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REVIEW 3 major objections 5 minor 133 references

Approximate Quantum Error Correction at Chiral Topological Edges

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Chiral topological edge states form a new family of approximate quantum error-correcting codes that beat their 1D counterparts under local erasure.

desk verdict A genuinely new AQEC construction with a clean relative-entropy reduction, but the headline 2D-vs-1D robustness comparison rests on an unproven dimensional-reduction assumption and on unpublished replica formulas. read the letter →

arxiv 2608.06258 v1 pith:MUZF3NFI submitted 2026-08-06 quant-ph cond-mat.str-elhep-th

Yuntai Song , Zejun Liu , Zhencheng Wang , Jong Yeon Lee , Bowen Shi This is my paper
classification quant-phcond-mat.str-elhep-th
keywords approximatequantumerrorcorrectionchiraltopologicalorderconformalfieldtheorycoherentinformationrelativeentropypower-lawexponentsedgecodeanyonsectors
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 introduces chiral edge codes: approximate quantum error-correcting codes built from the gapless edge modes of a two-dimensional topologically ordered phase on a cylinder. Each codeword is a cylinder primary state labeled by a bulk anyon sector and edge primary fields. The paper claims these codes are at least as robust to geometrically local erasure as the dimensionally reduced 1D conformal field theory (CFT) code, and strictly more robust in several examples. The central result is an exact formula equating the coherent-information loss of an erased region to an average relative entropy on that region, provided the complement contains a noncontractible bulk annulus. This leads to power-law scaling of information leakage, with exponents satisfying γ ≥ α ≥ min{α,β}.

What carries the argument

The central objects are cylinder primary states |a, φ^I_a(L), φ^J_{\bar a}(R)⟩ of a chiral topological order, representing one state per anyon sector. The key mechanism is a dimensional-reduction assumption that identifies UV-finite entanglement quantities (relative entropies and mutual information of full-column regions) computed on the cylinder with those of a 1D rational CFT. This assumption, combined with the orthogonality of anyon sectors detected by a noncontractible bulk annulus, converts the coherent-information loss into a CFT-computable relative entropy. The machinery also includes the universal twirled Petz map and the full-boundary entanglement bootstrap, which provide the power-law-range recovery map.

What would settle it

Compute the coherent-information loss or the average relative entropy Δ(Y; V) for a physical lattice model of a chiral topological order (e.g., the ν=1/2 bosonic Laughlin state on a cylinder) at several system sizes. If the fitted exponent γ does not satisfy γ ≥ α, or if the relative-entropy formula of Thm. 7 fails to reproduce Δ(A; V) when the complement contains a noncontractible bulk annulus, the central claim is falsified. A direct check: evaluate S(ρ^a_B ∥ ρ̄_B) - log D for an annulus region B that is not fully symmetric; if it is not exponentially small, the orthogonality assumption underlying the theorem breaks.

Watch

Extended reading notes

Core claim

For a chiral edge code, when the complement of the erased region contains a noncontractible bulk annulus, the coherent-information loss equals the average relative entropy between codeword states on the erased region: Δ(A; V) = (1/D) Σ_a S(ρ^a_A ∥ ρ̄_A). This reduces the recoverability problem to universal edge CFT data. Consequently, the loss scales as a power law in the angular size x of the erased region, with distinct exponents α, β, γ. The main robustness result is the hierarchy γ ≥ α ≥ min{α,β}, where γ governs truly 2D edge-local erasures and min{α,β} governs the dimensionally reduced 1D CFT code, implying the 2D edge code is never worse and often better. For Abelian code subspaces, the paper constructs a power-law-range recovery map supported on the erased region plus a subextensive buffer, with recovery infidelity decaying as O(1) $L_x^{{-γ*μ*/(γ*+μ*)}}$ scale.

Load-bearing premise

The whole quantitative comparison between 2D chiral edge codes and their 1D CFT counterparts rests on Assumption 5: that every UV-finite entanglement quantity computed on the cylinder using primary states and full-column regions equals the same quantity computed in the dimensionally reduced 1D RCFT with the corresponding intervals.

Editorial extensions

If this is right

  • If the claim is correct, chiral edge codes are the first explicit AQEC codes that combine a gapped, stable bulk with gapless edge degrees of freedom protected by topology, rather than fine-tuning, giving power-law robustness with computable exponents.
  • The hierarchy γ ≥ α ≥ min{α,β} implies that for local erasures, the 2D edge code has a larger effective code distance d*(δ) ~ (δ/c)^{1/γ} compared to the dimensional-reduced 1D code, so information leakage is smaller for a given erasure size.
  • The theorem establishing Δ = average relative entropy provides a direct, computable diagnostic for local recoverability: the coherent-information loss can be evaluated by CFT data alone, and bounds all local noise channel deficits.
  • The explicit power-law-range recovery map for Abelian sectors, independent of the encoded state, implies approximate error correction is possible with recovery supported within a subextensive buffer. For the chiral semion example the infidelity decays as L_x^{-0.697}, and for the chiral Ising example as L_x^{-0.276}.
  • If the edge exponents are universal, the same power-law scaling should be observed in all microscopic lattice realizations in the same chiral topological phase, making this a robust fingerprint of the edge CFT.

Reading between the lines

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

  • The dimensional-reduction assumption is stated as exact for UV-finite quantities; the paper's own discussion (Sec. VIII) openly lists deriving this correspondence with controlled errors as an open problem. A natural inference is that the exponents α, β, γ will receive finite-size and finite-correlation-length corrections, but the hierarchy γ ≥ α may still hold approximately in microscopic models.
  • The power-law-range recovery construction depends on the Markov property for Abelian sectors; extending this to non-Abelian sectors likely requires a formulation that tracks the fusion-space structure, and may yield weaker locality guarantees.
  • If the edge CFT codes are as robust as claimed, they provide a candidate platform for storing quantum information in fractional quantum Hall edge states, where the gapped bulk suppresses bulk errors and the edge handles the logical degrees of freedom.
  • A testable extension: the chiral edge code robustness under extensive IID noise, beyond the finite-size signs of favorable exponents seen in Appendix E, needs to determine whether a threshold exists controlled by γ or another edge datum.
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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 introduces a family of approximate quantum error-correcting codes, called chiral edge codes, whose code subspaces are spanned by cylinder primary states associated with distinct anyon sectors of a two-dimensional chiral topological order. The main technical results are: (i) an exact expression (Theorem 7) equating the coherent-information loss under erasure of a region A to an average of relative entropies on A whenever the complement contains a noncontractible bulk annulus; (ii) a power-law hierarchy (Proposition 8) relating the 2D local-erasure exponent γ to the 1D dimensional-reduction exponents α and β, namely γ ≥ α ≥ min{α, β}; and (iii) a power-law-range recovery theorem (Theorem 9) for Abelian code subspaces, with recovery supported on the erased region plus a subextensive buffer. The analytic predictions for α and β are tested numerically on lattice models realizing the chiral semion and Ising edge codes, and γ is estimated for the semion code using bosonic Laughlin wavefunctions. The paper is clearly written and the main logical steps are presented in detail, but the quantitative comparison between 2D and 1D codes rests on an unproven dimensional-reduction assumption and on replica-formula results deferred to a 'to appear' reference.

Significance. If the missing pieces are supplied, this is a valuable contribution that connects topological order, CFT-based approximate quantum error correction, and local-erasure diagnostics. The derivation of Theorem 7 is clean and follows from sector orthogonality plus the relative-entropy identity of Lemma 6; the monotonicity argument yielding γ ≥ α in Proposition 8 is sound. The numerical work is extensive and well documented (Appendices B, C, F), and the explicit construction of a code-subspace-dependent recovery map for Abelian sectors is a useful step beyond abstract recovery theorems. The main weakness is that the headline claim—that the 2D chiral edge code is at least as robust as its dimensionally reduced 1D counterpart—is conditional on Assumption 5 (Sec. IV D), which asserts an equivalence of all UV-finite entanglement quantities between cylinder primary states and 1D RCFT states. The paper itself lists the controlled derivation of this correspondence as an open problem.

major comments (3)
  1. [Sec. IV D, Assumption 5; used in Prop. 8, Eq. (53), and Thm. 9 (App. D)]
  2. [Appendix A 2, Eqs. (A20)-(A23)]
  3. [Sec. VII and Appendix D 3, Prop. 10 and proof of Thm. 9]
minor comments (5)
  1. [Sec. IV B, Definition 4]
  2. [Sec. V A, Eqs. (40)-(42)]
  3. [Sec. VI C, paragraph after Table I]
  4. [Appendix A 3]
  5. [Appendix B 1, Eq. (B1)]

Circularity Check

2 steps flagged · score 4.0 of 10

The robustness hierarchy is a proved inequality among defined exponents, but the 2D exponent is a definitional label of the same relative-entropy quantity and the analytic α,β values are deferred to an overlapping-authors 'to appear' paper; lattice numerics keep the central claim independently supported.

  1. self definitional [Sec. V C, Eq. (42) and Prop. 8 proof]
    "xγ ∼ 1/D ∑_{a∈A_code} S(ρ^a_Y||ρ̄_Y), ... The first claim, ∆(Y ; Vχ)∼x γ, follows immediately from Thm. 7 and the definition of γ in Eq. (42)."

    The exponent γ is defined by Eq. (42) as the leading small-x power of the average relative entropy on the region Y. Theorem 7 proves that the coherent-information loss ∆(Y;Vχ) is exactly equal to that same average relative entropy. Therefore the statement ∆(Y;Vχ)∼x^γ is not an independent prediction of the 2D robustness exponent; it is a relabeling of the assumed power-law scaling of the relative-entropy average, combined with Theorem 7. The genuinely nontrivial content is in Theorem 7 itself and in the monotonicity inequality γ≥α, not in the claim that the 2D loss is governed by the exponent γ, which holds by construction.

  2. self citation load bearing [Appendix A.2 (Relative entropy in RCFT); used in Sec. VI, Tables I and II]
    "the universal formulas are available for both the interval A and its complement ¯A, the details of which will be presented in future work [74]; here we summarize the formulas needed for the present work."

    The analytical values αth=(2,2) and βth=(1,5),(1/4,2) reported in Tables I and II come from Eqs. (A20)-(A23), whose derivation is deferred to Ref. [74], a paper with overlapping authorship (B. Shi, Y. Song, and Z. Wang are coauthors of the present manuscript). Thus the first-principles component of the exponent prediction is not self-contained: it appeals to an unpublished self-citation for the central formulas. However, the lattice numerics in Sec. VI independently estimate α and β for the same code subspaces, so the robustness comparison and the strict-enhancement examples do not depend on Ref. [74] alone. This limits the circularity to the analytic-theory layer rather than the main numerical claim.

full rationale

Theorem 7 is a genuine exact reduction: it proves that the coherent-information loss equals an average relative entropy, using the anyon-annulus orthogonality of the entanglement bootstrap (Ref. [47]), a published external result. The hierarchy γ≥α≥min{α,β} follows from the definitions of the exponents together with the monotonicity of relative entropy (Eq. (54)); it is a proved inequality among defined quantities, not a fitted prediction. The definitional component is that γ is introduced in Eq. (42) as the exponent of the very average relative entropy that Theorem 7 identifies with ∆(Y;Vχ), so asserting ∆∼x^γ is a labeling of an assumed scaling, while the nontrivial physical statement is the monotonicity inequality. The analytic values of α and β are deferred to Ref. [74], an overlapping-authors 'to appear' paper, so the analytic derivation is not self-contained; nevertheless the finite-size lattice computations of α, β, and γ in Sec. VI provide independent numerical support for the central comparison. The dimensional-reduction Assumption 5 is an explicit assumption whose controlled derivation the paper lists as an open problem in Sec. VIII; this is a limitation on the 2D/1D identification, not a hidden circular step. Overall, the main robustness comparison has independent numerical content, so the paper is only partially circular rather than fundamentally so.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central results rest on standard quantum-information inequalities, the topological-order input that annuli resolve anyon sectors, a dimensional-reduction matching assumption (Assumption 5), full-boundary entanglement bootstrap axioms, and replica relative-entropy formulas deferred to the authors' companion paper. The free parameters are the power-law exponents extracted from numerical fits; none of these is a universal constant justified by a first-principles derivation in this preprint. No new physical entities are introduced; the chiral edge code is a choice of code subspace using existing anyon and edge labels.

free parameters (5)
  • Power-law exponent α = 2.075 (semion {|0,0>,|s,s>}), 2.000 (semion {|0,0>,|3s,s>}), 2.007 (Ising I-σ), 3.787 (Ising I-ε)
    Fitted to the small-interval average relative entropy δα=c1 x^α in Sec. VI; appears in the hierarchy and in comparison with CFT predictions.
  • Power-law exponent β = 0.927, 4.737, 0.293, 1.946 for the same code subspaces
    Fitted to logD minus the average relative entropy on the complement (Fig. 10); determines the 1D code robustness through min{α,β}.
  • Power-law exponent γ = 2.774 and 2.471 (semion, for the two code subspaces)
    Fitted to relative entropy of edge-local regions Y using Laughlin wavefunctions (Fig. 11). No analytic prediction is provided, and Prop. 8 uses the definition of γ.
  • Disjoint-interval mutual information exponent µ* = ≈0.297 (Ising I-ε), ≈1.071 (semion)
    Fitted to I(a:c)∼η^µ in Appendix D.4; controls the recovery buffer exponent λ*=γ*/(γ*+µ*) and the recovery infidelity scaling in Thm. 9.
  • IID noise collapse exponents ζ_P = ζ_Z≈-0.653, ζ_X≈-0.760 (2D); ζ_Z≈-0.200, ζ_X≈0.000 (1D)
    Fitted in Appendix E to argue the 2D realization has a more favorable finite-size flow; the paper labels this as a finite-size observation rather than a demonstrated mechanism.
assumptions (6)
  • domain assumption Noncontractible bulk annuli perfectly resolve anyon sectors with orthogonal reduced density matrices (Eqs. (19)-(20))
    Used in Thm. 7 to conclude ρ^a_B have mutually orthogonal support. This relies on entanglement bootstrap theory from Ref. [47] and is approximate for lattice systems.
  • ad hoc to paper Assumption 5: dimensional reduction preserves UV-finite relative entropies and disjoint-interval mutual information between cylinder primary states and 1D RCFT states (Sec. IV D)
    This is a new assumption posited for this paper; it converts 2D cylinder quantities into 1D CFT quantities and underpins the α,β identification and the recovery theorem.
  • domain assumption Full boundary entanglement bootstrap axioms A0 and A1 hold for cylinder primary states, with A1 only for Abelian sectors (Appendix D 1 b)
    A1 is used in Prop. 10 to prove the conditional mutual information identity underlying the power-law-range recovery map. The paper says A1 is expected but not proven.
  • ad hoc to paper Replica formulas for relative entropy with probabilistic mixtures, Eqs. (A20) and (A23), are correct
    The proof is deferred to Ref. [74], an unpublished companion paper with overlapping authors; the present preprint only outlines the replica method.
  • domain assumption Lattice bosonic Laughlin wavefunctions at ν=1/2 realize the chiral semion edge primary states (Appendix B)
    Used for the γ and µ numerical estimates; relies on Ref. [45,76] for the wavefunction and expects finite-size deviations.
  • domain assumption puMPS tangent-space states approximate Ising CFT primaries (Appendix C)
    Used to compute Ising α, β, µ; the identification of the two lowest tangent vectors with σ and ε is assumed.

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Pith. "Pith review of Approximate Quantum Error Correction at Chiral Topological Edges." pith.science (2026). https://pith.science/paper/MUZF3NFI

@misc{pith2026260806258,
  author       = {Pith},
  title        = {Pith review of: Approximate Quantum Error Correction at Chiral Topological Edges},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MUZF3NFI}},
  note         = {Machine review of arXiv:2608.06258}
}
read the original abstract

Topologically ordered phases naturally realize quantum error correction through nonlocal encoding of quantum information. More recently, conformal field theories have been shown to realize approximate quantum error-correcting codes, but such constructions generally require fine tuning to criticality. Here we introduce a family of approximate quantum error-correcting codes realized by the chiral edges of two-dimensional topologically ordered phases. The proposed encoding combines the robustness of a gapped topological bulk with the flexibility of gapless edge conformal field theories. To characterize its robustness, we study coherent-information loss under local erasure. We derive an exact expression relating coherent-information loss to relative entropy, reducing the recoverability problem to universal properties of the edge theory. This leads to power-law scaling of coherent-information loss with the size of the erased region. We further show that, for geometrically local erasures near one edge, the two-dimensional chiral edge code is at least as robust as the dimensionally reduced CFT code, and is strictly more robust in several representative examples. For Abelian code subspaces, we further construct a power-law-range recovery map supported on the erased region together with a power-law-range buffer; this recovery map depends only on the code subspace, not on the unknown encoded state. We provide numerical calculations for lattice realizations of compact free boson and Ising CFT examples that support the theoretical predictions of the power-law exponents.

Figures

Figures reproduced from arXiv: 2608.06258 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Regions related to the definition of coherent informa [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Local decoherence of TQFT code can be fully re [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Anyon charges detected by an annulus surrounding it, [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Prepare a state in the code subspace of the chiral edge [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Three power law exponents defined by relative entropy behavior at small [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. A cylinder [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Regions on the cylinder used in Prop. [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Regions related to the identification of the power-law [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. The Euclidean path integral for computing [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Partitions of the cylinder viewed as a flat disk, useful [PITH_FULL_IMAGE:figures/full_fig_p026_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Partitions of a full column [PITH_FULL_IMAGE:figures/full_fig_p028_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p029_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. An illustration of the golden-cylinder lattice Λ [PITH_FULL_IMAGE:figures/full_fig_p030_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19 [PITH_FULL_IMAGE:figures/full_fig_p030_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20 [PITH_FULL_IMAGE:figures/full_fig_p031_20.png]

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Works this paper leans on

133 extracted references · 24 canonical work pages

  1. [74]

    Characterizing topological order by the information convex,

    B. Shi and Y.-M. Lu, “Characterizing topological order by the information convex,”Physical Review B99 (2019) 035112,1801.01519

    arXiv 2019

  2. [1]

    Beyond these examples, exact all-x results are rare

    Relative entropy between different primaries For relative entropy between two different primaries, there exist analytic formulas for any 0 < x <1 in free boson CFT [ 69, 70, 100] and Ising CFT [ 94], as we will discuss in the examples later. Beyond these examples, exact all-x results are rare. However, the leading small- interval (x≪ 1) behavior of the re...

  3. [2]

    Relative entropy with a probabilistic mixture Next, we present the relative entropy formulas between a primary and a mixture of primaries, which can be used to determine the exponentsαandβ. a. Analytic results ofα Before introducing the explicit results, for the small interval case, one may use joint convexity of relative entropy [103, 104] S( ∑ i λiρi|| ...

  4. [3]

    We then discuss the simplification 21 FIG

    Review of the replica method Here we give a brief review of the replica method used to compute the relative entropy in (1+1)D CFT, as first discussed in [69, 70]. We then discuss the simplification 21 FIG. 13. The Euclidean path integral for computing tr [ ρa A(ρb A)n−1] . Given region A, the replica manifold Mn is obtained by gluing n copies of Euclidean...

  5. [4]

    This conformal transformation makes the calculation more tractable

    Therefore, the above correlators can be equivalently computed on the complex plane. This conformal transformation makes the calculation more tractable. However, even after the conformal trans- formation, the relative entropy can only be computed explicitly in very special cases, for example, between two primaries in free boson CFT [ 69, 70, 100], or betwe...

  6. [5]

    charge number

    Analytical wave functions The chiral semion topological order can be realized by the bosonic fractional quantum Hall (FQH) with ν = 1/2. Ref. [45, 76] provides a wider class of FQH states, with ν = 1/q in which q is an even integer. (Similar models have been generalized to non-Abelian topological orders, see e.g. [105, 106].) Here, we discuss the wave fun...

  7. [6]

    11, we choose the lattice of N = LxLy sites as follows

    Cylinder with a square lattice For the cylinder used in Fig. 11, we choose the lattice of N = LxLy sites as follows. For integers Lx and Ly, sites are labeled by a row index j2 = 0,...,L y− 1 and a periodic coordinate j1 = 0,...,L x− 1, with j1∼j 1 +Lx. The corresponding row-major site label isj = 1+j2Lx+j1. The complex plane coordinates {zj} are generate...

  8. [7]

    To do so, we choose q = 2

    Spin chains We can use this analytical wave function to simulate the 1D compact free boson RCFT with compactification radius R = √ 2 and central charge c = 1. To do so, we choose q = 2. For the primary state |φ 1 2, 1 2 s ⟩ which corresponds to the vertex operator V(z,¯z) =: exp ( i√ 2ϕ(z) + i√ 2 ¯ϕ(¯z) ) :,(B6) we set w1 = 0 and w2 = 108. The qubits are ...

Show all 133 references
  1. [8]

    A puMPS with phys- ical dimension d = 2 and bond dimension D is specified by one tensorA s∈C D×D, repeated at every site: |Ψ(A)⟩= d∑ s1,...,sN=1 Tr [ As1As2···A sN ] |s1s2···s N⟩

    Hamiltonian and periodic uniform MPS ansatz The critical transverse-field Ising chain used in the simulation is HN =− N∑ j=1 XjXj+1− N∑ j=1 Zj,(C1) The ground-state variational ansatz is the periodic uni- form matrix product state (puMPS). A puMPS with phys- ical dimension d =...

  2. [9]

    For momentump= 2πk/N, define |Φp(B;A)⟩ = N∑ n=1 eipn∑ s Tr [As1···A sn−1BsnAsn+1···A sN ]|s⟩

    Construction of excited states|σ⟩and|ϵ⟩ Low-energy excited states are represented by Bloch- state tangent vectors built on the optimized ground-state tensorA. For momentump= 2πk/N, define |Φp(B;A)⟩ = N∑ n=1 eipn∑ s Tr [As1···A sn−1BsnAsn+1···A sN ]|s⟩. (C11) The sign of p is a...

  3. [10]

    Reduced density matrices from puMPS contractions The reduced density matrix of subsystem M can be obtained directly from puMPS transfer matrices, avoiding the dense vector of sizedN. For two local tensorsXs,Y s∈ CD×D, define the single-site double-layer transfer matrices Es,t ...

  4. [11]

    Tools for the proof We present a set of tools and necessary background knowledge for the proof of theorem 9. a. A universal recovery theorem There is a universal recovery theorem following from Eq. (15) in Ref. [ 50]. For a pair of states ρ,σ such that 25 supp(ρ)⊆supp (σ), the...

  5. [12]

    Furthermore, the recovery map R has an explicit form as worked out in Ref. [50]. We shall need the special context of tripartite states ρABC andσABC living on a tensor product of three finite- dimensional Hilbert spacesHA⊗HB⊗HC with a channel NA acting on A. Furthermore, we re...

  6. [13]

    Define C :=Q\ (AB); see Fig

    Proof of the theorem 9 Let A be the erased local disk near the edge and let B be a buffer region surrounding A. Define C :=Q\ (AB); see Fig. 15(a). We denote the channel on A asNA. Our first trick is to apply the variant of the universal recovery theorem from (D3), choosing σA...

  7. [14]

    Recall that,γ ab is defined according to S(ρa A||ρb A)∼x γab (D30) for regionAin Fig

    A trick for computingγ ab We present a useful proposition that allows us to com- pute γab exactly using the dimensional reduction picture. Recall that,γ ab is defined according to S(ρa A||ρb A)∼x γab (D30) for regionAin Fig. 16. We now defineα ab according to S(ρa ABC||ρb ABC)...

  8. [15]

    Computation of relevant exponentµ a In this appendix, we compute the exponent µa that is relevant to the power-law-range recovery channel in Sec. VII. Recall that µa is defined according to an empir- ical formula of mutual information of a certain 1D CFT state|φa⟩ obtained fro...

  9. [16]

    [31], we assume that in the weak-noise regime Ic(p) =f ( pNζP ) , p→0, N→∞.(E5) Here ζP is the IID scaling-collapse exponent for Pauli type P

    Weak-noise scaling ansatz Following the coherent-information scaling hypothesis of Ref. [31], we assume that in the weak-noise regime Ic(p) =f ( pNζP ) , p→0, N→∞.(E5) Here ζP is the IID scaling-collapse exponent for Pauli type P . (The exponent ζP corresponds to the exponent ...

  10. [17]

    We use the analytical wave function described in Appendix B 1, and we scale up the size of the cylinder N

    Finite size scaling with Golden-cylinder coordinate We now describe the finite-size scaling method for ν = 1/2 fractional quantum Hall states. We use the analytical wave function described in Appendix B 1, and we scale up the size of the cylinder N. We map the cylinder coordin...

  11. [18]

    In both cases the coherent information is computed using the purification algorithm described in Appendix F

    Numerical comparison:1D chain versus2D golden cylinder We compare IID Pauli dephasing for the dimensionally reduced 1D CFT code VCFT = Γ(Vχ( ) ) and the 2D chiral edge code Vχ( ) . In both cases the coherent information is computed using the purification algorithm described in...

    2000

  12. [19]

    Scheme for reducing decoherence in quantum computer memory,

    P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,”Phys. Rev. A52(Oct,

  13. [20]

    Simple approach to approximate quantum error correction based on the transpose channel,

    H. K. Ng and P. Mandayam, “Simple approach to approximate quantum error correction based on the transpose channel,”Phys. Rev. A81(June, 2010) 062342,0909.0931

    2010 arXiv

  14. [21]

    Theory of quantum error-correcting codes,

    E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,”Phys. Rev. A55(Feb., 1997) 900–911

    1997

  15. [22]

    Topological Orders in Rigid States,

    X. G. Wen, “Topological Orders in Rigid States,” International Journal of Modern Physics B4(Jan.,

  16. [23]

    Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces,

    X. G. Wen and Q. Niu, “Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces,” Phys. Rev. B41(May, 1990) 9377–9396, https://link.aps.org/doi/10.1103/PhysRevB.41.9377

    1990 doi

  17. [24]

    Fault-tolerant quantum computation by anyons,

    A. Y. Kitaev, “Fault-tolerant quantum computation by anyons,”Annals of Physics303(Jan., 2003) 2–30, quant-ph/9707021

    2003 arXiv

  18. [25]

    Topological quantum memory,

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory,”Journal of Mathematical Physics43(Sept., 2002) 4452–4505, quant-ph/0110143

    2002 arXiv

  19. [26]

    Quantum error correction for quantum memories,

    B. M. Terhal, “Quantum error correction for quantum memories,”Rev. Mod. Phys.87(2015), no. 2 307, 1302.3428

    2015 arXiv

  20. [27]

    Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance,

    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. B72(July, 2005) 045141, cond-mat/0503554. 33

    2005 arXiv

  21. [28]

    Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order,

    X. Chen, Z.-C. Gu, and X.-G. Wen, “Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order,” Phys. Rev. B82(Oct., 2010) 155138,1004.3835

    2010 arXiv

  22. [29]

    Topological quantum order: Stability under local perturbations,

    S. Bravyi, M. B. Hastings, and S. Michalakis, “Topological quantum order: Stability under local perturbations,”Journal of Mathematical Physics51 (Sept., 2010) 093512–093512,1001.0344

    2010 arXiv

  23. [30]

    Strong Resilience of Topological Codes to Depolarization,

    H. Bombin, R. S. Andrist, M. Ohzeki, H. G. Katzgraber, and M. A. Martin-Delgado, “Strong Resilience of Topological Codes to Depolarization,” Physical Review X2(Apr., 2012) 021004,1202.1852

    2012 arXiv

  24. [31]

    On the stability of topological order in tensor network states,

    D. J. Williamson, C. Delcamp, F. Verstraete, and N. Schuch, “On the stability of topological order in tensor network states,”Phys. Rev. B104(Dec., 2021) 235151,2012.15346

    2021 arXiv

  25. [32]

    Topological Quantum Spin Glass Order and its realization in qLDPC codes,

    B. Placke, T. Rakovszky, N. P. Breuckmann, and V. Khemani, “Topological Quantum Spin Glass Order and its realization in qLDPC codes,”arXiv e-prints (Dec., 2024) arXiv:2412.13248,2412.13248

    2024 arXiv

  26. [33]

    On stability of k-local quantum phases of matter,

    A. Lavasani, M. J. Gullans, V. V. Albert, and M. Barkeshli, “On stability of k-local quantum phases of matter,”arXiv e-prints(May, 2024) arXiv:2405.19412,2405.19412

    2024 arXiv

  27. [34]

    Low-Density Parity-Check Stabilizer Codes as Gapped Quantum Phases: Stability under Graph-Local Perturbations,

    W. De Roeck, V. Khemani, Y. Li, N. O’Dea, and T. Rakovszky, “Low-Density Parity-Check Stabilizer Codes as Gapped Quantum Phases: Stability under Graph-Local Perturbations,”PRX Quantum6(Aug.,

  28. [35]

    Approximate quantum error correction can lead to better codes,

    D. W. Leung, M. A. Nielsen, I. L. Chuang, and Y. Yamamoto, “Approximate quantum error correction can lead to better codes,”Phys. Rev. A56(Oct., 1997) 2567–2573,quant-ph/9704002

    1997 arXiv

  29. [36]

    Approximate Quantum Error Correction,

    B. Schumacher and M. D. Westmoreland, “Approximate Quantum Error Correction,”Quantum Information Processing1(Apr., 2002) 5–12,quant-ph/0112106

    2002 arXiv

  30. [37]

    Reversing quantum dynamics with near-optimal quantum and classical fidelity,

    H. Barnum and E. Knill, “Reversing quantum dynamics with near-optimal quantum and classical fidelity,” Journal of Mathematical Physics43(May, 2002) 2097–2106,quant-ph/0004088

    2002

  31. [38]

    The Information-Disturbance Tradeoff and the Continuity of Stinespring’s Representation,

    D. Kretschmann, D. Schlingemann, and R. F. Werner, “The Information-Disturbance Tradeoff and the Continuity of Stinespring’s Representation,”arXiv e-prints(Apr., 2006) quant–ph/0605009, quant-ph/0605009

    2006

  32. [39]

    Bootstrapping noninvertible symmetries,

    Y.-H. Lin and S.-H. Shao, “Bootstrapping noninvertible symmetries,”Phys. Rev. D107(June, 2023) 125025, 2302.13900

    2023 arXiv

  33. [40]

    General Conditions for Approximate Quantum Error Correction and Near-Optimal Recovery Channels,

    C. B´ eny and O. Oreshkov, “General Conditions for Approximate Quantum Error Correction and Near-Optimal Recovery Channels,”Phys. Rev. Lett. 104(Mar., 2010) 120501,0907.5391

    2010 arXiv

  34. [41]

    Limits on the storage of quantum information in a volume of space,

    S. T. Flammia, J. Haah, M. J. Kastoryano, and I. H. Kim, “Limits on the storage of quantum information in a volume of space,”Quantum1(Apr., 2017) 4, 1610.06169

    2017 arXiv

  35. [42]

    Complexity and order in approximate quantum error-correcting codes,

    J. Yi, W. Ye, D. Gottesman, and Z.-W. Liu, “Complexity and order in approximate quantum error-correcting codes,”Nature Physics20(Nov., 2024) 1798–1803,2310.04710

    2024 arXiv

  36. [43]

    Approximate Quantum Codes From Long Wormholes,

    G. Bentsen, P. Nguyen, and B. Swingle, “Approximate Quantum Codes From Long Wormholes,”Quantum8 (Aug., 2024) 1439,2310.07770

    2024 arXiv

  37. [44]

    Lov´ asz Meets Lieb-Schultz-Mattis: Complexity in Approximate Quantum Error Correction,

    J. Yi, R. Liu, and Z. Li, “Lov´ asz Meets Lieb-Schultz-Mattis: Complexity in Approximate Quantum Error Correction,”arXiv e-prints(Oct., 2025) arXiv:2510.04453,2510.04453

    2025

  38. [45]

    Continuous Symmetries and Approximate Quantum Error Correction,

    P. Faist, S. Nezami, V. V. Albert, G. Salton, F. Pastawski, P. Hayden, and J. Preskill, “Continuous Symmetries and Approximate Quantum Error Correction,”Physical Review X10(Oct., 2020) 041018, 1902.07714

    2020 arXiv

  39. [46]

    New perspectives on covariant quantum error correction,

    S. Zhou, Z.-W. Liu, and L. Jiang, “New perspectives on covariant quantum error correction,”Quantum5(Aug.,

  40. [47]

    Bulk locality and quantum error correction in AdS/CFT,

    A. Almheiri, X. Dong, and D. Harlow, “Bulk locality and quantum error correction in AdS/CFT,”Journal of High Energy Physics2015(Apr., 2015) 163, 1411.7041

    2015 arXiv

  41. [48]

    Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence,

    F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, “Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence,”Journal of High Energy Physics2015(June, 2015) 149, 1503.06237

    2015 arXiv

  42. [49]

    Towards Holography via Quantum Source-Channel Codes,

    F. Pastawski, J. Eisert, and H. Wilming, “Towards Holography via Quantum Source-Channel Codes,”Phys. Rev. Lett.119(July, 2017) 020501,1611.07528

    2017 arXiv

  43. [50]

    Approximate Quantum Error Correcting Codes from Conformal Field Theory,

    S. Sang, T. H. Hsieh, and Y. Zou, “Approximate Quantum Error Correcting Codes from Conformal Field Theory,”Phys. Rev. Lett.133(Nov., 2024) 210601, 2406.09555

    2024 arXiv

  44. [51]

    Eigenstate thermalization hypothesis and approximate quantum error correction,

    N. Bao and N. Cheng, “Eigenstate thermalization hypothesis and approximate quantum error correction,” Journal of High Energy Physics2019(Aug., 2019) 152, 1906.03669

    2019 arXiv

  45. [52]

    Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains,

    F. G. S. L. Brand˜ ao, E. Crosson, M. B. S ¸ahinoˇ glu, and J. Bowen, “Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains,”Phys. Rev. Lett.123(Sept., 2019) 110502,1710.04631

    2019 arXiv

  46. [53]

    Error Threshold of SYK Codes from Strong-to-Weak Parity Symmetry Breaking,

    J. Kim, E. Altman, and J. Y. Lee, “Error Threshold of SYK Codes from Strong-to-Weak Parity Symmetry Breaking,”arXiv e-prints(Oct., 2024) arXiv:2410.24225,2410.24225

    2024 arXiv

  47. [54]

    Conformal Fields and Operator Product Expansion in Critical Quantum Spin Chains,

    Y. Zou, A. Milsted, and G. Vidal, “Conformal Fields and Operator Product Expansion in Critical Quantum Spin Chains,”Phys. Rev. Lett.124(Jan., 2020) 040604, 1901.06439

    2020 arXiv

  48. [55]

    Critical theory of quantum spin chains,

    I. Affleck and F. D. M. Haldane, “Critical theory of quantum spin chains,”Phys. Rev. B36(Oct., 1987) 5291–5300

    1987

  49. [56]

    Symmetry Protection of Critical Phases and a Global Anomaly in 1 +1 Dimensions,

    S. C. Furuya and M. Oshikawa, “Symmetry Protection of Critical Phases and a Global Anomaly in 1 +1 Dimensions,”Phys. Rev. Lett.118(Jan., 2017) 021601

    2017

  50. [57]

    Anomaly Matching and Symmetry-Protected Critical Phases in S U (N ) Spin Systems in 1 +1 Dimensions,

    Y. Yao, C.-T. Hsieh, and M. Oshikawa, “Anomaly Matching and Symmetry-Protected Critical Phases in S U (N ) Spin Systems in 1 +1 Dimensions,”Phys. Rev. Lett.123(Nov., 2019) 180201,1805.06885

    2019 arXiv

  51. [58]

    A mathematical theory of gapless edges of 2d topological orders. Part I,

    L. Kong and H. Zheng, “A mathematical theory of gapless edges of 2d topological orders. Part I,”Journal of High Energy Physics2020(Feb., 2020) 150, 1905.04924

    2020 arXiv

  52. [59]

    Topological orders and edge excitations in fractional quantum Hall states,

    X.-G. Wen, “Topological orders and edge excitations in fractional quantum Hall states,”Advances in Physics 44(Sept., 1995) 405–473,cond-mat/9506066

    1995 arXiv

  53. [60]

    Anyons in an exactly solved model and beyond,

    A. Kitaev, “Anyons in an exactly solved model and beyond,”Annals of Physics321(Jan., 2006) 2–111, cond-mat/0506438

    2006 arXiv

  54. [61]

    Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally 34 Charged Excitations,

    R. B. Laughlin, “Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally 34 Charged Excitations,”Phys. Rev. Lett.50(May, 1983) 1395–1398

    1983

  55. [62]

    Equivalence of the resonating-valence-bond and fractional quantum Hall states,

    V. Kalmeyer and R. B. Laughlin, “Equivalence of the resonating-valence-bond and fractional quantum Hall states,”Phys. Rev. Lett.59(Nov, 1987) 2095–2098, https://link.aps.org/doi/10.1103/PhysRevLett.59.2095

    1987 doi

  56. [63]

    Nonabelions in the fractional quantum hall effect,

    G. Moore and N. Read, “Nonabelions in the fractional quantum hall effect,”Nuclear Physics B360(Aug.,

  57. [64]

    Laughlin Spin-Liquid States on Lattices Obtained from Conformal Field Theory,

    A. E. B. Nielsen, J. I. Cirac, and G. Sierra, “Laughlin Spin-Liquid States on Lattices Obtained from Conformal Field Theory,”Phys. Rev. Lett.108(June,

  58. [65]

    Generalizations of Kitaev’s honeycomb model from braided fusion categories,

    L. Eck and P. Fendley, “Generalizations of Kitaev’s honeycomb model from braided fusion categories,” SciPost Physics18(June, 2025) 170,2408.04006

    2025 arXiv

  59. [66]

    Fusion rules from entanglement,

    B. Shi, K. Kato, and I. H. Kim, “Fusion rules from entanglement,”Annals of Physics418(July, 2020) 168164,1906.09376

    2020 arXiv

  60. [67]

    Chiral Virasoro algebra from a single wavefunction,

    I. H. Kim, X. Li, T.-C. Lin, J. McGreevy, and B. Shi, “Chiral Virasoro algebra from a single wavefunction,” Annals of Physics471(Dec., 2024) 169849, 2403.18410

    2024 arXiv

  61. [68]

    Quantum Conditional Mutual Information and Approximate Markov Chains,

    O. Fawzi and R. Renner, “Quantum Conditional Mutual Information and Approximate Markov Chains,” Communications in Mathematical Physics340(Dec.,

  62. [69]

    Universal recovery maps and approximate sufficiency of quantum relative entropy,

    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(2018), no. 10 2955–2978,1509.07127

    2018 arXiv

  63. [70]

    Quantum data processing and error correction,

    B. Schumacher and M. A. Nielsen, “Quantum data processing and error correction,”Phys. Rev. A54(Oct.,

  64. [71]

    Relative entropy of excited states in two dimensional conformal field theories,

    G. S´ arosi and T. Ugajin, “Relative entropy of excited states in two dimensional conformal field theories,” Journal of High Energy Physics2016(July, 2016) 114, 1603.03057

    2016 arXiv

  65. [72]

    ρ′ QR =λ A⊗ρBR, whereλ A is an arbitrary density matrix andρ BR = TrA|ψQR⟩⟨ψQR|

    One realization of the strongest channel is such that it replaces the local density by a product, e.g. ρ′ QR =λ A⊗ρBR, whereλ A is an arbitrary density matrix andρ BR = TrA|ψQR⟩⟨ψQR|

  66. [73]

    Gottesman,Stabilizer codes and quantum error correction

    D. Gottesman,Stabilizer codes and quantum error correction. PhD thesis, California Institute of Technology, Jan., 1997

    1997

  67. [75]

    Seeing topological entanglement through the information convex,

    B. Shi, “Seeing topological entanglement through the information convex,”Physical Review Research1(Oct.,

  68. [76]

    Quantized thermal transport in the fractional quantum Hall effect,

    C. L. Kane and M. P. A. Fisher, “Quantized thermal transport in the fractional quantum Hall effect,”Phys. Rev. B55(June, 1997) 15832–15837, cond-mat/9603118

    1997 arXiv

  69. [77]

    Gapless edges of 2d topological orders and enriched monoidal categories,

    L. Kong and H. Zheng, “Gapless edges of 2d topological orders and enriched monoidal categories,”Nuclear Physics B927(Feb., 2018) 140–165,1705.01087

    2018 arXiv

  70. [78]

    Conformal Data and Renormalization Group Flow in Critical Quantum Spin Chains Using Periodic Uniform Matrix Product States,

    Y. Zou, A. Milsted, and G. Vidal, “Conformal Data and Renormalization Group Flow in Critical Quantum Spin Chains Using Periodic Uniform Matrix Product States,” Phys. Rev. Lett.121(Dec., 2018) 230402,1710.05397

    2018 arXiv

  71. [79]

    A mathematical theory of gapless edges of 2d topological orders. Part II,

    L. Kong and H. Zheng, “A mathematical theory of gapless edges of 2d topological orders. Part II,”arXiv e-prints(Dec., 2019) arXiv:1912.01760,1912.01760

    2019 arXiv

  72. [80]

    Conformal geometry from entanglement,

    I. H. Kim, X. Li, T.-C. Lin, J. McGreevy, and B. Shi, “Conformal geometry from entanglement,”arXiv e-prints(Apr., 2024) arXiv:2404.03725,2404.03725

    2024 arXiv

  73. [81]

    Protected Edge Modes without Symmetry,

    M. Levin, “Protected Edge Modes without Symmetry,” Physical Review X3(Apr., 2013) 021009,1301.7355

    2013 arXiv

  74. [82]

    Higher central charges and Witt groups,

    S.-H. Ng, E. C. Rowell, Y. Wang, and Q. Zhang, “Higher central charges and Witt groups,”arXiv e-prints (Feb., 2020) arXiv:2002.03570,2002.03570

    2020 arXiv

  75. [83]

    Higher central charges and topological boundaries in 2+1-dimensional TQFTs,

    J. Kaidi, Z. Komargodski, K. Ohmori, S. Seifnashri, and S.-H. Shao, “Higher central charges and topological boundaries in 2+1-dimensional TQFTs,”SciPost Physics13(Sept., 2022) 067,2107.13091

    2022 arXiv

  76. [84]

    Vertex operator algebras and operads,

    Y.-Z. Huang and J. Lepowsky, “Vertex operator algebras and operads,”arXiv e-prints(Jan., 1993) hep–th/9301009,hep-th/9301009

    1993

  77. [85]

    Vertex operator algebras, the Verlinde conjecture, and modular tensor categories,

    Y.-Z. Huang, “Vertex operator algebras, the Verlinde conjecture, and modular tensor categories,”Proceedings of the National Academy of Science102(Apr., 2005) 5352–5356,math/0412261

    2005 arXiv

  78. [86]

    Chiral topologically ordered states on a lattice from vertex operator algebra,

    N. Sopenko, “Chiral topologically ordered states on a lattice from vertex operator algebra,”Advances in Theoretical and Mathematical Physics28(Jan., 2024) 1–35,2301.08697

    2024 arXiv

  79. [87]

    On classification of modular tensor categories,

    E. Rowell, R. Stong, and Z. Wang, “On classification of modular tensor categories,”arXiv e-prints(Dec., 2007) arXiv:0712.1377,0712.1377

    2007 arXiv

  80. [88]

    General Relationship between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States,

    X.-L. Qi, H. Katsura, and A. W. W. Ludwig, “General Relationship between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States,”Phys. Rev. Lett.108(May, 2012) 196402, 1103.5437

    2012 arXiv

  81. [89]

    Relative Entropies in Conformal Field Theory,

    N. Lashkari, “Relative Entropies in Conformal Field Theory,”Phys. Rev. Lett.113(Aug., 2014) 051602, 1404.3216

    2014 arXiv

  82. [90]

    Modular Hamiltonian for Excited States in Conformal Field Theory,

    N. Lashkari, “Modular Hamiltonian for Excited States in Conformal Field Theory,”Phys. Rev. Lett.117(July,

  83. [91]

    Tessellation codes: encoded quantum gates by geometric rotation,

    Y. Wang, Y. Xu, and Z.-W. Liu, “Tessellation codes: encoded quantum gates by geometric rotation,”arXiv e-prints(Oct., 2024) arXiv:2410.18713,2410.18713

    2024

  84. [92]

    Relative entropy of excited states in conformal field theories of arbitrary dimensions,

    G. S´ arosi and T. Ugajin, “Relative entropy of excited states in conformal field theories of arbitrary dimensions,”Journal of High Energy Physics2017 (Feb., 2017) 60,1611.02959

    2017 arXiv

  85. [93]

    Entanglement in descendants,

    B. G. Chowdhury and J. R. David, “Entanglement in descendants,”Journal of High Energy Physics2022 (Feb., 2022) 3,2108.00898

    2022 arXiv

  86. [94]

    Lashkari, J

    N. Lashkari, J. Mohanty, B. Shi, Y. Song, and Z. Wang to appear(2026)

    2026

  87. [95]

    Codes for the Quantum Erasure Channel,

    M. Grassl, T. Beth, and T. Pellizzari, “Codes for the Quantum Erasure Channel,”Phys. Rev. A56(July,

  88. [96]

    Chiral Central Charge from a Single Bulk Wave Function,

    I. H. Kim, B. Shi, K. Kato, and V. V. Albert, “Chiral Central Charge from a Single Bulk Wave Function,” Phys. Rev. Lett.128(Apr., 2022) 176402,2110.06932

    2022 arXiv

  89. [97]

    Anyon braiding in semianalytical fractional quantum Hall lattice models,

    A. E. B. Nielsen, “Anyon braiding in semianalytical fractional quantum Hall lattice models,”Phys. Rev. B 91(Jan., 2015) 041106,1409.3073

    2015 arXiv

  90. [98]

    These standard errors quantify the uncertainty of the power-law fits but are not used as weights in the subsequent 1/Nregression

    In panels (c) and (d), the error bars on the filled finite-Nsymbols represent one standard error of the exponent obtained from the corresponding fixed-N log-log regression. These standard errors quantify the uncertainty of the power-law fits but are not used as weights in the ...

  91. [99]

    Many-body chirality of topological stabilizer states,

    T. D. Ellison, D. Lee, Z. Li, A. Moharramipour, Y. Panahi, and B. Yoshida, “Many-body chirality of topological stabilizer states,”arXiv e-prints(June,

  92. [100]

    Entanglement entropy of two disjoint intervals in conformal field theory,

    P. Calabrese, J. Cardy, and E. Tonni, “Entanglement entropy of two disjoint intervals in conformal field theory,”Journal of Statistical Mechanics: Theory and Experiment2009(Nov., 2009) 11001,0905.2069

    2009 arXiv

  93. [101]

    Entanglement entropy of two disjoint intervals in conformal field theory: II,

    P. Calabrese, J. Cardy, and E. Tonni, “Entanglement entropy of two disjoint intervals in conformal field theory: II,”Journal of Statistical Mechanics: Theory and Experiment2011(Jan., 2011) 01021,1011.5482

    2011 arXiv

  94. [102]

    Diagnostics of Mixed-State Topological Order and Breakdown of Quantum Memory,

    R. Fan, Y. Bao, E. Altman, and A. Vishwanath, “Diagnostics of Mixed-State Topological Order and Breakdown of Quantum Memory,”PRX Quantum5 (May, 2024) 020343,2301.05689

    2024 arXiv

  95. [103]

    Exact Calculations of Coherent Information for Toric Codes under Decoherence: Identifying the Fundamental Error Threshold,

    J. Y. Lee, “Exact Calculations of Coherent Information for Toric Codes under Decoherence: Identifying the Fundamental Error Threshold,”Phys. Rev. Lett.134 (Jun, 2025) 250601, https://link.aps.org/doi/10.1103/hlfh-86yz

    2025 doi

  96. [104]

    Coherent information for Calderbank-Shor-Steane codes under decoherence,

    R. Niwa and J. Y. Lee, “Coherent information for Calderbank-Shor-Steane codes under decoherence,” Phys. Rev. A111(Mar, 2025) 032402, https://link.aps.org/doi/10.1103/PhysRevA.111.032402

    2025 doi

  97. [105]

    Information Critical Phases under Decoherence,

    A. Vijay and J. Y. Lee, “Information Critical Phases under Decoherence,”arXiv e-prints(Dec., 2025) arXiv:2512.22121,2512.22121

    2025

  98. [106]

    Stability of Mixed-State Quantum Phases via Finite Markov Length,

    S. Sang and T. H. Hsieh, “Stability of Mixed-State Quantum Phases via Finite Markov Length,”Phys. Rev. Lett.134(Feb., 2025) 070403,2404.07251

    2025

  99. [107]

    Topological Mixed States: Phases of Matter from Axiomatic Approaches,

    T.-H. Yang, B. Shi, and J. Y. Lee, “Topological Mixed States: Phases of Matter from Axiomatic Approaches,” arXiv e-prints(June, 2025) arXiv:2506.04221, 2506.04221

    2025

  100. [108]

    Critical non-equilibrium phases from noisy topological memories,

    A.-R. Negari, S. Sahu, J. Behrends, B. B´ eri, and T. H. Hsieh, “Critical non-equilibrium phases from noisy topological memories,”arXiv e-prints(Jan., 2026) arXiv:2601.10792,2601.10792

    2026

  101. [109]

    Probing mixed-state phases on a quantum computer via Renyi correlators and variational decoding,

    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,”arXiv e-prints(May, 2025) arXiv:2505.02900,2505.02900

    2025 arXiv

  102. [110]

    Measuring Modular Matrices by Shearing Lattices,

    Y.-Z. You and M. Cheng, “Measuring Modular Matrices by Shearing Lattices,”arXiv e-prints(Feb., 2015) arXiv:1502.03192,1502.03192

    2015 arXiv

  103. [111]

    Instantaneous braids and Dehn twists in topologically ordered states,

    G. Zhu, A. Lavasani, and M. Barkeshli, “Instantaneous braids and Dehn twists in topologically ordered states,” arXiv e-prints(June, 2018) arXiv:1806.06078, 1806.06078

    2018 arXiv

  104. [113]

    Universal Quantum Computation with Gapped Boundaries,

    I. Cong, M. Cheng, and Z. Wang, “Universal Quantum Computation with Gapped Boundaries,”Phys. Rev. Lett.119(Oct., 2017) 170504,1707.05490

    2017 arXiv

  105. [114]

    Fractional quantum Hall states under density decoherence,

    Z. Wang, R. Fan, T. Wang, S. J. Garratt, and E. Altman, “Fractional quantum Hall states under density decoherence,”arXiv e-prints(Oct., 2025) arXiv:2510.08490,2510.08490

    2025

  106. [115]

    Numerical calculations on the relative entanglement entropy in critical spin chains,

    Y. O. Nakagawa and T. Ugajin, “Numerical calculations on the relative entanglement entropy in critical spin chains,”Journal of Statistical Mechanics: Theory and Experiment9(Sept., 2017) 093104,1705.07899

    2017 arXiv

  107. [116]

    Analytically Continuing the Randomized Measurement Toolbox,

    A. Vijay, A. Raj, J. Kudler-Flam, B. Vermersch, A. Elben, and L. Nie, “Analytically Continuing the Randomized Measurement Toolbox,”arXiv e-prints (Nov., 2025) arXiv:2511.02912,2511.02912

    2025 arXiv

  108. [118]

    Modular Commutators in Conformal Field Theory,

    Y. Zou, B. Shi, J. Sorce, I. T. Lim, and I. H. Kim, “Modular Commutators in Conformal Field Theory,” Phys. Rev. Lett.129(Dec., 2022) 260402,2206.00027

    2022 arXiv

  109. [119]

    Chirality, magic, and quantum correlations in multipartite quantum states,

    S. Vardhan, B. Shi, I. H. Kim, and Y. Zou, “Chirality, magic, and quantum correlations in multipartite quantum states,”SciPost Physics20(Mar., 2026) 066, 2503.10764

    2026 arXiv

  110. [122]

    Relative entanglement entropies in 1 + 1-dimensional conformal field theories,

    P. Ruggiero and P. Calabrese, “Relative entanglement entropies in 1 + 1-dimensional conformal field theories,” Journal of High Energy Physics2017(Feb., 2017) 39, 1612.00659

    2017 arXiv

  111. [123]

    Di Francesco, P

    P. Di Francesco, P. Mathieu, and D. Senechal, Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 1997

    1997

  112. [124]

    Fusion Category Symmetry II: Categoriosities at c = 1 and Beyond,

    R. Thorngren and Y. Wang, “Fusion Category Symmetry II: Categoriosities at c = 1 and Beyond,” arXiv e-prints(June, 2021) arXiv:2106.12577, 2106.12577

    2021 arXiv

  113. [125]

    Expectations and entropy inequalities for finite quantum systems,

    G. Lindblad, “Expectations and entropy inequalities for finite quantum systems,”Communications in Mathematical Physics39(June, 1974) 111–119

    1974

  114. [126]

    Trace inequalities and quantum entropy: an introductory course,

    E. Carlen, “Trace inequalities and quantum entropy: an introductory course,”Entropy and the quantum529 (2010), no. 73-140 146

    2010

  115. [127]

    Non-Abelian quasiholes in lattice Moore-Read states and parent Hamiltonians,

    S. Manna, J. Wildeboer, G. Sierra, and A. E. B. Nielsen, “Non-Abelian quasiholes in lattice Moore-Read states and parent Hamiltonians,”Phys. Rev. B98 (Oct., 2018) 165147,1807.11222

    2018 arXiv

  116. [128]

    Bridging conformal field theory and parton approaches to SU(n)k chiral spin liquids,

    T. Liu, Y.-H. Wu, H.-H. Tu, and T. Xiang, “Bridging conformal field theory and parton approaches to SU(n)k chiral spin liquids,”Phys. Rev. B111(May, 2025) 205137,2501.09567

    2025

  117. [129]

    Variational optimization algorithms for uniform matrix product states,

    V. Zauner-Stauber, L. Vanderstraeten, M. T. Fishman, F. Verstraete, and J. Haegeman, “Variational optimization algorithms for uniform matrix product states,”Phys. Rev. B97(Jan., 2018) 045145, 1701.07035

    2018 arXiv

  118. [130]

    Strict Area Law Entanglement versus Chirality,

    X. Li, T.-C. Lin, J. McGreevy, and B. Shi, “Strict Area Law Entanglement versus Chirality,”Phys. Rev. Lett. 134(May, 2025) 180402,2408.10306

    2025

  119. [131]

    Strict area law implies commuting parent Hamiltonian,

    I. H. Kim, T.-C. Lin, D. Ranard, and B. Shi, “Strict area law implies commuting parent Hamiltonian,”arXiv e-prints(Apr., 2024) arXiv:2404.05867,2404.05867

    2024 arXiv

  120. [132]

    Modular commutator in gapped quantum many-body systems,

    I. H. Kim, B. Shi, K. Kato, and V. V. Albert, “Modular commutator in gapped quantum many-body systems,” Phys. Rev. B106(Aug., 2022) 075147,2110.10400

    2022 arXiv

  121. [133]

    Testing the robustness of topological quantities evaluated from the modular Hamiltonian for a given wavefunction,

    S. Sharma and A. C. Balram, “Testing the robustness of topological quantities evaluated from the modular Hamiltonian for a given wavefunction,”arXiv e-prints (Apr., 2026) arXiv:2604.24058,2604.24058

    2026 arXiv

  122. [1995]

    R2493–R2496, https://link.aps.org/doi/10.1103/PhysRevA.52.R2493

    doi

  123. [1996]

    2629–2635,quant-ph/9604022

    arXiv

  124. [1997]

    33–38,quant-ph/9610042

    arXiv

  125. [2026]

    arXiv:2606.20472,2606.20472

    arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.