- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Topological Mixed States: Phases of Matter from Axiomatic Approaches

Tai-Hsuan Yang1, Bowen Shi1,2, and Jong Yeon Lee1,3

PDF

Phys. Rev. X 16, 041002 – Published 2 October, 2026

DOI: https://doi.org/10.1103/3bpt-f9pd

Abstract

For closed quantum systems, topological orders are understood through the equivalence classes of ground states of gapped local Hamiltonians. The generalization of this conceptual paradigm to open quantum systems, however, remains elusive, often relying on operational definitions without fundamental principles. Here, we fill this gap by proposing an approach based on three axioms: (i) local recoverability, (ii) absence of long-range correlations, and (iii) spatial uniformity. States that satisfy these axioms are fixed points; requiring the axioms only after coarse-graining promotes each fixed point to an equivalence class, i.e., a phase, presenting the first step toward the axiomatic classification of mixed-state phases of matter: the mixed-state bootstrap program. From these axioms, a rich set of topological data naturally emerges; importantly, these data are robust under relaxation of axioms. For example, each topological mixed state supports locally indistinguishable classical and/or quantum logical memories with distinct responses to topological operations. These data label distinct mixed-state phases, allowing one to distinguish them. We further uncover a hierarchy of secret-sharing constraints: In non-Abelian phases, reliable recovery—even of information that looks purely classical—demands a specific coordination among spatial subregions, a requirement different across non-Abelian classes. This originates from non-Abelian fusion rules that can stay robust under decoherence. Finally, we performed large-scale numerical simulations to corroborate stability: Weakly decohered fixed points respect the axioms once coarse-grained. Surprisingly, the axiom violations at different length scales exhibit an exact crossing at the critical point, enabling its identification without any rescaling or data-collapse ambiguities. These results lay the foundation for a systematic classification of topological states in open quantum systems.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Realistic quantum systems interact with their surroundings and are therefore described by mixed states, for which the usual theory of topological phases is incomplete. We address this conceptual gap by developing a quantum-information-based framework that defines when two mixed states belong to the same topological phase, enabling a systematic classification of mixed-state phases of matter. The framework identifies phase invariants—properties that remain unchanged within a phase—including the amount of quantum information that can be robustly stored (memory capacity) and the ways in which it can be distributed among spatial regions (secret sharing). It also provides a direct numerical diagnostic of phase transitions that avoids ambiguities associated with finite-size scaling. These results establish a general foundation for classifying topological order in open quantum systems.

Article Text

References (142)

  1. 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).
  2. 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).
  3. 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).
  4. 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).
  5. B. Zeng, X. Chen, D.-L. Zhou, X.-G. Wen et al., Quantum Information Meets Quantum Matter (Springer, New York, 2019).
  6. A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. (Amsterdam) 321, 2 (2006).
  7. M. A. Levin and X.-G. Wen, String-net condensation:A physical mechanism for topological phases, Phys. Rev. B 71, 045110 (2005).
  8. E. Rowell, R. Stong, and Z. Wang, On classification of modular tensor categories, Commun. Math. Phys. 292, 343 (2009).
  9. L. Fidkowski and A. Kitaev, Effects of interactions on the topological classification of free fermion systems, Phys. Rev. B 81, 134509 (2010).
  10. B. Shi, K. Kato, and I. H. Kim, Fusion rules from entanglement, Ann. Phys. (Amsterdam) 418, 168164 (2020).
  11. B. Shi and I. H. Kim, Entanglement bootstrap approach for gapped domain walls, Phys. Rev. B 103, 115150 (2021).
  12. J.-L. Huang, J. McGreevy, and B. Shi, Knots and entanglement, SciPost Phys. 14, 141 (2023).
  13. B. Shi, J.-L. Huang, and J. McGreevy, Remote detectability from entanglement bootstrap I: Kirby’s torus trick, SciPost Phys. 18, 126 (2025).
  14. T.-C. Lin and J. McGreevy, Conformal field theory ground states as critical points of an entropy function, Phys. Rev. Lett. 131, 251602 (2023).
  15. I. H. Kim, X. Li, T.-C. Lin, J. McGreevy, and B. Shi, Conformal geometry from entanglement, SciPost Phys. 18, 102 (2025).
  16. I. H. Kim and D. Ranard, Classifying 2D topological phases: Mapping ground states to string-nets, arXiv:2405.17379.
  17. X. Li, T.-C. Lin, J. McGreevy, and B. Shi, Strict area law entanglement versus chirality, Phys. Rev. Lett. 134, 180402 (2025).

  18. For chiral topological orders, we refer to Ref. [17] for more in-depth discussions.

  19. J. Y. Lee, Y.-Z. You, and C. Xu, Symmetry protected topological phases under decoherence, Quantum 9, 1607 (2025).
  20. 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).
  21. J. Y. Lee, C.-M. Jian, and C. Xu, Quantum criticality under decoherence or weak measurement, PRX Quantum 4, 030317 (2023).
  22. Y. Bao, R. Fan, A. Vishwanath, and E. Altman, Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions, Phys. Rev. Lett. 136, 220402 (2026).
  23. Z. Wang, Z. Wu, and Z. Wang, Intrinsic mixed-state topological order, PRX Quantum 6, 010314 (2025).
  24. Y. Zhao and D. E. Liu, Extracting error thresholds through the framework of approximate quantum error correction condition, Phys. Rev. Res. 6, 043258 (2024).
  25. J. Y. Lee, Exact calculations of coherent information for toric codes under decoherence: Identifying the fundamental error threshold, Phys. Rev. Lett. 134, 250601 (2025).
  26. S. Sang and T. H. Hsieh, Stability of mixed-state quantum phases via finite Markov length, Phys. Rev. Lett. 134, 070403 (2025).
  27. R. Niwa and J. Y. Lee, Coherent information for calderbank-shor-steane codes under decoherence, Phys. Rev. A 111, 032402 (2025).
  28. R. Ma and C. Wang, Average symmetry-protected topological phases, Phys. Rev. X 13, 031016 (2023).
  29. S. Sang, Y. Zou, and T. H. Hsieh, Mixed-state quantum phases: Renormalization and quantum error correction, Phys. Rev. X 14, 031044 (2024).
  30. A. Lyons, Understanding stabilizer codes under local decoherence through a general statistical mechanics mapping, arXiv:2403.03955.
  31. A. Coser and D. Pérez-García, Classification of phases for mixed states via fast dissipative evolution, Quantum 3, 174 (2019).
  32. H. Xue, J. Y. Lee, and Y. Bao, Tensor network formulation of symmetry protected topological phases in mixed states, arXiv:2403.17069.
  33. R. Ma and A. Turzillo, Symmetry protected topological phases of mixed states in the doubled space, arXiv:2403.13280.
  34. R. Sohal and A. Prem, A noisy approach to intrinsically mixed-state topological order, PRX Quantum 6, 010313 (2025).
  35. T. D. Ellison and M. Cheng, Toward a classification of mixed-state topological orders in two dimensions, PRX Quantum 6, 010315 (2025).
  36. Z. Zhang, U. Agrawal, and S. Vijay, Quantum communication and mixed-state order in decohered symmetry-protected topological states, Phys. Rev. B 111, 115141 (2025).
  37. Y.-H. Chen and T. Grover, Separability transitions in topological states induced by local decoherence, Phys. Rev. Lett. 132, 170602 (2024).
  38. Y.-H. Chen and T. Grover, Symmetry-enforced many-body separability transitions, PRX Quantum 5, 030310 (2024).
  39. Y. Guo and Y. Ashida, Two-dimensional symmetry-protected topological phases and transitions in open quantum systems, arXiv:2311.12619.
  40. K. Su, Z. Yang, and C.-M. Jian, Tapestry of dualities in decohered quantum error correction codes, Phys. Rev. B 110, 085158 (2024).
  41. T.-C. Lu, Disentangling transitions in topological order induced by boundary decoherence, Phys. Rev. B 110, 125145 (2024).
  42. P. Sala and R. Verresen, Stability and loop models from decohering non-Abelian topological order, Phys. Rev. Lett. 134, 250403 (2025).
  43. Z. Li, D. Lee, and B. Yoshida, How much entanglement is needed for emergent anyons and fermions?, Phys. Rev. X 15, 021090 (2025).
  44. Y. Xu and C.-M. Jian, Average-exact mixed anomalies and compatible phases, Phys. Rev. B 111, 125128 (2025).
  45. Z. Li and R. S. K. Mong, Replica topological order in quantum mixed states and quantum error correction, arXiv:2402.09516.
  46. Y.-H. Chen and T. Grover, Unconventional topological mixed-state transition and critical phase induced by self-dual coherent errors, arXiv:2403.06553.
  47. Z. Wang and L. Li, Anomaly in open quantum systems and its implications on mixed-state quantum phases, PRX Quantum 6, 010347 (2025).
  48. S. Lu, P. Zhu, and Y.-M. Lu, Bilayer construction for mixed state phenomena with strong, weak symmetries and symmetry breakings, arXiv:2411.07174.
  49. 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 [Phys. Rev. B (to be published)].
  50. Y. Ogata, Mixed state topological order: Operator algebraic approach, arXiv:2501.02398.
  51. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2010).
  52. E. H. Lieb and M. B. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys. (N.Y.) 14, 1938 (1973).
  53. D. Petz, Monotonicity of quantum relative entropy revisited, Rev. Math. Phys. 15, 79 (2003).
  54. O. Fawzi and R. Renner, Quantum conditional mutual information and approximate Markov chains, Commun. Math. Phys. 340, 575 (2015).
  55. M. Junge, R. Renner, D. Sutter, M. M. Wilde, and A. Winter, Universal recovery maps and approximate sufficiency of quantum relative entropy, Ann. Henri Poincare 19, 2955 (2018).
  56. O. Ogunnaike, J. Feldmeier, and J. Y. Lee, Unifying emergent hydrodynamics and lindbladian low-energy spectra across symmetries, constraints, and long-range interactions, Phys. Rev. Lett. 131, 220403 (2023).
  57. P. Sala, S. Gopalakrishnan, M. Oshikawa, and Y. You, Spontaneous strong symmetry breaking in open systems: Purification perspective, arXiv:2405.02402.
  58. 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).
  59. 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).
  60. J. Kim, E. Altman, and J. Y. Lee, Error threshold of syk codes from strong-to-weak parity symmetry breaking, arXiv:2410.24225.
  61. 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, arXiv:2409.17530.
  62. B. Buča and T. Prosen, A note on symmetry reductions of the Lindblad equation: Transport in constrained open spin chains, New J. Phys. 14, 073007 (2012).
  63. V. V. Albert and L. Jiang, Symmetries and conserved quantities in Lindblad master equations, Phys. Rev. A 89, 022118 (2014).

  64. By defects, we mean both pointlike defects in the sense of Refs. [65, 66] and line defects, such as domain walls [67, 68, 69]; we discuss these further in Sec. 4.

  65. H. Bombin and M. A. Martin-Delgado, Family of non-Abelian Kitaev models on a lattice: Topological condensation and confinement, Phys. Rev. B 78, 115421 (2008).
  66. B. J. Brown, S. D. Bartlett, A. C. Doherty, and S. D. Barrett, Topological entanglement entropy with a twist, Phys. Rev. Lett. 111, 220402 (2013).
  67. S. Beigi, P. W. Shor, and D. Whalen, The quantum double model with boundary: Condensations and symmetries, Commun. Math. Phys. 306, 663 (2011).
  68. B. Shi and I. H. Kim, Domain wall topological entanglement entropy, Phys. Rev. Lett. 126, 141602 (2021).
  69. M. Buican, R. Geiko, M. Moses, and B. Shi, An algebraic theory of gapped domain wall partons, arXiv:2506.22544.
  70. L. Zou and J. Haah, Spurious long-range entanglement and replica correlation length, Phys. Rev. B 94, 075151 (2016).
  71. L. H. Santos, J. Cano, M. Mulligan, and T. L. Hughes, Symmetry-protected topological interfaces and entanglement sequences, Phys. Rev. B 98, 075131 (2018).
  72. D. J. Williamson, A. Dua, and M. Cheng, spurious topological entanglement entropy from subsystem symmetries, Phys. Rev. Lett. 122, 140506 (2019).
  73. D. T. Stephen, H. Dreyer, M. Iqbal, and N. Schuch, Detecting subsystem symmetry protected topological order via entanglement entropy, Phys. Rev. B 100, 115112 (2019).
  74. 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).
  75. B. Shi, A softer notion of circuit invariant for topological phases (to be published).
  76. 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).

  77. Using purification and strong subadditivity, one can prove that S(BC)+S(C)−S(B)≥I(A:C),I(A:C|B).

  78. A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96, 110404 (2006).
  79. J.-L. Huang, J. McGreevy, and B. Shi, Knots and entanglement, SciPost Phys. 14, 141 (2023).
  80. B. Shi, Immersed figure-8 Annuli and Anyons, Ann. Henri Poincaré 27, 51 (2026).
  81. S. Bravyi, M. B. Hastings, and S. Michalakis, Topological quantum order: Stability under local perturbations, J. Math. Phys. (N.Y.) 51, 093512 (2010).
  82. B. Shi and Y.-M. Lu, Characterizing topological order by the information convex, Phys. Rev. B 99, 035112 (2019).
  83. I. H. Kim, T.-C. Lin, D. Ranard, and B. Shi, Strict area law implies commuting parent Hamiltonian, arXiv:2404.05867.
  84. M. Horodecki, P. W. Shor, and M. B. Ruskai, Entanglement breaking channels, Rev. Math. Phys. 15, 629 (2003).
  85. C. Castelnovo and C. Chamon, Topological order and topological entropy in classical systems, Phys. Rev. B 76, 174416 (2007).
  86. M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96, 110405 (2006).

  87. In the context of the mixed-state phase, the Levin-Wen topological entropy is well defined due to axiom M1. On the other hand, the Kitaev-Preskill topological entanglement entropy [78] stops being a topological invariant in the mixed-state context; see Appendix pp5.

  88. I. H. Kim, Long-range entanglement is necessary for a topological storage of quantum information, Phys. Rev. Lett. 111, 080503 (2013).

  89. It means that the only anyon that has trivial braiding statistics (“transparent”) with all nontrivial anyons is the vacuum.

  90. B. Shi, Verlinde formula from entanglement, Phys. Rev. Res. 2, 023132 (2020).
  91. I. H. Kim, B. Shi, K. Kato, and V. V. Albert, Modular commutator in gapped quantum many-body systems, Phys. Rev. B 106, 075147 (2022).
  92. Y. Zhang, T. Grover, A. Turner, M. Oshikawa, and A. Vishwanath, Quasiparticle statistics and braiding from ground-state entanglement, Phys. Rev. B 85, 235151 (2012).
  93. I. H. Kim, X. Li, T.-C. Lin, J. McGreevy, and B. Shi, Chiral Virasoro algebra from a single wavefunction, Ann. Phys. (Amsterdam) 471, 169849 (2024).

  94. Note that, when we generalize this to non-Abelian cases, charge and flux labels correspond to irreducible representation and conjugacy classes, respectively.

  95. D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd ed. (Wiley, New York, 2004).
  96. M. Levin, Physical proof of the topological entanglement entropy inequality, Phys. Rev. B 110, 165154 (2024).
  97. H.-S. Kim, I. H. Kim, and D. Ranard, Learning state preparation circuits for quantum phases of matter, arXiv:2410.23544.
  98. D. Ranard (2025), private communication about on-going work with Isaac H. Kim, Alexei Kitaev and Milo Moses.
  99. C.-F. Chen and C. Rouzé, Quantum Gibbs states are locally Markovian, arXiv:2504.02208.
  100. J. M. Hammersley and P. Clifford, Markov fields on finite graphs and lattices (1971), https://ora.ox.ac.uk/objects/uuid:4ea849da-1511-4578-bb88-6a8d02f457a6.
  101. H. Casini, M. Huerta, and R. C. Myers, Towards a derivation of holographic entanglement entropy, J. High Energy Phys. 05 (2011) 036.
  102. A. W. Ludwig, Infinite hierarchies of exponents in a diluted ferromagnet and their interpretation, Nucl. Phys. B330, 639 (1990).
  103. J. Cardy, Topics in the conformal field theory of disordered systems, in New Theoretical Approaches to Strongly Correlated Systems, edited by A. M. Tsvelik (Springer, Netherlands, Dordrecht, 2001), pp. 163–171.
  104. V. Gurarie and A. W. W. Ludwig, Conformal algebras of two-dimensional disordered systems, J. Phys. A 35, L377 (2002).

  105. There is another approach closely related to FDLC but it uses local Lindbladian [26]. It may not be completely equivalent to the local channel condition.

  106. M. Levin, Protected edge modes without symmetry, Phys. Rev. X 3, 021009 (2013).
  107. I. H. Kim and B. J. Brown, Ground-state entanglement constrains low-energy excitations, Phys. Rev. B 92, 115139 (2015).
  108. B. Shi, Seeing topological entanglement through the information convex, Phys. Rev. Res. 1, 033048 (2019).
  109. T. Lan, J. C. Wang, and X.-G. Wen, Gapped domain walls, gapped boundaries, and topological degeneracy, Phys. Rev. Lett. 114, 076402 (2015).
  110. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. (N.Y.) 43, 4452 (2002).
  111. A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. (Amsterdam) 303, 2 (2003).

  112. A mixed unitary channel is a quantum channel that can be expressed as a convex combination of unitary operations.

  113. A. Shamir, How to share a secret, Commun. ACM 22, 612 (1979).
  114. R. Cleve, D. Gottesman, and H.-K. Lo, How to share a quantum secret, Phys. Rev. Lett. 83, 648 (1999).
  115. K. Kato, F. Furrer, and M. Murao, Information-theoretical analysis of topological entanglement entropy and multipartite correlations, Phys. Rev. A 93, 022317 (2016).
  116. L. Fiedler, P. Naaijkens, and T. J. Osborne, Jones index, secret sharing and total quantum dimension, New J. Phys. 19, 023039 (2017).
  117. W. P. Thurston and S. Levy, Three-Dimensional Geometry and Topology (Princeton University Press, Princeton, NJ, 2000), Vol. 1.
  118. T. Grover, A. M. Turner, and A. Vishwanath, Entanglement entropy of gapped phases and topological order in three dimensions, Phys. Rev. B 84, 195120 (2011).
  119. B. Swingle and J. McGreevy, Mixed s -sourcery: Building many-body states using bubbles of nothing, Phys. Rev. B 94, 155125 (2016).
  120. N. Tantivasadakarn, A. Vishwanath, and R. Verresen, Hierarchy of topological order from finite-depth unitaries, measurement, and feedforward, PRX Quantum 4, 020339 (2023).
  121. N. Tantivasadakarn, R. Verresen, and A. Vishwanath, Shortest route to non-Abelian topological order on a quantum processor, Phys. Rev. Lett. 131, 060405 (2023).
  122. M. Iqbal, N. Tantivasadakarn, R. Verresen, S. L. Campbell, J. M. Dreiling, C. Figgatt, J. P. Gaebler, J. Johansen, M. Mills, S. A. Moses, J. M. Pino, A. Ransford, M. Rowe, P. Siegfried, R. P. Stutz, M. Foss-Feig, A. Vishwanath, and H. Dreyer, Non-Abelian topological order and anyons on a trapped-ion processor, Nature (London) 626, 505 (2024).
  123. P. Sala, J. Alicea, and R. Verresen, Decoherence and wavefunction deformation of d4 non-abelian topological order, arXiv:2409.12948.
  124. A. Kitaev and L. Kong, Models for gapped boundaries and domain walls, Commun. Math. Phys. 313, 351 (2012).
  125. L. Kong, Anyon condensation and tensor categories, Nucl. Phys. B886, 436 (2014).
  126. L. Kong, Z.-H. Zhang, J. Zhao, and H. Zheng, Higher condensation theory, arXiv:2403.07813.
  127. I. H. Kim, B. Shi, K. Kato, and V. V. Albert, Chiral central charge from a single bulk wave function, Phys. Rev. Lett. 128, 176402 (2022).
  128. S. Vardhan, B. Shi, I. H. Kim, and Y. Zou, Chirality, magic, and quantum correlations in multipartite quantum states, arXiv:2503.10764.
  129. A. Vijay and J. Y. Lee, Information critical phases under decoherence, arXiv:2512.22121.
  130. S. Sang, L. A. Lessa, R. S. K. Mong, T. Grover, C. Wang, and T. H. Hsieh, Mixed-state phases from local reversibility, arXiv:2507.02292.
  131. T.-H. Yang, B. Shi, and J. Y. Lee, Mixed-state topological phases: figures and calculations, GitHub repository, https://github.com/xiaotai-yang/mixed_topo (2026).

  132. The proof of the following statement is nontrivial, but it is essentially identical to the argument in Appendix C in Ref. [10]. We, thus, omit the proof here.

  133. O. Fawzi and R. Renner, Quantum conditional mutual information and approximate markov chains, Commun. Math. Phys. 340, 575 (2015).
  134. K. M. R. Audenaert, A sharp continuity estimate for the von Neumann entropy, J. Phys. A 40, 8127 (2007).
  135. A. Winter, Tight uniform continuity bounds for quantum entropies: Conditional entropy, relative entropy distance and energy constraints, Commun. Math. Phys. 347, 291 (2016).
  136. C. A. Fuchs and J. van de Graaf, Cryptographic distinguishability measures for quantum mechanical states, arXiv:quant-ph/9712042.
  137. I. H. Kim (unpublished).
  138. B. Shi, T. H. Yang, and J. Y. Lee (unpublished).
  139. A. Kitaev, Toward a topological classification of many-body quantum states with short-range entanglement, Simons Center for Geometry and Physics, https://scgp.stonybrook.edu/video_portal/video.php?id=1835 (2011).
  140. A. Hamma, R. Ionicioiu, and P. Zanardi, Bipartite entanglement and entropic boundary law in lattice spin systems, Phys. Rev. A 71, 022315 (2005).
  141. Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, and T. Xiang, Coarse-graining renormalization by higher-order singular value decomposition, Phys. Rev. B 86, 045139 (2012).
  142. M. Levin and C. P. Nave, Tensor renormalization group approach to two-dimensional classical lattice models, Phys. Rev. Lett. 99, 120601 (2007).

Outline

Information

Phys. Rev. X 16, 041002– Published 2 October, 2026
  • Received 8 July 2025
  • Revised 4 April 2026
  • Accepted 20 July 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation