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Mixed-state phases from local reversibility

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proposes that mixed-state phases be defined by locally reversible channel circuits, making the 2D classical loop ensemble a nontrivial phase despite two-way equivalence to a trivial state.

desk verdict A genuinely useful refinement of mixed-state phase equivalence; the loop-state example is compelling, and the one questionable proof step is likely repairable through a missing uniformity lemma. read the letter →

arxiv 2507.02292 v2 pith:SVRWPIPQ submitted 2025-07-03 quant-ph cond-mat.stat-mech

Shengqi Sang , Leonardo A. Lessa , Roger S. K. Mong , Tarun Grover , Chong Wang , Timothy H. Hsieh This is my paper
classification quant-phcond-mat.stat-mech
keywords mixed-statephaseslocalreversibilityMarkovlengthtopologicaldegeneracyclassicalloopensemble1-formsymmetrychannelentropy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that the usual two-way channel definition of mixed-state phases is too coarse, and proposes a refined notion: two mixed states are in the same phase only when a local channel circuit connects them while the evolving state never develops a diverging Markov length. The added condition, local reversibility, preserves properties the old definition discards. In particular, topological degeneracy, measured by the shape of the locally indistinguishable set Q(ρ), is shown to be a phase invariant. Applied to the 2D classical loop ensemble, the results show this purely classical state occupies a nontrivial phase on a torus: its Q(ρ) is a tetrahedron, while a trivial product state has a single-point Q(ρ), so no locally reversible circuit can connect them. The significance is that classical topological memories and strong/weak 1-form symmetry anomalies are stable phase data rather than artifacts of the coarser equivalence relation.

What carries the argument

The central object is the Markov length ξ(ρ), defined through exponential decay of the conditional mutual information I(A:C|B) as the buffer region B widens; Definition 2 requires the evolving state to keep finite Markov length through every layer of a local channel circuit, which guarantees gate-by-gate local reversibility. The locally indistinguishable set Q(ρ), the collection of states that agree with ρ on all simply connected regions and have bounded Markov length, carries the phase invariant: locally reversible circuits give an isometric bijection Q(ρ) ⇌ Q(C[ρ]).

What would settle it

A concrete falsifying test: on a small torus, pick σ0 in Q(ρcl), for instance one of the four winding-parity loop states, run an explicit Def. 2 circuit C whose reversal circuit eC was constructed to reverse C on ρcl, and check whether eC C[σ0] equals σ0 up to superpolynomially small error; if the two differ at polynomial distance, Eq. (30) fails and the claimed bijection Q(ρ) ⇌ Q(C[ρ]) does not follow. Running this check for all four extremal states under random local reversible circuits would settle whether topological degeneracy is genuinely invariant under Def. 2.

Watch

Extended reading notes

Core claim

The paper's central discovery is that replacing global two-way reversibility with local reversibility, enforced by requiring the evolving state's Markov length to stay finite at every step of a local channel circuit, turns topological degeneracy into a mixed-state phase invariant. Concretely, it proves that a locally reversible circuit C induces an isometric bijection between the locally indistinguishable sets Q(ρ) and Q(C[ρ]) for any two states in the same phase. For the 2D classical loop ensemble on a torus, Q(ρcl) is exactly the tetrahedron conv{ρ00cl, ρ01cl, ρ10cl, ρ11cl} of fixed-winding-parity loop states, whereas the trivial product state has Q = {|0...0⟩⟨0...0|}; hence the loop state lies in a different phase under the refined definition, despite explicit two-way local channels connecting it to the trivial state. The same machinery shows that weak 1-form symmetries survive as locally dressed channel symmetries, that the mutual strong-weak anomaly, equivalent to spontaneous breaking of the weak 1-form symmetry, is preserved through the phase, and that every state in the phase carries at least log 2 topological entropy on an annulus, so the non-triviality does not require the system to have nonzero genus.

Load-bearing premise

The argument assumes that the local reversal circuit built gate-by-gate from the reference state also undoes the forward circuit on every other state in its locally indistinguishable set Q(ρ), without separately proving that local reversibility is uniform across Q.

Editorial extensions

If this is right

  • The classical loop state ρcl belongs to a nontrivial mixed-state phase under the Markov length definition, even though explicit O(1)-depth two-way channels relate it to the trivial product state, so the two definitions genuinely disagree.
  • Topological degeneracy is a phase invariant: any state σ in the same phase as ρ has Q in bijection with Q(ρ), so on a torus every state in the phase has the same tetrahedral memory space, and a (M, E, D) topological memory can be transported through the phase by composing with the forward and reversal circuits.
  • Weak unitary 1-form symmetries are locally dressed into channel symmetries, quantum channels supported on smeared loops, and the mutual anomaly or spontaneous breaking with the strong symmetry is preserved, keeping string-like excitations well-defined throughout the phase.
  • Any state in the loop phase has at least log 2 topological entropy on an annulus-shaped region and a nontrivial locally indistinguishable set there, giving a signature that does not require the whole system to have nonzero genus.
  • For abelian higher-form symmetries generally, anomalies and spontaneous symmetry breaking are phase invariants under the refined definition, extending the pure-state results to open quantum systems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's logic implies that two-way equivalence actively erases stable classical memory: since ρcl and the trivial state are two-way connected, any definition based only on global reversibility must place a phase transition somewhere along the dephasing path, and the Markov-length divergence at finite dephasing is the natural location of that transition.
  • The same Q(ρ)-bijection argument should apply to any mixed state with a strong-weak mutual anomaly, not just the loop ensemble; dephased toric code states and measurement-prepared gauge states are natural places to look for the same locally reversible classification.
  • Channel-symmetry dressing suggests that in open systems symmetry is intrinsically process-dependent: a weak symmetry of one state can become a non-unitary channel symmetry of an equivalent state, so anomaly matching conditions should be formulated for pairs of locally reversible processes rather than for states alone.
  • A practical testable extension is to measure Markov conditional mutual information and annulus topological degeneracy in a decohered loop ensemble; the paper's results predict that the topological lower bound log 2 appears only at the critical dephasing strength, not for intermediate couplings.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a refined definition of mixed-state phase equivalence based on locally reversible channel circuits, implemented through a finite Markov length condition (Def. 2). It claims that topological degeneracy—defined via locally indistinguishable sets Q(ρ)—is invariant under this equivalence, and that weak 1-form symmetries are dressed into channel symmetries, preserving anomalies. The central example is the 2D classical loop ensemble, which is shown to be non-trivial under Def. 2 despite being two-way connected to a trivial state. The paper also provides a lower bound on topological entropy for states in the same phase and numerical evidence for the phase transition along the two-way path.

Significance. The proposed definition is a natural and potentially important refinement of the two-way channel definition, as it distinguishes classical topological memory from trivial states. The concepts of channel symmetry and local reversibility as a phase condition are likely to be useful beyond the example. The paper is careful to provide auxiliary numerical checks and explicit constructions for the loop ensemble. However, the proof of the central invariance theorem has a gap that needs to be filled.

major comments (3)
  1. [III.C, Eq. (30)] The claim that the reversal circuit eC satisfies eC C[σ0] = σ0 for every σ0 ∈ Q(ρ0) is not proven by the preceding argument. The argument establishes that each σ_t has finite Markov length and is locally indistinguishable from ρ_t, but finite Markov length alone guarantees only the existence of some local reversal for each gate on σ_t, not that the particular gates of eC (constructed via Petz recovery maps on ρ_t) also reverse the forward gates on σ_t. Local indistinguishability on simply connected regions implies equality of the reduced states that enter the Petz recovery construction, so a uniform reversal lemma should hold; please provide this step explicitly. Without it, the bijection Q(ρ) ⇌ Q(C[ρ]) is not established.
  2. [III.B, Eq. (25) and following] The argument that conv{ρ^{sx,sy}} equals Q(ρcl) only examines stabilizer states in Q(ρcl). Since Q(ρcl) as defined includes arbitrary states with finite Markov length, one must also exclude non-stabilizer candidates, such as coherent superpositions of loop configurations, which may be locally indistinguishable from ρcl. The main conclusion that ρcl is non-trivial only requires Q(ρcl) to contain at least two states, so this gap does not affect the primary no-go result, but the stated equality needs a complete proof.
  3. [III.C, Eq. (27)] The manipulation σ1 = C1 D[ρ0] = C1 D eC1 C1[ρ0] assumes the local-partition identity of Eq. (16) but this is not stated; please clarify the use of the localizability property.
minor comments (5)
  1. [II.B, footnote 2] The expression "eO(1)2" appears to be a typo for "e^{O(1)}" or a similar notation; please fix.
  2. [IV.A, Eq. (40) and text] The decomposition C = CR CR should distinguish the two factors (e.g., C = C_R C_{\bar R}) to avoid confusion.
  3. [III.A, Def. 3] The dependence on ξ0 of Q(ρ; ξ0) is dropped later; please state explicitly that all results hold for any sufficiently large but fixed ξ0.
  4. [Appendix B, Eq. (B17)] The statement that ξ(q) measures a correlation area is confusing; consider rewording to clarify that the divergence is in the area scale.
  5. [IV, Eq. (35)] In the second line of Eq. (35), the multiplication order C†[Zω] σ is ambiguous; please clarify that this is a strong symmetry condition (i.e., C†[Zω] σ = σ).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new definition and TD invariant are not rigged to the loop-state example; the disputed Eq. (30) is a proof gap (uniformity of eC on Q(ρ0)), not a circular reduction.

full rationale

The paper's central derivation is self-contained in the relevant sense. Definition 2 introduces local reversibility via finite Markov length, relying on Ref. [29] (a published theorem by two coauthors) and the Petz recovery maps [42]. Both are external, parameter-free results whose stated assumptions do not include topological degeneracy invariance or the nontriviality of the classical loop state, so citing them is independent evidence rather than circularity. The computation of Q(ρcl) as conv{ρ00,ρ01,ρ10,ρ11} is derived from stabilizer generators and the conditional mutual information formula; it is not a fitted input, a renamed known result, or a parameter chosen to force the answer. The phase-invariance argument does contain an unsupported uniformity step: after Eq. (29) proves only ξ(σ_t) ≤ ξ(ρ_t), Eq. (30) asserts that the same reversal circuit eC satisfies eC C[σ0]=σ0 for every σ0∈Q(ρ0). Finite Markov length of each σ_t guarantees the existence of some local reverse for that state, not equality with the particular eC built from ρ_t, and local indistinguishability alone does not force this equality (ρcl and ρ^{sx,sy}_{cl} are locally indistinguishable yet differ globally by non-contractible homology). This is a rigor gap in an otherwise non-circular derivation, not an instance of an equation reducing to its own input by construction: the conclusion is not contained in the premises by definition. No parameter is fitted and then renamed a prediction, and no uniqueness claim from the authors' prior work is invoked to forbid alternatives. Hence the circularity score is 0, with the noted correctness caveat about Eq. (30).

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the local reversibility theorem of Ref [29] plus standard quantum information inequalities. No free parameters are fitted; the only hand-chosen constant is the Markov length cutoff ξ0 in the definition, which is argued to be immaterial. No new physical entities are introduced; channel symmetry is a concept, not a postulated object.

free parameters (1)
  • Markov length cutoff ξ0
    Definitional cutoff in Def. 2 and Def. 3; assumed O(1) and sufficiently large. The paper argues results are independent of the precise value in the thermodynamic limit, so it is not fitted to data.
assumptions (5)
  • domain assumption Local reversibility theorem (Ref [29]): any local channel gate acting on a finite-Markov-length state can be locally reversed by another local channel.
    Invoked throughout Secs. II.B and III.C to build the reversal circuit eC and to define the refined phase equivalence. Proven in the authors' prior work, not re-derived here.
  • standard math Data processing inequality for conditional mutual information and for trace distance.
    Used in Sec. III.C Eq. (28) to bound Markov length and in Eq. (31) to preserve distances between states in Q(ρ).
  • standard math Stabilizer CMI formula: I(A:C|B) is the minimum number of stabilizer generators (over equivalent generating sets) whose support intersects both A and C.
    Used in Sec. III.B to compute Markov length of the loop stabilizer states and to characterize Q(ρcl). Citation [45].
  • standard math Existence of universal recovery maps (Petz map, Ref [42]) that approximately reverse local channels on finite-Markov-length states.
    Underlies the local reversibility theorem and is used in comment 4 under Def. 3 to construct local channels D mapping ρ to σ.
  • domain assumption Physical mixed states of interest have finite Markov length.
    Motivates Def. 2. The paper notes Gibbs states of commuting Hamiltonians have zero Markov length and quantum Gibbs states have finite Markov length (Ref [50]).

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Pith. "Pith review of Mixed-state phases from local reversibility." pith.science (2026). https://pith.science/paper/SVRWPIPQ

@misc{pith2026250702292,
  author       = {Pith},
  title        = {Pith review of: Mixed-state phases from local reversibility},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SVRWPIPQ}},
  note         = {Machine review of arXiv:2507.02292}
}
read the original abstract

We propose a refined definition of mixed-state phase equivalence based on locally reversible channel circuits. We show that such circuits preserve topological degeneracy and the locality of all operators including both strong and weak symmetries. Under a locally reversible channel, weak unitary symmetries are locally dressed into channel symmetries, a new generalization of symmetry for open quantum systems. For abelian higher-form symmetries, we show the refined definition preserves anomalies and spontaneous breaking of such symmetries within a phase. As a primary example, a two-dimensional classical loop ensemble is trivial under the previously adopted definition of mixed-state phases. However, it has non-trivial topological degeneracy arising from a mutual anomaly between strong and weak 1-form symmetries, and our results show that it is not connected to a trivial state via locally reversible channel circuits.

Figures

Figures reproduced from arXiv: 2507.02292 by the authors.

Figure 1
Figure 1. (a) A channel circuit acting on ρ is locally re￾versible if each of its gates (e.g. the one highlighted in red) is reversible, meaning that the gate’s action upon the state at the time can be reversed locally by some other gate (the gate containing ∼). As such, local reversibility is a property of both the circuit and the input state ρ. Locally reversible circuits are natural extensions of local unitary circuits, wh… view at source ↗
Figure 2
Figure 2. (a) Illustration of local reversibility of a single gate. We say a channel gate [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Construction of a stabilizer state σ belonging to Q(ρcl). We now argue that this inclusion is an equality. Con￾sider any stabilizer state in Q(ρcl), which must be locally indistinguishable from ρcl and have finite Markov length. All Av terms must be retained as stabilizers; otherwise, the loop constraint would be violated at some vertices, contradicting local indistinguishability Moreover, no new stabilizer whose su… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Illustration of ’t Hooft anomaly / SSB of 1-form [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Entropic measures of ρ tr→cl(q) that signal the phase transition at qc = 1/2: (Left) the non-extensive correction ∆SA(q) to the entanglement entropy, whose magnitude matches the topological entropy γ in the thermodynamic limit; and (Right) the CMI I(A:C|B) in the Marko…

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Forward citations

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Reference graph

Works this paper leans on

71 extracted references · 37 canonical work pages · cited by 4 Pith papers

  1. [29]

    Coser and D

    A. Coser and D. P´ erez-Garc ´ ıa, Classification of phases for mixed states via fast dissipative evolution, Quantum 3, 174 (2019)

    work page 2019

  2. [1]

    From the trivial state to the classical loop state 14

    work page

  3. [2]

    Critical behavior 14

    work page

  4. [3]

    INTRODUCTION Quantum information insights have led to substantial progress in understanding quantum many-body phenom- ena

    Local non-reversibility and anomaly breaking 15 References 16 I. INTRODUCTION Quantum information insights have led to substantial progress in understanding quantum many-body phenom- ena. Among them, a circuit-based definition of quantum phases of matter has become a cornerstone in classify- ing and characterizing gapped quantum phases of mat- ter [1, 2]....

    arXiv 2025

  5. [4]

    III the topological degeneracy of mixed-states via local indistinguishability, which gen- eralizes the topological degeneracy in topologically or- dered ground states

    We define in Sec. III the topological degeneracy of mixed-states via local indistinguishability, which gen- eralizes the topological degeneracy in topologically or- dered ground states. We prove that topological degen- eracy is invariant in each mixed-state phase defined using locally reversible channel circuits

    work page

  6. [5]

    IV that the spontaneous symme- try breaking of 1-form symmetries, or equivalently the ’t Hooft anomaly associated with the symmetries, is also a phase invariant

    We prove in Sec. IV that the spontaneous symme- try breaking of 1-form symmetries, or equivalently the ’t Hooft anomaly associated with the symmetries, is also a phase invariant. More concretely, if ρ has spontaneously broken 1-form symmetries, then any other state σ in the same phase as ρ also possesses a dressed 1-form symmetry that is spontaneously bro...

    work page

  7. [6]

    topological entanglement entropy

    We show in Sec. IV A how the broken weak 1-form symmetry can be detected by a non-zero topological entropy in the Levin-Wen partition [32], as well as by a non-trivial topological degeneracy on an annulus- shaped subregion. (a) (b) Figure 1. (a) A channel circuit acting on ρ is locally re- versible if each of its gates ( e.g. the one highlighted in red) i...

    work page

  8. [7]

    As we will illustrate with the ρcl example, the constraint removes ‘unphys- ical’ locally indistinguishable states

    The finite-Markov-length constraint is necessary for the definition to be meaningful. As we will illustrate with the ρcl example, the constraint removes ‘unphys- ical’ locally indistinguishable states. These unphysi- cal degeneracies are unstable and do not carry over to other states in the same phase. We expect that Q(ρ; ξ0)’s dependence on ξ0 = O(1) is ...

    work page

Show all 71 references
  1. [8]

    To see this, it suffices to check that σ3 is locally indistinguishable from ρ and has a finite Markov length

    Q(ρ) defines a convex set, i.e., if σ1, σ2 ∈ Q(ρ), then ∀p ∈ [0, 1], σ3 ≡ pσ1 + (1 − p)σ2 ∈ Q(ρ). To see this, it suffices to check that σ3 is locally indistinguishable from ρ and has a finite Markov length. The former follows from: σ3,A = pσ1,A + (1 − p)σ2,A = ρA for any A. T...

  2. [9]

    Reflexivity: If ρ ∈ Q(σ), then σ ∈ Q(ρ)

  3. [10]

    fidelity observable

    A state σ belongs to Q(ρ) if and only if for any simply connected region A away from Λ’s boundary, there exists a channel D acting on A only satisfying D[ρ] = σ. To prove the ‘if’ claim, we let B be a width r buffer region surrounding A, and C be the rest of the system. Then b...

  4. [11]

    Balasubramanian, M

    S. Balasubramanian, M. Davydova, and E. Lake, A local automaton for the 2d toric code (2025), arXiv:2412.19803 [quant-ph]

    2025 arXiv

  5. [12]

    tr(E †[O] · ρ) = tr(O · E[ρ]) = λ Proof. (1⇔2) Using the Stinespring dilation of the chan- nel E: E[ρ] = trA[U (ρ ⊗ |0⟩⟨0|A)U †] (A1) Then we have: O · E[ρ] = λE[ρ] ⇔ O · U (ρ ⊗ |0⟩⟨0|A)U † = λU (ρ ⊗ |0⟩⟨0|A)U † ⇔ U †OU · (ρ ⊗ |0⟩⟨0|A) = λρ ⊗ |0⟩⟨0|A ⇔ A⟨0|U †OU |0⟩A · ρ = λρ ...

  6. [13]

    For q = 1 /2, the classical loop state with no non-contractible loops ρcl = ρ00 cl is reached

    F rom the trivial state to the classical loop state Alternatively, to continuously transition from ρtr = I/ dim(H) to ρcl, we first replace ρtr with the state |0⟩ ≡ |00 · · ·0⟩, also in the trivial phase, via the replacement channel R|0⟩ ≡ |0⟩⟨0| tr[·], and then dephase the pl...

  7. [14]

    Critical behavior With more work, we can also analyze the critical be- havior near the critical point qc = 1/2. Because of our in- terest in the CMI I(A:C|B) for both the Markov and the Levin-Wen partitions, we will focus on the deviations of the entropy SA(q) from the extensi...

  8. [15]

    (B17) not only gives the correct critical behavior at q ≈ qc, but is also accurate even for q/qc ≪ 1 (and |A| large), far from the regime ( q ≈ qc) it was initially derived from

    We were able to verify that the approximation given by Eq. (B17) not only gives the correct critical behavior at q ≈ qc, but is also accurate even for q/qc ≪ 1 (and |A| large), far from the regime ( q ≈ qc) it was initially derived from

  9. [16]

    Local non-reversibility and anomaly breaking We finish the appendix by explaining in more details how ρtr and ρcl are two-way connected, even though they are in different phases. Apart from the divergence of the Markov length that was already discussed, another way to see that...

  10. [17]

    Chen, Z.-C

    X. Chen, Z.-C. Gu, and X.-G. Wen, Local unitary trans- formation, long-range quantum entanglement, wave func- tion renormalization, and topological order, Physical Re- view B—Condensed Matter and Materials Physics 82, 155138 (2010)

    2010

  11. [18]

    M. B. Hastings and X.-G. Wen, Quasiadiabatic contin- uation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance, Physical review b 72, 045141 (2005)

    2005

  12. [19]

    Verstraete, M

    F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, Quan- tum computation and quantum-state engineering driven by dissipation, Nature Physics 5, 633 (2009)

    2009

  13. [20]

    Diehl, A

    S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. B¨ uchler, and P. Zoller, Quantum states and phases in driven open quantum systems with cold atoms, Nature Physics 4, 878 (2008)

    2008

  14. [21]

    symmetry pullback

    and [22], with former calling it “symmetry pullback”. There, however, the dressed symmetry is defined on an enlarged Hilbert space, preserving its unitarity. After tracing out the ancilla, it coincides with our definition via the dual channel. 11 when the two states are approx...

    2020

  15. [22]

    R. Fan, Y. Bao, E. Altman, and A. Vishwanath, Diag- nostics of mixed-state topological order and breakdown of quantum memory, arXiv preprint arXiv:2301.05689 (2023)

    2023 arXiv

  16. [23]

    J. Y. Lee, C.-M. Jian, and C. Xu, Quantum criticality under decoherence or weak measurement, arXiv preprint arXiv:2301.05238 (2023)

    2023 arXiv

  17. [24]

    Y. Zou, S. Sang, and T. H. Hsieh, Channeling quantum criticality, Physical Review Letters 130, 250403 (2023)

    2023

  18. [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)

    2023

  19. [26]

    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 (2022), arXiv:2208.11699 [cond-mat.str-el]

    2022 arXiv

  20. [27]

    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)

    2023

  21. [28]

    M. B. Hastings, Topological order at nonzero tempera- ture, Physical review letters 107, 210501 (2011)

    2011

  22. [30]

    Ma and C

    R. Ma and C. Wang, Average symmetry-protected topo- logical phases, Physical Review X 13, 031016 (2023)

    2023

  23. [31]

    Ma, J.-H

    R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Topological phases with average symmetries: the de- cohered, the disordered, and the intrinsic (2023), arXiv:2305.16399 [cond-mat.str-el]

    2023 arXiv

  24. [32]

    Ma and A

    R. Ma and A. Turzillo, Symmetry protected topologi- cal phases of mixed states in the doubled space, arXiv preprint arXiv:2403.13280 (2024)

    2024 arXiv

  25. [33]

    Chen and T

    Y.-H. Chen and T. Grover, Separability transitions in topological states induced by local decoherence, arXiv preprint arXiv:2309.11879 (2023)

    2023 arXiv

  26. [34]

    Chen and T

    Y.-H. Chen and T. Grover, Symmetry-enforced many- body separability transitions (2023), arXiv:2310.07286 [quant-ph]

    2023 arXiv

  27. [35]

    H. Xue, J. Y. Lee, and Y. Bao, Tensor network formula- tion of symmetry protected topological phases in mixed states (2024), arXiv:2403.17069 [cond-mat]

    2024 arXiv

  28. [36]

    L. A. Lessa, M. Cheng, and C. Wang, Mixed-state quantum anomaly and multipartite entanglement (2024), arXiv:2401.17357 [cond-mat.str-el]

    2024 arXiv

  29. [37]

    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 preprint arXiv:2503.12792 (2025)

    2025 arXiv

  30. [38]

    Ellison and M

    T. Ellison and M. Cheng, Towards a classification of mixed-state topological orders in two dimensions, arXiv preprint arXiv:2405.02390 (2024)

    2024 arXiv

  31. [39]

    Z. Wang, Z. Wu, and Z. Wang, Intrinsic mixed- state topological order without quantum memory, arXiv preprint arXiv:2307.13758 (2023)

    2023 arXiv

  32. [40]

    Wang and L

    Z. Wang and L. Li, Anomaly in open quantum systems and its implications on mixed-state quantum phases, arXiv preprint arXiv:2403.14533 (2024)

    2024 arXiv

  33. [41]

    T.-T. Wang, M. Song, Z. Y. Meng, and T. Grover, An analog of topological entanglement entropy for mixed states (2024), arXiv:2407.20500

    2024 arXiv

  34. [42]

    de Groot, A

    C. de Groot, A. Turzillo, and N. Schuch, Symmetry pro- tected topological order in open quantum systems, Quan- tum 6, 856 (2022)

    2022

  35. [43]

    S. Sang, Y. Zou, and T. H. Hsieh, Mixed-state quantum phases: Renormalization and quantum error correction, Phys. Rev. X 14, 031044 (2024)

    2024

  36. [44]

    Rakovszky, S

    T. Rakovszky, S. Gopalakrishnan, and C. von Keyser- lingk, Defining stable phases of open quantum systems, Phys. Rev. X 14, 041031 (2024)

    2024

  37. [45]

    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)

    2025

  38. [46]

    S.-T. Zhou, M. Cheng, T. Rakovszky, C. von Keyser- lingk, and T. D. Ellison, Finite-temperature quantum topological order in three dimensions, arXiv preprint arXiv:2503.02928 (2025)

    2025 arXiv

  39. [47]

    Bravyi, M

    S. Bravyi, M. B. Hastings, and F. Verstraete, Lieb- robinson bounds and the generation of correlations and topological quantum order, Physical review letters 97, 050401 (2006)

    2006

  40. [48]

    Levin and X.-G

    M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96, 110405 (2006)

    2006

  41. [49]

    Castelnovo and C

    C. Castelnovo and C. Chamon, Topological order and topological entropy in classical systems, Physical Review B—Condensed Matter and Materials Physics 76, 174416 (2007)

    2007

  42. [50]

    Zhang, Y

    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)

    2025

  43. [51]

    Kitaev and J

    A. Kitaev and J. Preskill, Topological entanglement en- tropy, Phys. Rev. Lett. 96, 110404 (2006)

    2006

  44. [52]

    Castelnovo and C

    C. Castelnovo and C. Chamon, Entanglement and topo- logical entropy of the toric code at finite temperature, Phys. Rev. B 76, 184442 (2007). 17

    2007

  45. [53]

    T.-C. Lu, T. H. Hsieh, and T. Grover, Detecting topo- logical order at finite temperature using entanglement negativity, Phys. Rev. Lett. 125, 116801 (2020)

    2020

  46. [54]

    Brown and D

    W. Brown and D. Poulin, Quantum markov net- works and commuting hamiltonians, arXiv preprint arXiv:1206.0755 (2012)

    2012 arXiv

  47. [55]

    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), arXiv:2405.03639 [quant-ph]

    2025 arXiv

  48. [56]

    P. Sala, S. Gopalakrishnan, M. Oshikawa, and Y. You, Spontaneous strong symmetry breaking in open systems: Purification perspective, Physical Review B 110, 155150 (2024)

    2024

  49. [57]

    J. Y. Lee, C.-M. Jian, and C. Xu, Quantum Criticality Under Decoherence or Weak Measurement, PRX Quan- tum 4, 030317 (2023)

    2023

  50. [58]

    Junge, R

    M. Junge, R. Renner, D. Sutter, M. M. Wilde, and A. Winter, Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy, Annales Henri Poincar´ e19, 2955 (2018)

    2018

  51. [59]

    Li and R

    Z. Li and R. S. Mong, Replica topological order in quan- tum mixed states and quantum error correction, arXiv preprint arXiv:2402.09516 (2024)

    2024 arXiv

  52. [60]

    B. Shi, K. Kato, and I. H. Kim, Fusion rules from entan- glement, Annals of Physics 418, 168164 (2020)

    2020

  53. [61]

    S. Sang, Z. Li, T. H. Hsieh, and B. Yoshida, Ultrafast entanglement dynamics in monitored quantum circuits, PRX Quantum 4, 040332 (2023)

    2023

  54. [62]

    M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, 2010)

    2010

  55. [63]

    J. Y. Lee, Exact Calculations of Coherent Information for Toric Codes under Decoherence: Identifying the Fun- damental Error Threshold, Physical Review Letters 134, 250601 (2025)

    2025

  56. [64]

    Levin, A physical proof of the topological entangle- ment entropy inequality (2024)

    M. Levin, A physical proof of the topological entangle- ment entropy inequality (2024)

    2024

  57. [65]

    I. H. Kim, Long-range entanglement is necessary for a topological storage of quantum information, Physical re- view letters 111, 080503 (2013)

    2013

  58. [66]

    Chen and C

    C.-F. Chen and C. Rouz´ e, Quantum gibbs states are locally markovian, arXiv preprint arXiv:2504.02208 (2025)

    2025 arXiv

  59. [67]

    Kato and F

    K. Kato and F. G. Brandao, Quantum approximate markov chains are thermal, Communications in Math- ematical Physics 370, 117 (2019)

    2019

  60. [68]

    T.-H. Yang, B. Shi, and J. Y. Lee, Topological mixed states: Axiomatic approaches and phases of matter, arXiv preprint arXiv:2506.04221 (2025)

    2025

  61. [69]

    Placke, V

    B. Placke, V. Khemani, and T. Rakovszky, Emergent symmetries in open quantum systems, to be appeared (2025)

    2025

  62. [70]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics 43, 4452 (2002)

    2002

  63. [71]

    Y. Bao, R. Fan, A. Vishwanath, and E. Altman, Mixed- state topological order and the errorfield double formu- lation of decoherence-induced transitions, arXiv preprint arXiv:2301.05687 (2023)

    2023 arXiv

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