (2025).

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

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

[26] 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).

[27] Rahul Trivedi, Adrian Franco Rubio, and J. Ignacio Cirac, "Quantum advantage and stability to errors in analogue quantum simulators", Nature Communications 15 1, 6507 (2024).

[28] 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).

[29] Yoshiko Ogata, "Mixed State Topological Order: Operator Algebraic Approach", Communications in Mathematical Physics 406 12, 305 (2025).

[30] Tibor Rakovszky, Sarang Gopalakrishnan, and Curt von Keyserlingk, "Defining Stable Phases of Open Quantum Systems", Physical Review X 14 4, 041031 (2024).

[31] Cambyse Rouzé, Daniel Stilck França, and Álvaro M. Alhambra, "Efficient thermalization and universal quantum computing with quantum Gibbs samplers", Nature Physics 22 7, 1123 (2026).

[32] Sanket Chirame, Abhinav Prem, Sarang Gopalakrishnan, and Fiona J. Burnell, "Stabilizing Non-Abelian Topological Order Against Heralded Noise via Local Lindbladian Dynamics", PRX Quantum 6 3, 030363 (2025).

[33] Jinmin Yi, Kangle Li, Chuan Liu, Zixuan Li, and Liujun Zou, "Universal Decay of Mutual Information and Conditional Mutual Information in Gapped Pure- and Mixed-State Quantum Matter", Physical Review Letters 136 11, 116604 (2026).

[34] Shuo Liu, Shao-Kai Jian, and Shi-Xin Zhang, "Noisy monitored quantum circuits", Journal of Physics: Condensed Matter 38 20, 203001 (2026).

[35] David Gunn, Georgios Styliaris, Tristan Kraft, and Barbara Kraus, "Phases of matrix product states with symmetric quantum circuits and symmetric measurements with feedforward", Physical Review B 111 11, 115110 (2025).

[36] Vikram Kashyap, Georgios Styliaris, Sara Mouradian, J. Ignacio Cirac, and Rahul Trivedi, "Accuracy Guarantees and Quantum Advantage in Analog Open Quantum Simulation with and without Noise", Physical Review X 15 2, 021017 (2025).

[37] Yuhan Liu, Alberto Ruiz-de-Alarcón, Georgios Styliaris, Xiao-Qi Sun, David Pérez-García, and J. Ignacio Cirac, "Parent Lindbladians for matrix product density operators", Physical Review Research 8 1, 013210 (2026).

[38] Sanket Chirame, Fiona J. Burnell, Sarang Gopalakrishnan, and Abhinav Prem, "Stable Symmetry-Protected Topological Phases in Systems with Heralded Noise", Physical Review Letters 134 1, 010403 (2025).

[39] Ze-Min Huang, Sebastian Diehl, and Xiao-Qi Sun, "Topological Response in Open Quantum Systems with Weak Symmetries", Physical Review Letters 135 21, 213002 (2025).

[40] Shengqi Sang and Timothy H. Hsieh, "Stability of Mixed-State Quantum Phases via Finite Markov Length", Physical Review Letters 134 7, 070403 (2025).

[41] Brett Min, Yuxuan Zhang, Yuxuan Guo, Dvira Segal, and Yuto Ashida, "Mixed-state phase transitions in spin-Holstein models", Physical Review B 111 11, 115123 (2025).

[42] Daniele Toniolo and Sougato Bose, "Dynamical α -Rényi Entropies of Local Hamiltonians Grow at Most Linearly in Time", Physical Review X 15 3, 031046 (2025).

[43] Shu-Tong Zhou, Meng Cheng, Tibor Rakovszky, Curt von Keyserlingk, and Tyler D. Ellison, "Finite-Temperature Quantum Topological Order in Three Dimensions", Physical Review Letters 135 4, 040402 (2025).

[44] Tyler D. Ellison and Meng Cheng, "Toward a Classification of Mixed-State Topological Orders in Two Dimensions", PRX Quantum 6 1, 010315 (2025).

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

[46] Amir-Reza Negari, Tyler D. Ellison, and Timothy H. Hsieh, "Spacetime Markov length: A diagnostic for fault tolerance via mixed-state phases", Physical Review B 114 4, 045130 (2026).

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

[48] Zack Weinstein, "Efficient Detection of Strong-to-Weak Spontaneous Symmetry Breaking via the Rényi-1 Correlator", Physical Review Letters 134 15, 150405 (2025).

[49] 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).

[50] Zijian Wang and Linhao Li, "Anomaly in Open Quantum Systems and its Implications on Mixed-State Quantum Phases", PRX Quantum 6 1, 010347 (2025).

[51] David T. Stephen, Rahul Nandkishore, and Jian-Hao Zhang, "Many-Body Quantum Catalysts for Transforming Between Phases of Matter", PRX Quantum 6 2, 020353 (2025).

[52] Lin Su, Rahul Sahay, Michal Szurek, Alexander Douglas, Ognjen Marković, Ceren B. Dag, Ruben Verresen, and Markus Greiner, "Topological phase transitions and mixed-state order in a Hubbard quantum simulator", Nature Physics (2026).

[53] 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).

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

[55] Leonardo A. Lessa, Meng Cheng, and Chong Wang, "Mixed-State Quantum Anomaly and Multipartite Entanglement", Physical Review X 15 1, 011069 (2025).

[56] Federico Garcia-Gaitan and Branislav K Nikolić, "Fate of entanglement in open quantum spin liquid: time evolution of its genuine multipartite negativity upon sudden coupling to a dissipative bosonic environment", New Journal of Physics 28 8, 083503 (2026).

[57] 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).

[58] Shengqi Sang, Yijian Zou, and Timothy H. Hsieh, "Mixed-State Quantum Phases: Renormalization and Quantum Error Correction", Physical Review X 14 3, 031044 (2024).

[59] Xu-Dong Dai, Zijian Wang, He-Ran Wang, and Zhong Wang, "Steady-state topological order", Physical Review B 111 11, 115142 (2025).

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

[61] Ruochen Ma, Vedika Khemani, and Shengqi Sang, "Circuit-based characterization of finite-temperature quantum phases and self-correcting quantum memory", arXiv:2509.15204, (2025).

[62] 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).

[63] 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).

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

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

[66] Léo Le-Nestour, David Pérez-García, and Alberto Ruiz-de-Alarcón, "Renormalization flows for 1D mixed states and a quantum Goursat lemma", arXiv:2607.08568, (2026).

[67] Paul Rosa-Ruiz, Matteo Scandi, Ángela Capel, and Álvaro M. Alhambra, "Static features from mixing in short- and long-range Lindbladians: Markov property and correlations", arXiv:2606.28054, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-10 13:25:01) and SAO/NASA ADS (last updated successfully 2026-08-10 13:25:03). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.