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arxiv: 2510.22542 · v4 · pith:4GIFO5U7new · submitted 2025-10-26 · 🪐 quant-ph · cond-mat.stat-mech· hep-th

Krylov Complexity and Mixed-State Phase Transition

Hung-Hsuan Teh , Takahiro Orito This is my paper

Pith reviewed 2026-05-25 07:47 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechhep-th
keywords Krylov complexitymixed-state phase transitionSWSSBdecoherencequantum channelsimaginary-time evolutionerror proliferationarea-to-volume transition
0
0 comments X

The pith

Krylov complexity stays nonsingular across SWSSB crossovers but develops a singular area-to-volume transition exactly at genuine mixed-state phase transitions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a mapping that turns any decoherence process on a density matrix into an imaginary-time evolution of a pure state in a doubled Hilbert space. Expanding that evolution in the Krylov basis produces a direct correspondence in which the nth basis vector is an n-error state created by the decoherence. When this construction is applied to two concrete dephasing channels, the resulting Krylov complexity remains smooth through strong-to-weak spontaneous symmetry-breaking crossovers yet exhibits a sharp change from area-law to volume-law scaling precisely when a genuine SWSSB phase transition occurs in the mixed state.

Core claim

By vectorizing the density matrix and expanding the resulting imaginary-time evolution in Krylov space, the nth Krylov basis vector is identified with an n-error state; this identification shows that Krylov complexity is nonsingular for SWSSB crossovers while displaying a singular area-to-volume-law transition that marks genuine mixed-state SWSSB phase transitions.

What carries the argument

The vectorization of the density matrix into a pure state in the doubled Hilbert space, followed by its expansion in the Krylov basis where each successive vector corresponds to an additional error generated by decoherence.

If this is right

  • Krylov complexity functions as a diagnostic that is smooth at crossovers but singular at genuine mixed-state phase transitions.
  • The rate of complexity growth is tied directly to the rate of error proliferation under decoherence.
  • Genuine SWSSB transitions produce an abrupt change in the scaling of complexity with system size, from area law to volume law.
  • The same framework applies to any decoherence channel that can be written as imaginary-time evolution after vectorization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same error-to-Krylov mapping could be used to track how other noise channels affect complexity growth beyond the two dephasing examples.
  • If the singularity is confirmed, complexity-based diagnostics might distinguish intrinsic mixed-state transitions from finite-size or crossover effects in larger open systems.
  • The construction suggests a route to relate operator growth under open dynamics to the proliferation of logical errors in quantum error correction.

Load-bearing premise

The nth Krylov basis vector corresponds to an n-error state generated by the decoherence process.

What would settle it

Compute the Krylov complexity versus subsystem size or evolution time in one of the paper's dephasing-channel models and check whether a non-analytic jump from area-law to volume-law scaling appears exactly at the parameter value identified as the SWSSB transition point.

Figures

Figures reproduced from arXiv: 2510.22542 by Hung-Hsuan Teh, Takahiro Orito.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic illustration of the wave-packet dy [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Time evolution of the wave-packets in the Krylov [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Time evolution of the wave packets in the Krylov [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

We establish a unified framework connecting decoherence and quantum complexity. By vectorizing the density matrix into a pure state in a double Hilbert space, a decoherence process is mapped to an imaginary-time evolution. Expanding this evolution in the Krylov space, we find that the $n$-th Krylov basis corresponds to an $n$-error state generated by the decoherence, providing a natural bridge between error proliferation and complexity growth. Using two dephasing quantum channels as concrete examples, we show that the Krylov complexity remains nonsingular for strong-to-weak spontaneous symmetry-breaking (SWSSB) crossovers, while it exhibits a singular area-to-volume-law transition for genuine SWSSB phase transitions, intrinsic to mixed states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a unified framework connecting decoherence and quantum complexity: vectorizing the density matrix maps a decoherence process to imaginary-time evolution in a doubled Hilbert space; expanding this evolution in Krylov space yields basis vectors that the authors identify with n-error states, thereby linking error proliferation to complexity growth. Using two concrete dephasing channels as examples, the paper reports that Krylov complexity remains nonsingular across strong-to-weak spontaneous symmetry-breaking (SWSSB) crossovers but exhibits a singular area-to-volume-law transition precisely at genuine SWSSB phase transitions intrinsic to mixed states.

Significance. If the claimed correspondence between Krylov vectors and error sectors can be established rigorously and independently of the specific channels, the work would supply a new, complexity-based diagnostic capable of distinguishing genuine mixed-state phase transitions from crossovers, extending standard vectorization and Krylov techniques to open-system diagnostics.

major comments (2)
  1. [Abstract] Abstract (and the paragraph introducing the Krylov expansion): the identification of the n-th Krylov basis vector with an n-error state generated by the decoherence process is presented as providing the 'natural bridge' between error proliferation and complexity growth, yet the manuscript supplies no general derivation showing that higher-order Krylov vectors remain confined to definite error sectors under the Liouvillian superoperator; if mixing or phase factors appear, the reported singularity cannot be attributed to the mixed-state transition itself.
  2. [Examples section] The two dephasing-channel examples: the distinction between nonsingular behavior for crossovers and singular area-to-volume transition for genuine SWSSB is load-bearing for the central claim, but the text does not demonstrate that this distinction survives when the error-counting interpretation is relaxed or when the channels are deformed; an explicit check that the Krylov basis vectors track physical error number without sector mixing is required.
minor comments (2)
  1. Notation for the doubled Hilbert space and the precise definition of the vectorized imaginary-time evolution should be stated once in a dedicated preliminary section rather than introduced piecemeal.
  2. [Abstract] The abstract would be clearer if it named the two dephasing channels explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the insightful comments, which help clarify the scope of our results. We address each major comment below and plan revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the paragraph introducing the Krylov expansion): the identification of the n-th Krylov basis vector with an n-error state generated by the decoherence process is presented as providing the 'natural bridge' between error proliferation and complexity growth, yet the manuscript supplies no general derivation showing that higher-order Krylov vectors remain confined to definite error sectors under the Liouvillian superoperator; if mixing or phase factors appear, the reported singularity cannot be attributed to the mixed-state transition itself.

    Authors: The manuscript establishes the correspondence explicitly for the dephasing channels under consideration, where the Liouvillian superoperator preserves the error sectors due to its diagonal structure in the error basis. A fully general derivation for arbitrary channels is not provided and would require additional assumptions on the form of the decoherence. We will revise the abstract to specify that the identification holds in the studied models and add a brief discussion of the conditions for sector preservation. revision: yes

  2. Referee: [Examples section] The two dephasing-channel examples: the distinction between nonsingular behavior for crossovers and singular area-to-volume transition for genuine SWSSB is load-bearing for the central claim, but the text does not demonstrate that this distinction survives when the error-counting interpretation is relaxed or when the channels are deformed; an explicit check that the Krylov basis vectors track physical error number without sector mixing is required.

    Authors: We agree that an explicit demonstration is necessary. In the revised manuscript, we will include calculations verifying that the Krylov basis vectors correspond to states with definite error number, showing no mixing between sectors for both dephasing channels. This will support that the observed singularity is tied to the mixed-state transition. We note that the distinction is shown within the dephasing models; deforming the channels or relaxing the interpretation is outside the current scope but could be explored in future work. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses standard mappings without self-referential reduction.

full rationale

The paper begins with the standard vectorization of the density matrix to convert decoherence into imaginary-time evolution under the Liouvillian, then expands that evolution in the Krylov basis. The stated correspondence between the n-th Krylov vector and an n-error state is presented as a derived observation from that expansion rather than a definitional premise or fitted input. The subsequent distinction between nonsingular complexity for SWSSB crossovers and singular area-to-volume transitions for genuine phase transitions is obtained by direct application to two explicit dephasing channels. No self-citations, ansatzes smuggled via prior work, or uniqueness theorems are invoked as load-bearing steps in the abstract or described chain. The central claim therefore remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two domain-level assumptions: the vectorization mapping that converts decoherence into imaginary-time evolution, and the identification of Krylov basis order with error count. No free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Vectorization of the density matrix into a pure state in the doubled Hilbert space converts a decoherence process into imaginary-time evolution.
    This is the foundational step that allows the subsequent Krylov expansion.
  • ad hoc to paper The n-th Krylov basis vector corresponds to an n-error state generated by the decoherence.
    This identification supplies the direct link between error proliferation and complexity growth.

pith-pipeline@v0.9.0 · 5649 in / 1326 out tokens · 39076 ms · 2026-05-25T07:47:24.926036+00:00 · methodology

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Reference graph

Works this paper leans on

78 extracted references · 78 canonical work pages · 1 internal anchor

  1. [1]

    C. W. Gardiner and P. Zoller,Quantum Noise, 2nd ed., edited by H. Haken (Springer, 2000)

    work page 2000

  2. [2]

    W. H. Zurek, Decoherence and the transition from quantum to classical – revisited (2003), arXiv:quant- ph/0306072 [quant-ph]

    work page arXiv 2003

  3. [3]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topo- logical quantum memory, J. Math. Phys.43, 4452–4505 (2002)

    work page 2002

  4. [4]

    Preskill, Quantum computing in the nisq era and be- yond, Quantum2, 79 (2018)

    J. Preskill, Quantum computing in the nisq era and be- yond, Quantum2, 79 (2018)

    work page 2018

  5. [5]

    Bluvstein, S

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kali- nowski, D. Hangleiter, J. P. Bonilla Ataides, N. Maskara, I. Cong, X. Gao, P. Sales Rodriguez, T. Karolyshyn, G. Semeghini, M. J. Gullans, M. Greiner, V. Vladan, and M. D. Lukin, Logical quantum processor based on reconfigurable atom arrays, Nature626, 58 (2024)

    work page 2024

  6. [6]

    Ebadi, T

    S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Se- meghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pich- ler, W. W. Ho, S. Choi, S. Sachdev, M. Greiner, V. Vladan, and M. D. Lukin, Quantum phases of matter on a 256-atom programmable quantum simulator, Nature 595, 227 (2021)

    work page 2021

  7. [7]

    R. Fan, Y. Bao, E. Altman, and A. Vishwanath, Diag- nostics of mixed-state topological order and breakdown of quantum memory, PRX Quantum5, 020343 (2024)

    work page 2024

  8. [8]

    J. Y. Lee, Y.-Z. You, and C. Xu, Symmetry protected topological phases under decoherence, Quantum9, 1607 (2025)

    work page 2025

  9. [9]

    Ma and C

    R. Ma and C. Wang, Average symmetry-protected topo- logical phases, Phys. Rev. X13, 031016 (2023)

    work page 2023

  10. [10]

    Z. Wang, Z. Wu, and Z. Wang, Intrinsic mixed-state topological order, PRX Quantum6, 010314 (2025)

    work page 2025

  11. [11]

    Sohal and A

    R. Sohal and A. Prem, Noisy approach to intrinsically mixed-state topological order, PRX Quantum6, 010313 (2025)

    work page 2025

  12. [12]

    T. D. Ellison and M. Cheng, Toward a classification of mixed-state topological orders in two dimensions, PRX Quantum6, 010315 (2025)

    work page 2025

  13. [13]

    S. Sang, Y. Zou, and T. H. Hsieh, Mixed-state quantum phases: Renormalization and quantum error correction, Phys. Rev. X14, 031044 (2024)

    work page 2024

  14. [14]

    Zhang, U

    Z. Zhang, U. Agrawal, and S. Vijay, Quantum commu- nication and mixed-state order in decohered symmetry- protected topological states, Phys. Rev. B111, 115141 (2025)

    work page 2025

  15. [15]

    Buˇ ca and T

    B. Buˇ ca and T. Prosen, A note on symmetry reductions of the lindblad equation: transport in constrained open spin chains, New J. of Phys.14, 073007 (2012)

    work page 2012

  16. [16]

    V. V. Albert and L. Jiang, Symmetries and conserved quantities in lindblad master equations, Phys. Rev. A 89, 022118 (2014)

    work page 2014

  17. [17]

    de Groot, A

    C. de Groot, A. Turzillo, and N. Schuch, Symmetry pro- tected topological order in open quantum systems, Quan- tum6, 856 (2022)

    work page 2022

  18. [18]

    Guo and Y

    Y. Guo and Y. Ashida, Two-dimensional symmetry- protected topological phases and transitions in open quantum systems, Phys. Rev. B109, 195420 (2024). 6

    work page 2024

  19. [19]

    Orito, Y

    T. Orito, Y. Kuno, and I. Ichinose, Strong and weak symmetries and their spontaneous symmetry breaking in mixed states emerging from the quantum ising model under multiple decoherence, Phys. Rev. B111, 054106 (2025)

    work page 2025

  20. [20]

    Chen and T

    Y.-H. Chen and T. Grover, Zipping many-body quantum states: a scalable approach to diagonal entropy (2025), arXiv:2502.18898 [quant-ph]

    work page arXiv 2025

  21. [21]

    S´ a and B

    L. S´ a and B. B´ eri, Exactly solvable dissipative dynam- ics and one-form strong-to-weak spontaneous symme- try breaking in interacting two-dimensional spin systems (2025), arXiv:2505.11501 [quant-ph]

    work page arXiv 2025

  22. [22]

    Y. Xue, Z. Cheng, and M. Ippoliti, Simulation of bilayer hamiltonians based on monitored quantum trajectories (2025), arXiv:2509.13440 [quant-ph]

    work page arXiv 2025

  23. [23]

    Luo, Y.-N

    R. Luo, Y.-N. Wang, and Z. Bi, Topological hologra- phy for mixed-state phases and phase transitions (2025), arXiv:2507.06218 [cond-mat.str-el]

    work page arXiv 2025

  24. [24]

    Y. Guo, S. Yang, and X.-J. Yu, Quantum strong-to-weak spontaneous symmetry breaking in decohered one di- mensional critical states (2025), arXiv:2503.14221 [cond- mat.str-el]

    work page arXiv 2025

  25. [25]

    Zhang, T

    Y. Zhang, T. H. Hsieh, Y. B. Kim, and Y. Zou, Probing mixed-state phases on a quantum computer via renyi correlators and variational decoding (2025), arXiv:2505.02900 [quant-ph]

    work page arXiv 2025

  26. [26]

    J. Y. Lee, C.-M. Jian, and C. Xu, Quantum criticality under decoherence or weak measurement, Phys. Rev. X Quantum4, 030317 (2023)

    work page 2023

  27. [27]

    L. A. Lessa, R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Strong-to-weak spontaneous symmetry break- ing in mixed quantum states, PRX Quantum6, 010344 (2025)

    work page 2025

  28. [28]

    Weinstein, Efficient detection of strong-to-weak spon- taneous symmetry breaking via the r´ enyi-1 correlator, Phys

    Z. Weinstein, Efficient detection of strong-to-weak spon- taneous symmetry breaking via the r´ enyi-1 correlator, Phys. Rev. Lett.134, 150405 (2025)

    work page 2025

  29. [29]

    P. Sala, S. Gopalakrishnan, M. Oshikawa, and Y. You, Spontaneous strong symmetry breaking in open sys- tems: Purification perspective, Phys. Rev. B110, 155150 (2024)

    work page 2024

  30. [30]

    Z. Liu, L. Chen, Y. Zhang, S. Zhou, and P. Zhang, Diag- nosing strong-to-weak symmetry breaking via wightman correlators, Communications Physics8, 274 (2025)

    work page 2025

  31. [31]

    L. Chen, N. Sun, and P. Zhang, Strong-to-weak symme- try breaking and entanglement transitions, Phys. Rev. B 111, L060304 (2025)

    work page 2025

  32. [32]

    Zhang, C

    J.-H. Zhang, C. Xu, and Y. Xu, Fluctuation-dissipation theorem and information geometry in open quantum sys- tems (2024), arXiv:2409.18944 [quant-ph]

    work page arXiv 2024

  33. [33]

    Sachdev,Quantum Phase Transitions 2nd edition (Cambridge University Press, 2011)

    S. Sachdev,Quantum Phase Transitions 2nd edition (Cambridge University Press, 2011)

    work page 2011

  34. [34]

    X. Feng, Z. Cheng, and M. Ippoliti, Hardness of observing strong-to-weak symmetry breaking (2025), arXiv:2504.12233 [quant-ph]

    work page arXiv 2025

  35. [35]

    Choi, Completely positive linear maps on complex matrices, Lin

    M.-D. Choi, Completely positive linear maps on complex matrices, Lin. Alg. Appl.10, 285 (1975)

    work page 1975

  36. [36]

    Jamio lkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep

    A. Jamio lkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep. Math. Phys.3, 275 (1972)

    work page 1972

  37. [37]

    M¨ uck and Y

    W. M¨ uck and Y. Yang, Krylov complexity and orthogo- nal polynomials, Nuclear Physics B984, 115948 (2022)

    work page 2022

  38. [38]

    Nandy, A

    P. Nandy, A. S. Matsoukas-Roubeas, P. Mart´ ınez- Azcona, A. Dymarsky, and A. del Campo, Quantum dynamics in krylov space: Methods and applications, Physics Reports1125–1128, 1–82 (2025)

    work page 2025

  39. [39]

    Krylov Complexity

    E. Rabinovici, A. S´ anchez-Garrido, R. Shir, and J. Son- ner, Krylov complexity (2025), arXiv:2507.06286 [hep- th]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  40. [40]

    Baiguera, V

    S. Baiguera, V. Balasubramanian, P. Caputa, S. Chap- man, J. Haferkamp, M. P. Heller, and N. Y. Halpern, Quantum complexity in gravity, quantum field theory, and quantum information science (2025), arXiv:2503.10753 [hep-th]

    work page arXiv 2025

  41. [41]

    Balasubramanian, P

    V. Balasubramanian, P. Caputa, J. M. Magan, and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D106, 046007 (2022)

    work page 2022

  42. [42]

    Erdmenger, S.-K

    J. Erdmenger, S.-K. Jian, and Z.-Y. Xian, Universal chaotic dynamics from krylov space, Journal of High En- ergy Physics2023, 10.1007/jhep08(2023)176 (2023)

    work page doi:10.1007/jhep08(2023)176 2023

  43. [43]

    Alishahiha, S

    M. Alishahiha, S. Banerjee, and M. J. Vasli, Krylov com- plexity as a probe for chaos, The European Physical Jour- nal C85, 749 (2025)

    work page 2025

  44. [44]

    D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Physical Review X9, 10.1103/physrevx.9.041017 (2019)

    work page doi:10.1103/physrevx.9.041017 2019

  45. [45]

    Lee and E.-G

    S. Lee and E.-G. Moon, Robust mixed-state cluster states and spurious topological entanglement negativity (2025), arXiv:2504.16165 [quant-ph]

    work page arXiv 2025

  46. [46]

    B. Min, Y. Zhang, Y. Guo, D. Segal, and Y. Ashida, Mixed-state phase transitions in spin-holstein models, Phys. Rev. B111, 115123 (2025)

    work page 2025

  47. [47]

    Nishimori and G

    H. Nishimori and G. Ortiz,Elements of phase transitions and critical phenomena(Oxford university press, 2011)

    work page 2011

  48. [48]

    See supplemental material for (A) the definition of strong and weak symmetries, (B) mapping from a quantum channel to an imaginary time evolution, (C) the deriva- tion of Lanzcos coefficients forH NN, (D) the derivation of Krylov complexity forH NN, (E) numerical method to calculate R´ enyi-2 correlator, (F) the definition of infnite- range dephasing chan...

    work page

  49. [49]

    Viswanath and G

    V. Viswanath and G. Muller,The Recursion Method: Ap- plications to Many-body Dynamics(Springer, New York, 2008)

    work page 2008

  50. [50]

    L. Li, M. Oshikawa, and Y. Zheng, Noninvertible duality transformation between symmetry-protected topological and spontaneous symmetry breaking phases, Phys. Rev. B108, 214429 (2023)

    work page 2023

  51. [51]

    Caputa, J

    P. Caputa, J. M. Magan, and D. Patramanis, Geometry of krylov complexity, Phys. Rev. Res.4, 013041 (2022)

    work page 2022

  52. [52]

    Hu, W.-Y

    Q. Hu, W.-Y. Zhang, Y. Han, and W.-L. You, Krylov complexity in quantum many-body scars of spin-1 mod- els, Phys. Rev. B111, 165106 (2025)

    work page 2025

  53. [53]

    Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011), january 2011 Special Issue

    U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011), january 2011 Special Issue

    work page 2011

  54. [54]

    Fishman, S

    M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor Software Library for Tensor Network Calcula- tions, SciPost Phys. Codebases , 4 (2022)

    work page 2022

  55. [55]

    N. Sun, P. Zhang, and L. Feng, Scheme to detect the strong-to-weak symmetry breaking via randomized mea- surements, Phys. Rev. Lett.135, 090403 (2025)

    work page 2025

  56. [56]

    Bhattacharyya, D

    A. Bhattacharyya, D. Ghosh, and P. Nandi, Opera- tor growth and krylov complexity in bose-hubbard model, Journal of High Energy Physics2023, 10.1007/jhep12(2023)112 (2023). 7

    work page doi:10.1007/jhep12(2023)112 2023

  57. [57]

    Lipkin, N

    H. Lipkin, N. Meshkov, and A. Glick, Validity of many- body approximation methods for a solvable model: (i). exact solutions and perturbation theory, Nuclear Physics 62, 188 (1965)

    work page 1965

  58. [58]

    Afrasiar, J

    M. Afrasiar, J. K. Basak, B. Dey, K. Pal, and K. Pal, Time evolution of spread complexity in quenched lip- kin–meshkov–glick model, Journal of Statistical Mechan- ics: Theory and Experiment2023, 103101 (2023)

    work page 2023

  59. [59]

    P. H. S. Bento, A. del Campo, and L. C. C´ eleri, Krylov complexity and dynamical phase transition in the quenched lipkin-meshkov-glick model, Phys. Rev. B109, 224304 (2024)

    work page 2024

  60. [60]

    K. Takahashi, Dynamical quantum phase transition, metastable state, and dimensionality reduction: Krylov analysis of fully connected spin models, Physical Review B112, 10.1103/m4jf-7svp (2025)

    work page doi:10.1103/m4jf-7svp 2025

  61. [61]

    Holstein and H

    T. Holstein and H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev.58, 1098 (1940)

    work page 1940

  62. [62]

    J. J. Sakurai and J. Napolitano,Modern quantum me- chanics(Cambridge University Press, 2020)

    work page 2020

  63. [63]

    Y. Kuno, T. Orito, and I. Ichinose, Intrinsic mixed-state topological order in a stabilizer system under stochas- tic decoherence: Strong-to-weak spontaneous symmetry breaking from a percolation point of view, Phys. Rev. B 111, 064111 (2025)

    work page 2025

  64. [64]

    Ma, J.-H

    R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Topological phases with average symmetries: The deco- hered, the disordered, and the intrinsic, Phys. Rev. X15, 021062 (2025)

    work page 2025

  65. [65]

    Guo and S

    Y. Guo and S. Yang, Strong-to-weak spontaneous sym- metry breaking meets average symmetry-protected topo- logical order, Phys. Rev. B111, L201108 (2025)

    work page 2025

  66. [66]

    Y. Kuno, T. Orito, and I. Ichinose, Strong-to-weak spon- taneous symmetry breaking and average symmetry pro- tected topological order in the doubled hilbert space, Phys. Rev. B111, 174110 (2025)

    work page 2025

  67. [67]

    Sang and T

    S. Sang and T. H. Hsieh, Stability of mixed-state quan- tum phases via finite markov length, Phys. Rev. Lett. 134, 070403 (2025)

    work page 2025

  68. [68]

    T.-H. Yang, B. Shi, and J. Y. Lee, Topological mixed states: Axiomatic approaches and phases of matter (2025), arXiv:2506.04221 [cond-mat.str-el]

    work page arXiv 2025

  69. [69]

    S. Sang, L. A. Lessa, R. S. K. Mong, T. Grover, C. Wang, and T. H. Hsieh, Mixed-state phases from local reversibil- ity (2025), arXiv:2507.02292 [quant-ph]

    work page arXiv 2025

  70. [70]

    Y. Kuno, T. Orito, and I. Ichinose, R´ enyi markov length in one-dimensional non-trivial mixed state phases and mixed state phase transitions (2025), arXiv:2505.02125 [quant-ph]

    work page arXiv 2025

  71. [71]

    Colmenarez, S

    L. Colmenarez, S. Kim, and M. M¨ uller, Fundamental thresholds for computational and erasure errors via the coherent information (2025), arXiv:2412.16727 [quant- ph]

    work page arXiv 2025

  72. [72]

    Niwa and J

    R. Niwa and J. Y. Lee, Coherent information for calderbank-shor-steane codes under decoherence, Phys. Rev. A111, 032402 (2025)

    work page 2025

  73. [73]

    J. Y. Lee, Exact calculations of coherent information for toric codes under decoherence: Identifying the fun- damental error threshold, Phys. Rev. Lett.134, 250601 (2025)

    work page 2025

  74. [74]

    Zhou and C

    Y.-N. Zhou and C. Liu, Generalized loschmidt echo and information scrambling in open systems (2024), arXiv:2412.01851 [quant-ph]

    work page arXiv 2024

  75. [75]

    D. J. Yates and A. Mitra, Strong and almost strong modes of floquet spin chains in krylov subspaces, Phys. Rev. B104, 195121 (2021)

    work page 2021

  76. [76]

    Yeh and A

    H.-C. Yeh and A. Mitra, Universal model of floquet op- erator krylov space, Phys. Rev. B110, 155109 (2024)

    work page 2024

  77. [77]

    Nahum, S

    A. Nahum, S. Vijay, and J. Haah, Operator spreading in random unitary circuits, Phys. Rev. X8, 021014 (2018)

    work page 2018

  78. [78]

    Gautschi,Orthogonal Polynomials: Computation and Approximation(Oxford University Press, 2004)

    W. Gautschi,Orthogonal Polynomials: Computation and Approximation(Oxford University Press, 2004). 1 Supplemental Material — Krylov Complexity and Mixed-State Phase Transition Hung-Hsuan Teh The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba, 277-8581, Japan Graduate School of Informatics, Nagoya University, Nagoya, 464-0814, Ja...

    work page 2004