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Symmetry Protected Topological Phases of Mixed States in the Doubled Space (2403.13280v3)

Published 20 Mar 2024 in quant-ph, cond-mat.stat-mech, and cond-mat.str-el

Abstract: The interplay of symmetry and topology in quantum many-body mixed states has recently garnered significant interest. In a phenomenon not seen in pure states, mixed states can exhibit average symmetries -- symmetries that act on component states while leaving the ensemble invariant. In this work, we systematically characterize symmetry protected topological (SPT) phases of short-range entangled (SRE) mixed states of spin systems -- protected by both average and exact symmetries -- by studying their pure Choi states in a doubled Hilbert space, where the familiar notions and tools for SRE and SPT pure states apply. This advantage of the doubled space comes with a price: extra symmetries as well as subtleties around how hermiticity and positivity of the original density matrix constrain the possible SPT invariants. Nevertheless, the doubled space perspective allows us to obtain a systematic classification of mixed-state SPT (MSPT) phases. We also investigate the robustness of MSPT invariants under symmetric finite-depth quantum channels, the bulk-boundary correspondence for MSPT phases, and the consequences of the MSPT invariants for the separability of mixed states and the symmetry-protected sign problem. In addition to MSPT phases, we study the patterns of spontaneous symmetry breaking (SSB) of mixed states, including the phenomenon of exact-to-average SSB, and the order parameters that detect them. Mixed state SSB is related to an ingappability constraint on symmetric Lindbladian dynamics.

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References (74)
  1. John McGreevy, “Generalized Symmetries in Condensed Matter,” Annual Review of Condensed Matter Physics 14, 57–82 (2023), arXiv:2204.03045 [cond-mat.str-el] .
  2. Hannes Bernien, Sylvain Schwartz, Alexander Keesling, Harry Levine, Ahmed Omran, Hannes Pichler, Soonwon Choi, Alexander S. Zibrov, Manuel Endres, Markus Greiner, Vladan Vuletić,  and Mikhail D. Lukin, “Probing many-body dynamics on a 51-atom quantum simulator,” Nature (London) 551, 579–584 (2017), arXiv:1707.04344 [quant-ph] .
  3. John Preskill, “Quantum Computing in the NISQ era and beyond,” Quantum 2, 79 (2018), arXiv:1801.00862 [quant-ph] .
  4. Caroline de Groot, Alex Turzillo,  and Norbert Schuch, “Symmetry Protected Topological Order in Open Quantum Systems,” Quantum 6, 856 (2022), arXiv:2112.04483 [quant-ph] .
  5. Ruochen Ma and Chong Wang, “Average Symmetry-Protected Topological Phases,” Physical Review X 13, 031016 (2023), arXiv:2209.02723 [cond-mat.str-el] .
  6. Ruochen Ma, Jian-Hao Zhang, Zhen Bi, Meng Cheng,  and Chong Wang, “Topological Phases with Average Symmetries: the Decohered, the Disordered, and the Intrinsic,” arXiv e-prints , arXiv:2305.16399 (2023), arXiv:2305.16399 [cond-mat.str-el] .
  7. Jong Yeon Lee, Yi-Zhuang You,  and Cenke Xu, “Symmetry protected topological phases under decoherence,” arXiv e-prints , arXiv:2210.16323 (2022a), arXiv:2210.16323 [cond-mat.str-el] .
  8. Yimu Bao, Ruihua Fan, Ashvin Vishwanath,  and Ehud Altman, “Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions,” arXiv e-prints , arXiv:2301.05687 (2023), arXiv:2301.05687 [quant-ph] .
  9. Ruihua Fan, Yimu Bao, Ehud Altman,  and Ashvin Vishwanath, “Diagnostics of mixed-state topological order and breakdown of quantum memory,” arXiv e-prints , arXiv:2301.05689 (2023), arXiv:2301.05689 [quant-ph] .
  10. Jong Yeon Lee, Chao-Ming Jian,  and Cenke Xu, “Quantum Criticality Under Decoherence or Weak Measurement,” PRX Quantum 4, 030317 (2023), arXiv:2301.05238 [cond-mat.stat-mech] .
  11. Tsung-Cheng Lu, Zhehao Zhang, Sagar Vijay,  and Timothy H. Hsieh, “Mixed-State Long-Range Order and Criticality from Measurement and Feedback,” PRX Quantum 4, 030318 (2023), arXiv:2303.15507 [cond-mat.str-el] .
  12. Xie Chen, Zheng-Cheng Gu,  and Xiao-Gang Wen, “Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order,” Phys. Rev. B 82, 155138 (2010), arXiv:1004.3835 [cond-mat.str-el] .
  13. Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu,  and Xiao-Gang Wen, “Symmetry protected topological orders and the group cohomology of their symmetry group,” Phys. Rev. B 87, 155114 (2013), arXiv:1106.4772 [cond-mat.str-el] .
  14. M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Reviews of Modern Physics 82, 3045–3067 (2010), arXiv:1002.3895 [cond-mat.mes-hall] .
  15. Xiao-Liang Qi and Shou-Cheng Zhang, “Topological insulators and superconductors,” Reviews of Modern Physics 83, 1057–1110 (2011), arXiv:1008.2026 [cond-mat.mes-hall] .
  16. Ian Affleck, Tom Kennedy, Elliott H Lieb,  and Hal Tasaki, “Valence bond ground states in isotropic quantum antiferromagnets,” Communications in Mathematical Physics 115, 477–528 (1988).
  17. M. B. Hastings and Xiao-Gang Wen, “Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance,” Phys. Rev. B 72, 045141 (2005), arXiv:cond-mat/0503554 [cond-mat.str-el] .
  18. Yu-Hsueh Chen and Tarun Grover, “Separability transitions in topological states induced by local decoherence,” arXiv e-prints , arXiv:2309.11879 (2023a), arXiv:2309.11879 [quant-ph] .
  19. Yu-Hsueh Chen and Tarun Grover, “Symmetry-enforced many-body separability transitions,” arXiv e-prints , arXiv:2310.07286 (2023b), arXiv:2310.07286 [quant-ph] .
  20. Yi-Zhuang You, Zhen Bi, Alex Rasmussen, Kevin Slagle,  and Cenke Xu, “Wave Function and Strange Correlator of Short-Range Entangled States,” Phys. Rev. Lett.  112, 247202 (2014), arXiv:1312.0626 [cond-mat.str-el] .
  21. Xiao-Liang Qi, Taylor L. Hughes,  and Shou-Cheng Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B 78, 195424 (2008), arXiv:0802.3537 [cond-mat.mes-hall] .
  22. M. Kliesch, D. Gross,  and J. Eisert, “Matrix-Product Operators and States: NP-Hardness and Undecidability,” Phys. Rev. Lett.  113, 160503 (2014), arXiv:1404.4466 [quant-ph] .
  23. Tyler D. Ellison, Kohtaro Kato, Zi-Wen Liu,  and Timothy H. Hsieh, “Symmetry-protected sign problem and magic in quantum phases of matter,” Quantum 5, 612 (2021), arXiv:2010.13803 [cond-mat.str-el] .
  24. Andrea Coser and David Pérez-García, “Classification of phases for mixed states via fast dissipative evolution,” Quantum 3, 174 (2019), arXiv:1810.05092 [quant-ph] .
  25. Shengqi Sang, Yijian Zou,  and Timothy H. Hsieh, “Mixed-state Quantum Phases: Renormalization and Quantum Error Correction,” arXiv e-prints , arXiv:2310.08639 (2023), arXiv:2310.08639 [quant-ph] .
  26. Tibor Rakovszky, Sarang Gopalakrishnan,  and Curt von Keyserlingk, “Defining stable phases of open quantum systems,” arXiv e-prints , arXiv:2308.15495 (2023), arXiv:2308.15495 [quant-ph] .
  27. Eduardo Fradkin, “Disorder Operators and Their Descendants,” Journal of Statistical Physics 167, 427–461 (2017), arXiv:1610.05780 [cond-mat.stat-mech] .
  28. John Preskill, ‘‘Lecture notes for physics 229: Quantum information and computation,” California Institute of Technology 16, 1–8 (1998).
  29. Berislav Buča and Tomaž Prosen, “A note on symmetry reductions of the lindblad equation: transport in constrained open spin chains,” New Journal of Physics 14, 073007 (2012).
  30. Norbert Schuch, David Pérez-García,  and Ignacio Cirac, “Classifying quantum phases using matrix product states and projected entangled pair states,” Physical Review B 84 (2011), 10.1103/physrevb.84.165139.
  31. F. Verstraete and J. I. Cirac, “Matrix product states represent ground states faithfully,” Phys. Rev. B 73, 094423 (2006), arXiv:cond-mat/0505140 [cond-mat.str-el] .
  32. F. Verstraete, V. Murg,  and J. I. Cirac, “Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems,” Advances in Physics 57, 143–224 (2008), arXiv:0907.2796 [quant-ph] .
  33. Ulrich Schollwöck, “The density-matrix renormalization group in the age of matrix product states,” Annals of Physics 326, 96–192 (2011), arXiv:1008.3477 [cond-mat.str-el] .
  34. Ignacio Cirac, David Perez-Garcia, Norbert Schuch,  and Frank Verstraete, “Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems,” arXiv e-prints , arXiv:2011.12127 (2020), arXiv:2011.12127 [quant-ph] .
  35. Michael M. Wolf, Frank Verstraete, Matthew B. Hastings,  and J. Ignacio Cirac, “Area Laws in Quantum Systems: Mutual Information and Correlations,” Phys. Rev. Lett.  100, 070502 (2008), arXiv:0704.3906 [quant-ph] .
  36. J. I. Cirac, D. Pérez-García, N. Schuch,  and F. Verstraete, “Matrix product density operators: Renormalization fixed points and boundary theories,” Annals of Physics 378, 100–149 (2017), arXiv:1606.00608 [quant-ph] .
  37. Paolo Zanardi, “Entanglement of quantum evolutions,” Phys. Rev. A 63, 040304 (2001), arXiv:quant-ph/0010074 [quant-ph] .
  38. Gemma De las Cuevas, Norbert Schuch, David Pérez-García,  and J. Ignacio Cirac, “Purifications of multipartite states: limitations and constructive methods,” New Journal of Physics 15, 123021 (2013), arXiv:1308.1914 [quant-ph] .
  39. Xie Chen, Yuan-Ming Lu,  and Ashvin Vishwanath, “Symmetry-protected topological phases from decorated domain walls,” Nature Communications 5, 3507 (2014), arXiv:1303.4301 [cond-mat.str-el] .
  40. 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), arXiv:1702.08469 [cond-mat.str-el] .
  41. Frank Pollmann, Ari M. Turner, Erez Berg,  and Masaki Oshikawa, “Entanglement spectrum of a topological phase in one dimension,” Phys. Rev. B 81, 064439 (2010), arXiv:0910.1811 [cond-mat.str-el] .
  42. Frank Pollmann, Erez Berg, Ari M. Turner,  and Masaki Oshikawa, “Symmetry protection of topological phases in one-dimensional quantum spin systems,” Phys. Rev. B 85, 075125 (2012), arXiv:0909.4059 [cond-mat.str-el] .
  43. Frank Pollmann and Ari M. Turner, “Detection of symmetry-protected topological phases in one dimension,” Phys. Rev. B 86, 125441 (2012), arXiv:1204.0704 [cond-mat.str-el] .
  44. Ken Shiozaki and Shinsei Ryu, “Matrix product states and equivariant topological field theories for bosonic symmetry-protected topological phases in (1+1) dimensions,” Journal of High Energy Physics 2017, 100 (2017a), arXiv:1607.06504 [cond-mat.str-el] .
  45. Ari M. Turner, Frank Pollmann,  and Erez Berg, “Topological phases of one-dimensional fermions: An entanglement point of view,” Phys. Rev. B 83, 075102 (2011), arXiv:1008.4346 [cond-mat.str-el] .
  46. Qing-Rui Wang, Shang-Qiang Ning,  and Meng Cheng, “Domain Wall Decorations, Anomalies and Spectral Sequences in Bosonic Topological Phases,” arXiv e-prints , arXiv:2104.13233 (2021), arXiv:2104.13233 [cond-mat.str-el] .
  47. Davide Gaiotto and Theo Johnson-Freyd, “Symmetry protected topological phases and generalized cohomology,” Journal of High Energy Physics 2019, 7 (2019), arXiv:1712.07950 [hep-th] .
  48. Hassan Shapourian, Ken Shiozaki,  and Shinsei Ryu, “Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases,” Phys. Rev. Lett.  118, 216402 (2017), arXiv:1607.03896 [cond-mat.str-el] .
  49. Xie Chen and Ashvin Vishwanath, “Towards Gauging Time-Reversal Symmetry: A Tensor Network Approach,” Physical Review X 5, 041034 (2015), arXiv:1401.3736 [cond-mat.str-el] .
  50. Angina Seng, “Group cohomology of product with swapping (twisting) factors,” Mathematics Stack Exchange, uRL:https://math.stackexchange.com/q/3051498 (version: 2018-12-24), https://math.stackexchange.com/q/3051498 .
  51. Arun Debray, “Bordism for the 2-group symmetries of the heterotic and CHL strings,” arXiv e-prints , arXiv:2304.14764 (2023), arXiv:2304.14764 [math.AT] .
  52. Michael Levin and Zheng-Cheng Gu, “Braiding statistics approach to symmetry-protected topological phases,” Phys. Rev. B 86, 115109 (2012), arXiv:1202.3120 [cond-mat.str-el] .
  53. Ken Shiozaki and Shinsei Ryu, “Matrix product states and equivariant topological field theories for bosonic symmetry-protected topological phases in (1+1) dimensions,” Journal of High Energy Physics 2017, 100 (2017b), arXiv:1607.06504 [cond-mat.str-el] .
  54. Jutho Haegeman, David Pérez-García, Ignacio Cirac,  and Norbert Schuch, “Order Parameter for Symmetry-Protected Phases in One Dimension,” Phys. Rev. Lett.  109, 050402 (2012), arXiv:1201.4174 [cond-mat.str-el] .
  55. Xie Chen, Zheng-Xin Liu,  and Xiao-Gang Wen, “Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations,” Phys. Rev. B 84, 235141 (2011), arXiv:1106.4752 [cond-mat.str-el] .
  56. M. B. Hastings, “How quantum are non-negative wavefunctions?” Journal of Mathematical Physics 57, 015210 (2016), arXiv:1506.08883 [quant-ph] .
  57. Lorenzo Piroli and J. Ignacio Cirac, “Quantum Cellular Automata, Tensor Networks, and Area Laws,” Phys. Rev. Lett.  125, 190402 (2020), arXiv:2007.15371 [quant-ph] .
  58. Robert Raussendorf and Hans J Briegel, “A one-way quantum computer,” Physical review letters 86, 5188 (2001).
  59. F. Verstraete, M. M. Wolf, D. Perez-Garcia,  and J. I. Cirac, “Criticality, the Area Law, and the Computational Power of Projected Entangled Pair States,” Phys. Rev. Lett.  96, 220601 (2006), arXiv:quant-ph/0601075 [quant-ph] .
  60. Lukasz Fidkowski and Alexei Kitaev, “Topological phases of fermions in one dimension,” Phys. Rev. B 83, 075103 (2011), arXiv:1008.4138 [cond-mat.str-el] .
  61. Dominic V. Else and Chetan Nayak, “Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge,” Phys. Rev. B 90, 235137 (2014), arXiv:1409.5436 [cond-mat.str-el] .
  62. M. Wolf, “Quantum channels and operations: Guided tour,” Lecture Notes  (2012).
  63. Michael Levin, “Constraints on Order and Disorder Parameters in Quantum Spin Chains,” Communications in Mathematical Physics 378, 1081–1106 (2020), arXiv:1903.09028 [cond-mat.str-el] .
  64. M. B. Hastings, “Locality in Quantum and Markov Dynamics on Lattices and Networks,” Phys. Rev. Lett.  93, 140402 (2004a), arXiv:cond-mat/0405587 [cond-mat.stat-mech] .
  65. 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), arXiv:1409.7809 [quant-ph] .
  66. Toby S. Cubitt, Angelo Lucia, Spyridon Michalakis,  and David Perez-Garcia, “Stability of Local Quantum Dissipative Systems,” Communications in Mathematical Physics 337, 1275–1315 (2015), arXiv:1303.4744 [quant-ph] .
  67. Fei Song, Shunyu Yao,  and Zhong Wang, “Non-Hermitian skin effect and chiral damping in open quantum systems,” arXiv e-prints , arXiv:1904.08432 (2019), arXiv:1904.08432 [cond-mat.quant-gas] .
  68. David Poulin, “Lieb-Robinson Bound and Locality for General Markovian Quantum Dynamics,” Phys. Rev. Lett.  104, 190401 (2010), arXiv:1003.3675 [quant-ph] .
  69. Elliott Lieb, Theodore Schultz,  and Daniel Mattis, “Two soluble models of an antiferromagnetic chain,” Annals of Physics 16, 407–466 (1961).
  70. Masaki Oshikawa, “Commensurability, Excitation Gap, and Topology in Quantum Many-Particle Systems on a Periodic Lattice,” Phys. Rev. Lett.  84, 1535–1538 (2000), arXiv:cond-mat/9911137 [cond-mat.str-el] .
  71. M. B. Hastings, ‘‘Lieb-Schultz-Mattis in higher dimensions,” Phys. Rev. B 69, 104431 (2004b), arXiv:cond-mat/0305505 [cond-mat.str-el] .
  72. Kohei Kawabata, Ramanjit Sohal,  and Shinsei Ryu, “Lieb-Schultz-Mattis Theorem in Open Quantum Systems,” arXiv e-prints , arXiv:2305.16496 (2023), arXiv:2305.16496 [cond-mat.stat-mech] .
  73. Jutho Haegeman, Karel Van Acoleyen, Norbert Schuch, J. Ignacio Cirac,  and Frank Verstraete, “Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems,” Physical Review X 5, 011024 (2015), arXiv:1407.1025 [quant-ph] .
  74. Jacob C. Bridgeman and Christopher T. Chubb, “Hand-waving and interpretive dance: an introductory course on tensor networks,” Journal of Physics A Mathematical General 50, 223001 (2017), arXiv:1603.03039 [quant-ph] .
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