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  • Open Access

Holographic Duality Between Bulk Topological Order and Boundary Mixed-State Order

Tsung-Cheng Lu1,*, Yu-Jie Liu2,†, Sarang Gopalakrishnan3,‡, and Yizhi You4,§

  • *Contact author: tclu@umd.edu
  • †Contact author: yujieliu@mit.edu
  • ‡Contact author: sgopalakrishnan@princeton.edu
  • §Contact author: y.you@northeastern.edu
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PRX Quantum 7, 033041 – Published 28 August, 2026

DOI: https://doi.org/10.1103/lrqp-pnl1

Abstract

We introduce a holographic framework for analyzing the steady states of repeated quantum channels with strong symmetries. Using channel-state duality, we show that the steady state of a d-dimensional quantum channel is holographically mapped to the boundary reduced density matrix of a (d+1)-dimensional wavefunction generated by a sequential unitary circuit. From this perspective, strong-to-weak spontaneous symmetry breaking (SW-SSB) in the steady state arises from the anyon condensation on the boundary of a topological order in one higher dimension. The conditional mutual information (CMI) associated with SW-SSB is then inherited from the bulk topological entanglement entropy. We make this duality explicit using isometric tensor network states (isoTNS) by identifying the channel’s time evolution with the transfer matrix of a higher-dimensional isoTNS. Built on isoTNS, we further construct continuously tunable quantum channels that exhibit distinct mixed-state phases and transitions in the steady states.

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Physics Subject Headings (PhySH)

Popular Summary

What forms of collective behavior can emerge in an open quantum many-body system subject to environmental dissipation? Generically, correlations and information in the system will eventually leak into the environment at sufficiently long times. However, when a global symmetry is preserved throughout the dynamics, a novel type of quantum phenomenon known as strong-to-weak spontaneous symmetry breaking (SW-SSB) can emerge in the long-time limit. Remarkably, this phenomenon is intrinsic to open systems and has no analog in isolated systems. In particular, SW-SSB challenges the conventional paradigm of symmetry-breaking order, since the associated order is invisible to standard experimental probes based on local observables. As such, our understanding of SW-SSB is still far from complete. In this work, we establish a new holographic framework for understanding SW-SSB. Within this framework, the universal structure underlying SW-SSB is encoded into the environmental degrees of freedom, which form a quantum many-body wave function living in one higher spatial dimension. Interestingly, this higher-dimensional wave function generically exhibits topological order, an exotic form of quantum matter characterized by emergent quasiparticles known as anyons. These anyons are governed by loop-like deformable symmetries, which reduce to the symmetries associated with SW-SSB in the open system. Our holographic approach therefore provides a unified framework connecting novel phenomena in open systems with topological quantum matter in isolated systems and opens a new route toward characterizing nonequilibrium quantum phases through higher-dimensional entanglement structures.

Article Text

References (148)

  1. X.-G. Wen, Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons (Oxford University Press, New York, 2004).
  2. M. B. Hastings and X.-G. Wen, Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance, Phys. Rev. B 72, 045141 (2005).
  3. X. Chen, Z.-C. Gu, and X.-G. Wen, Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order, Phys. Rev. B 82, 155138 (2010).
  4. X. Chen, Z.-C. Gu, and X.-G. Wen, Classification of gapped symmetric phases in one-dimensional spin systems, Phys. Rev. B 83, 035107 (2011).
  5. X. Chen, Z.-C. Gu, and X.-G. Wen, Complete classification of one-dimensional gapped quantum phases in interacting spin systems, Phys. Rev. B 84, 235128 (2011).
  6. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. 43, 4452 (2002).
  7. R. Fan, Y. Bao, E. Altman, and A. Vishwanath, Diagnostics of mixed-state topological order and breakdown of quantum memory, PRX Quantum 5, 020343 (2024).
  8. C. de Groot, A. Turzillo, and N. Schuch, Symmetry protected topological order in open quantum systems, Quantum 6, 856 (2022).
  9. J. Y. Lee, Y.-Z. You, and C. Xu, Symmetry protected topological phases under decoherence, Quantum 9, 1607 (2025).
  10. J. Y. Lee, Exact calculations of coherent information for toric codes under decoherence: Identifying the fundamental error threshold, Phys. Rev. Lett. 134, 250601 (2025).
  11. R. Niwa and J. Y. Lee, Coherent information for Calderbank-Shor-Steane codes under decoherence, Phys. Rev. A 111, 032402 (2025).
  12. R. Ma and C. Wang, Average symmetry-protected topological phases, Phys. Rev. X 13, 031016 (2023).
  13. R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Topological phases with average symmetries: The decohered, the disordered, and the intrinsic, Phys. Rev. X 15, 021062 (2025).
  14. J.-H. Zhang, K. Ding, S. Yang, and Z. Bi, Fractonic higher-order topological phases in open quantum systems, Phys. Rev. B 108, 155123 (2023).
  15. R. Ma and A. Turzillo, Symmetry-protected topological phases of mixed states in the doubled space, PRX Quantum 6, 010348 (2025).
  16. H. Xue, J. Y. Lee, and Y. Bao, Tensor network formulation of symmetry protected topological phases in mixed states, arXiv:2403.17069.
  17. Y. Guo, J.-H. Zhang, H.-R. Zhang, S. Yang, and Z. Bi, Locally purified density operators for symmetry-protected topological phases in mixed states, Phys. Rev. X 15, 021060 (2025).
  18. Y.-H. Chen and T. Grover, Separability transitions in topological states induced by local decoherence, Phys. Rev. Lett. 132, 170602 (2024).
  19. Y.-H. Chen and T. Grover, Unconventional topological mixed-state transition and critical phase induced by self-dual coherent errors, Phys. Rev. B 110, 125152 (2024).
  20. Y.-H. Chen and T. Grover, Symmetry-enforced many-body separability transitions, PRX Quantum 5, 030310 (2024).
  21. L. A. Lessa, M. Cheng, and C. Wang, Mixed-state quantum anomaly and multipartite entanglement, Phys. Rev. X 15, 011069 (2025).
  22. Z. Wang and L. Li, Anomaly in open quantum systems and its implications on mixed-state quantum phases, PRX Quantum 6, 010347 (2025).
  23. T. D. Ellison and M. Cheng, Toward a classification of mixed-state topological orders in two dimensions, PRX Quantum 6, 010315 (2025).
  24. R. Sohal and A. Prem, Noisy approach to intrinsically mixed-state topological order, PRX Quantum 6, 010313 (2025).
  25. T.-C. Lu, Z. Zhang, S. Vijay, and T. H. Hsieh, Mixed-state long-range order and criticality from measurement and feedback, PRX Quantum 4, 030318 (2023).
  26. T.-C. Lu, T. H. Hsieh, and T. Grover, Detecting topological order at finite temperature using entanglement negativity, Phys. Rev. Lett. 125, 116801 (2020).
  27. S. Chirame, F. J. Burnell, S. Gopalakrishnan, and A. Prem, Stable symmetry-protected topological phases in systems with heralded noise, Phys. Rev. Lett. 134, 010403 (2025).
  28. V. V. Albert and L. Jiang, Symmetries and conserved quantities in Lindblad master equations, Phys. Rev. A 89, 022118 (2014).
  29. C. Zhang, Y. Xu, J.-H. Zhang, C. Xu, Z. Bi, and Z.-X. Luo, Strong-to-weak spontaneous breaking of 1-form symmetry and intrinsically mixed topological order, Phys. Rev. B 111, 115137 (2025).
  30. J.-H. Zhang, C. Xu, and Y. Xu, Fluctuation-dissipation theorem and information geometry in open quantum systems, arXiv:2409.18944.
  31. R. Raussendorf, J. Harrington, and K. Goyal, Topological fault-tolerance in cluster state quantum computation, New J. Phys. 9, 199 (2007).
  32. F. Eckstein, B. Han, S. Trebst, and G.-Y. Zhu, Robust teleportation of a surface code and cascade of topological quantum phase transitions, PRX Quantum 5, 040313 (2024).
  33. J. Y. Lee, W. Ji, Z. Bi, and M. P. A. Fisher, Decoding measurement-prepared quantum phases and transitions: From Ising model to gauge theory, and beyond, arXiv:2208.11699.
  34. G.-Y. Zhu, N. Tantivasadakarn, A. Vishwanath, S. Trebst, and R. Verresen, Nishimori’s cat: Stable long-range entanglement from finite-depth unitaries and weak measurements, Phys. Rev. Lett. 131, 200201 (2023).
  35. F. Fang, K. Wang, V. S. Liu, Y. Wang, R. Cimmino, J. Wei, M. Bintz, A. Parr, J. Kemp, K.-K. Ni, and N. Y. Yao, Probing critical phenomena in open quantum systems using atom arrays, arXiv:2402.15376.
  36. A. Coser and D. Pérez-García, Classification of phases for mixed states via fast dissipative evolution, Quantum 3, 174 (2019).
  37. Y. Bao, R. Fan, A. Vishwanath, and E. Altman, Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions, arXiv:2301.05687.
  38. S. Sang, Y. Zou, and T. H. Hsieh, Mixed-state quantum phases: Renormalization and quantum error correction, Phys. Rev. X 14, 031044 (2024).
  39. J.-H. Zhang, Y. Qi, and Z. Bi, Strange correlation function for average symmetry-protected topological phases, arXiv:2210.17485.
  40. B. J. Brown, D. Loss, J. K. Pachos, C. N. Self, and J. R. Wootton, Quantum memories at finite temperature, Rev. Mod. Phys. 88, 045005 (2016).
  41. Z. Wang, Z. Wu, and Z. Wang, Intrinsic mixed-state topological order, PRX Quantum 6, 010314 (2025).
  42. A. Antinucci, G. Galati, G. Rizi, and M. Serone, Symmetries and topological operators, on average, SciPost Phys. 15, 125 (2023).
  43. T.-C. Lu and S. Vijay, Characterizing long-range entanglement in a mixed state through an emergent order on the entangling surface, Phys. Rev. Res. 5, 033031 (2023).
  44. Y. You and M. Oshikawa, Intrinsic symmetry-protected topological mixed state from modulated symmetries and hierarchical structure of boundary anomaly, Phys. Rev. B 110, 165160 (2024).
  45. T.-C. Lu, Disentangling transitions in topological order induced by boundary decoherence, Phys. Rev. B 110, 125145 (2024).
  46. T.-C. Lu, S. Gopalakrishnan, and Y. You, Spacetime duality between sequential and measurement-feedback circuits, arXiv:2507.12523.
  47. K. Su, N. Myerson-Jain, C. Wang, C.-M. Jian, and C. Xu, Higher-form symmetries under weak measurement, Phys. Rev. Lett. 132, 200402 (2024).
  48. L. A. Lessa, S. Sang, T.-C. Lu, T. H. Hsieh, and C. Wang, Higher-form anomaly and long-range entanglement of mixed states, arXiv:2503.12792.
  49. S. Sang and T. H. Hsieh, Stability of mixed-state quantum phases via finite Markov length, Phys. Rev. Lett. 134, 070403 (2025).
  50. A. Moharramipour, L. A. Lessa, C. Wang, T. H. Hsieh, and S. Sahu, Symmetry-enforced entanglement in maximally mixed states, PRX Quantum 5, 040336 (2024).
  51. S. Sang, T. H. Hsieh, and Y. Zou, Approximate quantum error correcting codes from conformal field theory, Phys. Rev. Lett. 133, 210601 (2024).
  52. Y. Guo, K. Ding, and S. Yang, A new framework for quantum phases in open systems: Steady state of imaginary-time Lindbladian evolution, Rep. Prog. Phys. 88, 118001 (2025).
  53. D. Gu, Z. Wang, and Z. Wang, Spontaneous symmetry breaking in open quantum systems: Strong, weak, and strong-to-weak, Phys. Rev. B 112, 245123 (2025).
  54. J. Y. Lee, C.-M. Jian, and C. Xu, Quantum criticality under decoherence or weak measurement, PRX Quantum 4, 030317 (2023).
  55. P. Sala, S. Gopalakrishnan, M. Oshikawa, and Y. You, Spontaneous strong symmetry breaking in open systems: Purification perspective, Phys. Rev. B 110, 155150 (2024).
  56. L. A. Lessa, R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Strong-to-weak spontaneous symmetry breaking in mixed quantum states, PRX Quantum 6, 010344 (2025).
  57. Z. Song and J.-H. Zhang, Strong-to-weak spontaneous symmetry breaking of higher-form non-invertible symmetries in Kitaev’s quantum double model, arXiv:2509.24179.
  58. X. Huang, M. Qi, J.-H. Zhang, and A. Lucas, Hydrodynamics as the effective field theory of strong-to-weak spontaneous symmetry breaking, Phys. Rev. B 111, 125147 (2025).
  59. P. Sala, F. Pollmann, M. Oshikawa, and Y. You, Entanglement holography in quantum phases via twisted Rényi-N correlators, Quantum Sci. Technol. 11, 025017 (2026).
  60. Y.-J. Liu and S. Lieu, Dissipative phase transitions and passive error correction, Phys. Rev. A 109, 022422 (2024).
  61. X. Feng, Z. Cheng, and M. Ippoliti, Hardness of observing strong-to-weak symmetry breaking, Phys. Rev. Lett. 135, 200402 (2025).
  62. J. McGreevy, Generalized symmetries in condensed matter, Annu. Rev. Condens. Matter Phys. 14, 57 (2023).
  63. T. Ando, S. Ryu, and M. Watanabe, Gauge theory and mixed state criticality, Phys. Rev. B 113, 115106 (2026).
  64. J. Kim, E. Altman, and J. Y. Lee, Error threshold of syk codes from strong-to-weak parity symmetry breaking, arXiv:2410.24225.
  65. Z. Wang, X.-D. Dai, H.-R. Wang, and Z. Wang, Topologically ordered steady states in open quantum systems, SciPost Phys. 17, 167 (2024).
  66. X.-D. Dai, Z. Wang, H.-R. Wang, and Z. Wang, Steady-state topological order, Phys. Rev. B 111, 115142 (2025).
  67. B. Aldossari, S. Blinov, and Z.-X. Luo, Tensor network representations for intrinsically mixed-state topological orders, arXiv:2507.22989.
  68. T.-H. Yang, B. Shi, and J. Y. Lee, Topological mixed states: Axiomatic approaches and phases of matter, arXiv:2506.04221.
  69. S. Sang, L. A. Lessa, R. S. Mong, T. Grover, C. Wang, and T. H. Hsieh, Mixed-state phases from local reversibility, arXiv:2507.02292.
  70. Y. Liu, A. Ruiz-de Alarcón, G. Styliaris, X.-Q. Sun, D. Pérez-García, and J. I. Cirac, Parent Lindbladians for matrix product density operators, arXiv:2501.10552.
  71. N. Ziereis, S. Moudgalya, and M. Knap, Strong-to-weak symmetry breaking phases in steady states of quantum operations, arXiv:2509.09669.
  72. T. Rakovszky, S. Gopalakrishnan, and C. von Keyserlingk, Defining stable phases of open quantum systems, Phys. Rev. X 14, 041031 (2024).
  73. X. Chen, M. Hermele, and D. T. Stephen, Sequential adiabatic generation of chiral topological states, arXiv:2402.03433.
  74. X. Chen, A. Dua, M. Hermele, D. T. Stephen, N. Tantivasadakarn, R. Vanhove, and J.-Y. Zhao, Sequential quantum circuits as maps between gapped phases, Phys. Rev. B 109, 075116 (2024).
  75. M. P. Zaletel and F. Pollmann, Isometric tensor network states in two dimensions, Phys. Rev. Lett. 124, 037201 (2020).
  76. S. Gopalakrishnan, Push-down automata as sequential generators of highly entangled states, J. Phys. A 58, 055301 (2025).
  77. S. Sun, J.-H. Zhang, Z. Bi, and Y. You, Holographic view of mixed-state symmetry-protected topological phases in open quantum systems, PRX Quantum 6, 020333 (2025).
  78. T. Soejima, K. Siva, N. Bultinck, S. Chatterjee, F. Pollmann, and M. P. Zaletel, Isometric tensor network representation of string-net liquids, Phys. Rev. B 101, 085117 (2020).
  79. Y.-J. Liu, K. Shtengel, A. Smith, and F. Pollmann, Methods for simulating string-net states and anyons on a digital quantum computer, PRX Quantum 3, 040315 (2022).
  80. R. Luo, Y.-N. Wang, and Z. Bi, Topological holography for mixed-state phases and phase transitions, PRX Quantum 6, 040358 (2025).
  81. M. Qi, R. Sohal, X. Chen, D. T. Stephen, and A. Prem, The symmetry taco: Equivalences between gapped, gapless, and mixed-state spts, arXiv:2507.05335.
  82. S. Schafer-Nameki, A. Tiwari, A. Warman, and C. Zhang, SymTFT approach for mixed states with non-invertible symmetries, arXiv:2507.05350.
  83. M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96, 110405 (2006).
  84. A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96, 110404 (2006).
  85. S. Papanikolaou, K. S. Raman, and E. Fradkin, Topological phases and topological entropy of two-dimensional systems with finite correlation length, Phys. Rev. B 76, 224421 (2007).
  86. C. Castelnovo and C. Chamon, Quantum topological phase transition at the microscopic level, Phys. Rev. B 77, 054433 (2008).
  87. R. Raussendorf and H. J. Briegel, A one-way quantum computer, Phys. Rev. Lett. 86, 5188 (2001).
  88. R. Raussendorf, D. E. Browne, and H. J. Briegel, Measurement-based quantum computation on cluster states, Phys. Rev. A 68, 022312 (2003).
  89. H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest, Measurement-based quantum computation, Nat. Phys. 5, 19 (2009).
  90. M. Van den Nest, A. Miyake, W. Dür, and H. J. Briegel, Universal resources for measurement-based quantum computation, Phys. Rev. Lett. 97, 150504 (2006).
  91. A. Bolt, G. Duclos-Cianci, D. Poulin, and T. M. Stace, Foliated quantum error-correcting codes, Phys. Rev. Lett. 117, 070501 (2016).
  92. T. Okuda, A. P. Mana, and H. Sukeno, Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement, SciPost Phys. 17, 113 (2024).
  93. Z. Li, D. Lee, and B. Yoshida, How much entanglement is needed for topological codes and mixed states with anomalous symmetry?, Phys. Rev. X 15, 021090 (2025).
  94. S. D. Pace, Emergent generalized symmetries in ordered phases and applications to quantum disordering, SciPost Phys. 17, 080 (2024).
  95. J. Haah, An invariant of topologically ordered states under local unitary transformations, Commun. Math. Phys. 342, 771 (2016).
  96. S. D. Pace and X.-G. Wen, Exact emergent higher-form symmetries in bosonic lattice models, Phys. Rev. B 108, 195147 (2023).
  97. Y.-J. Liu, W.-T. Xu, F. Pollmann, and M. Knap, Detecting emergent 1-form symmetries with quantum error correction, arXiv:2502.17572.
  98. I. H. Kim, M. Levin, T.-C. Lin, D. Ranard, and B. Shi, Universal lower bound on topological entanglement entropy, Phys. Rev. Lett. 131, 166601 (2023).
  99. M. Levin, Physical proof of the topological entanglement entropy inequality, Phys. Rev. B 110, 165154 (2024).
  100. L. Zou and J. Haah, Spurious long-range entanglement and replica correlation length, Phys. Rev. B 94, 075151 (2016).
  101. D. J. Williamson, A. Dua, and M. Cheng, Spurious topological entanglement entropy from subsystem symmetries, Phys. Rev. Lett. 122, 140506 (2019).
  102. K. Kato and F. G. S. L. Brandão, Toy model of boundary states with spurious topological entanglement entropy, Phys. Rev. Res. 2, 032005(R) (2020).
  103. A. Uhlmann, The “transition probability” in the state space of a *-algebra, Rep. Math. Phys. 9, 273 (1976).
  104. Y.-J. Liu, K. Shtengel, and F. Pollmann, Simulating two-dimensional topological quantum phase transitions on a digital quantum computer, Phys. Rev. Res. 6, 043256 (2024).
  105. J. Boesl, Y.-J. Liu, W.-T. Xu, F. Pollmann, and M. Knap, Quantum phase transitions between symmetry-enriched fracton phases, Phys. Rev. B 112, 125152 (2025).
  106. J. Boesl, Y.-J. Liu, F. Pollmann, and M. Knap, Skeleton of isometric tensor network states for Abelian string-net models, arXiv:2511.13821.

  107. This follows directly from trace preservation [28]. Given a quantum channel E, viewed as a linear map acting on operators in the Hilbert space, trace preservation ensures that the identity operator I is always a left eigenvector of E with eigenvalue 1(E†[I]=I). Let Pλ denote the projector onto the eigensubspace of the strong symmetry U with distinct eigenvalue λ, satisfying ∑sPs=I. Since each Pλ commutes with E†, the operators Pλ·I form linearly independent left eigenvectors associated with the same eigenvalue 1. Consequently, there must exist the same number of right eigenvectors, i.e., steady states, with eigenvalue 1.

  108. N. Schuch, I. Cirac, and D. Pérez-García, Peps as ground states: Degeneracy and topology, Ann. Phys. (Amsterdam) 325, 2153 (2010).
  109. N. Schuch, D. Pérez-García, and I. Cirac, Classifying quantum phases using matrix product states and projected entangled pair states, Phys. Rev. B 84, 165139 (2011).
  110. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982).

  111. Note that although the Z2 symmetry here is expressed in a different basis from the one used in Sec. 3, generated by ∏iXi, the two are related by a virtual Hadamard rotation H satisfying HXH=Z and H2=I. This basis change cancels out when the virtual legs are contracted and therefore does not alter the bulk isoTNS wavefunction. In other words,the basis change is merely a “gauge” transformation of the 2d tensor-network state.

  112. At g=−1,Wijmn=1 for any i,j,m,n, so the corresponding 2d tensor-network state is a superposition of all Z-basis product state,or equivalently, |++···⟩.

  113. C. Xu and J. E. Moore, Strong-weak coupling self-duality in the two-dimensional quantum phase transition of p+ip superconducting arrays, Phys. Rev. Lett. 93, 047003 (2004).
  114. Y. You and F. von Oppen, Majorana quantum lego, a route towards fracton matter, arXiv:1812.06091.
  115. Y. You, T. Devakul, F. J. Burnell, and S. L. Sondhi, Subsystem symmetry protected topological order, Phys. Rev. B 98, 035112 (2018).
  116. W. Shirley, K. Slagle, and X. Chen, Fractional excitations in foliated fracton phases, Ann. Phys. (Amsterdam) 410, 167922 (2019).
  117. M. Levin and X.-G. Wen, Fermions, strings, and gauge fields in lattice spin models, Phys. Rev. B 67, 245316 (2003).
  118. C. W. von Keyserlingk and F. J. Burnell, Walker-Wang models and axion electrodynamics, Phys. Rev. B 91, 045134 (2015).
  119. Y.-A. Chen and A. Kapustin, Bosonization in three spatial dimensions and a 2-form gauge theory, Phys. Rev. B 100, 245127 (2019).
  120. Y.-A. Chen, T. D. Ellison, and N. Tantivasadakarn, Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions, Phys. Rev. Res. 3, 013056 (2021).
  121. S.-T. Zhou, M. Cheng, T. Rakovszky, C. von Keyserlingk, and T. D. Ellison, Finite-temperature quantum topological order in three dimensions, Phys. Rev. Lett. 135, 040402 (2025).
  122. J. Hauser, Y. Bao, S. Sang, A. Lavasani, U. Agrawal, and M. Fisher, Information dynamics in decohered quantum memory with repeated syndrome measurements: A dual approach, arXiv:2407.07882.
  123. Q. Wang, R. Vasseur, S. Trebst, A. W. Ludwig, and G.-Y. Zhu, Decoherence-induced self-dual criticality in topological states of matter, arXiv:2502.14034.
  124. A.-R. Negari, T. D. Ellison, and T. H. Hsieh, Spacetime Markov length: A diagnostic for fault tolerance via mixed-state phases, arXiv:2412.00193.
  125. M.-Y. Li and P. Ye, Subdimensional entanglement entropy: From virtual response to mixed-state holography, arXiv:2510.15766.
  126. R. Sahay, C. von Keyserlingk, R. Verresen, and C. Zhang, Enforced gaplessness from states with exponentially decaying correlations, arXiv:2503.01977.
  127. W. Ji and X.-G. Wen, Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions, Phys. Rev. Res. 2, 033417 (2020).
  128. D. Gaiotto and J. Kulp, Orbifold groupoids, J. High Energy Phys. 02 (2021) 132.
  129. F. Apruzzi, F. Bonetti, Iñaki García Etxebarria, S. S. Hosseini, and S. Schäfer-Nameki, Symmetry TFTs from string theory, Commun. Math. Phys. 402, 895 (2023).
  130. D. S. Freed, G. W. Moore, and C. Teleman, Topological symmetry in quantum field theory, Quantum Topol. 15, 779 (2024).
  131. K. Su, A. Sarma, M. Bintz, T. Kiely, Y. Bao, M. P. A. Fisher, and C. Xu, Spin liquid and superconductivity emerging from steady states and measurements, Phys. Rev. Lett. 135, 050403 (2025).
  132. Y. Li, F. Pollmann, N. Read, and P. Sala, Highly entangled stationary states from strong symmetries, Phys. Rev. X 15, 011068 (2025).
  133. C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Mixed-state entanglement and quantum error correction, Phys. Rev. A 54, 3824 (1996).
  134. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  135. M. Horodecki, P. Horodecki, and R. Horodecki, Separability of mixed states: Necessary and sufficient conditions, Phys. Lett. A 223, 1 (1996).
  136. J. Eisert and M. B. Plenio, A comparison of entanglement measures, J. Mod. Opt. 46, 145 (1999).
  137. G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).
  138. R. Fan, Y. Wu, Y. Bao, and Z. Dai, No-go theorems for sequential preparation of two-dimensional chiral states via channel-state correspondence, arXiv:2511.19612.
  139. A. Vijay and J. Y. Lee, arXiv:2511.21685.
  140. C. Shu, K. Zhang, Z.-X. Luo, Y. You, and K. Sun, Universal dynamical scaling of strong-to-weak spontaneous symmetry breaking in open quantum systems, arXiv:2603.06363.
  141. D. J. Williamson, C. Delcamp, F. Verstraete, and N. Schuch, On the stability of topological order in tensor network states, Phys. Rev. B 104, 235151 (2021).

  142. In fact, any GHZ-type state |{σ1,σ2,⋯,σLx⟩+∏i=1LxXi|{σ1,σ2,⋯,σLx⟩, with {σi} being a computational Z-basis state, is a valid steady state that exhibit SN-SSB instead of SW-SSB.

  143. R. Peierls, On Ising’s model of ferromagnetism, in Mathematical Proceedings of the Cambridge Philosophical Society (Cambridge University Press, 1936), Vol. 32, pp. 477–481.

  144. This means that it contains all the irreducible representations of G

  145. The left regular representation is defined on a vector space of dimension |G| and satisfies Ug|h⟩=|gh⟩ for g,h∈G.

  146. D. J. Williamson, N. Bultinck, M. Mariën, M. B. Şahinoğlu, J. Haegeman, and F. Verstraete, Matrix product operators for symmetry-protected topological phases: Gauging and edge theories, Phys. Rev. B 94, 205150 (2016).
  147. M. B. Şahinoğlu, D. Williamson, N. Bultinck, M. Mariën, J. Haegeman, N. Schuch, and F. Verstraete, Characterizing topological order with matrix product operators, Ann. Henri Poincaré 22, 563 (2021).
  148. N. Bultinck, M. Mariën, D. J. Williamson, M. B. Şahinoğlu, J. Haegeman, and F. Verstraete, Anyons and matrix product operator algebras, Ann. Phys. (Amsterdam) 378, 183 (2017).

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PRX Quantum 7, 033041– Published 28 August, 2026
  • Received 17 January 2026
  • Accepted 20 July 2026
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