{"id":"6e751167-1f22-4380-8d7d-3967a631ce62","arxiv_id":"2607.26707","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Quasi-local topological entanglement negativity maps decoherence-driven phase transitions and reveals that color-code and toric-code states respond differently to explosive percolation.","lead":"This paper introduces a local map of topological entanglement, called QLTEN, and uses it to watch quantum decoherence destroy topological order in two quantum codes. It finds the two codes respond differently to biased 'explosive' decoherence, which matters for error-resistant quantum computing and for theories of mixed-state topological order.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"QLTEN's identification rule is unproven for general configurations; if wrong, all QLTEN-based percolation thresholds lose meaning.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing concern: the QLTEN classification rule is heuristic, verified only for special configurations, and the paper explicitly acknowledges the difficulty of general analysis. This is the single most critical point because QLTEN is the paper's new observable and all percolation-based conclusions—cluster growth, threshold extraction, EP sensitivity—depend on its correct labeling of local topological regions. The proposed concrete test would settle the issue by directly comparing the heuristic rule against exact computations for many configurations. Agreement is 'agree' because the reader's concern matches mine. The verdict remains CONDITIONAL: the paper is promising and internally consistent, but this verification is needed before the central claim can be fully accepted. No reason to change the reader's verdict to ACCEPT or REJECT; the concern is real but not yet demonstrated fatal.","tokens_in":20061,"tokens_out":2530,"duration_ms":37462,"concrete_test":"On a small lattice (e.g., 12×12), exhaustively enumerate or randomly sample a large set of decoherence configurations for each 7-plaquette hexagon and its surrounding stabilizers. For each configuration, compute the exact γ_N via Eq. (18) using the stabilizer rank formalism (Eqs. 15–16), and independently evaluate the encircling rule (whether any A, B, or C hexagon is enclosed by a quasi-closed chain of merged S^Z stabilizers). Tabulate the fraction of configurations where the rule predicts γ_N=1 but the exact value differs, and vice versa, near the percolation threshold. If this mismatch rate is non-negligible (e.g., >1%), the QLTEN classification is unreliable and the percolation analysis must be redone with a corrected local rule.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central method—QLTEN snapshots—labels a 7-plaquette subsystem as a 'TC region' (γ_N=1) iff at least one of its A, B, C hexagons is encircled by a (quasi-)closed chain of merged S^Z stabilizers. This rule is verified only for selected configurations (Fig. 8) and the paper explicitly concedes: 'General cases are rather difficult to examine analytically, but the above consideration provides us with a heuristic reflection' (Sec. III.C). The local TEN (Eq. 18) combines contributions from the X and Z parts (Eq. 20), so configurations with partial merging on multiple boundaries, or with a quasi-closed loop that does not fully encircle a subsystem, might also yield γ_N=1 or fail to yield it. If the rule is not exact, QLTEN mislabels spatial regions, and the extracted thresholds p_QL, p_UF and the claimed contrast in EP sensitivity between color code and toric code rest on an invalid local observable. This is a load-bearing unproven lemma about the definition of the new measure, not a minor numerical caveat.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies decoherence-induced transitions from the color code to an emergent toric code, and from the toric code to a Higgs-like phase, using the stabilizer formalism on lattices of up to a few hundred qubits. It introduces QLTEN, the topological entanglement negativity of a 7-plaquette hexagon, as a local probe, and interprets global TEN and the 1-form disorder parameter as first- and zeroth-homology observables. Under uniform and n-trial explosive-percolation (EP) biased decoherence, the authors report that in the color code the QLTEN clusters and global observables are nearly EP-insensitive, while in the toric code QLTEN clusters, TEN, and logical loop operators show clear n-dependence. The central claim is that QLTEN clusters react before the 1-form symmetry and logical-order parameters, and that the two models' different responses to EP indicate the absence of a simple universal microscopic description of these transitions.","tokens_in":20321,"tokens_out":6239,"duration_ms":95411,"significance":"The paper's exact stabilizer-formalism numerics are a strength, as are the multiple independent observables (TEN, string operator, logical loops, QLTEN cluster statistics) and the explicit finite-size scaling with data collapse for the disorder parameter. The percolation/homology analogy is appealing and the QLTEN proposal is a natural way to visualize spatial structure in a topological transition. If the interpretation is substantiated, the work could provide a practical diagnostic for quantum error mitigation in topological codes. However, the central interpretive step—identifying QLTEN γ_N=1 regions with quasi-closed loops—is only heuristic, and the extracted event-based thresholds are quoted without uncertainties. These issues are fixable and do not undermine the exactness of the computed QLTEN values, but they must be addressed before the microscopic claims can be regarded as established.","major_comments":[{"comment":"The identification of QLTEN γ_N=1 with 'at least one of A,B,C is encircled by a (quasi-)closed loop' is verified only for selected configurations; the text concedes that 'general cases are rather difficult to examine analytically' and calls the consideration a 'heuristic reflection.' This is load-bearing because the percolation interpretation, the extraction of p_QL/p_UF, and the claimed color-code/toric-code contrast all use QLTEN clusters as TC/Higgs regions. Since the QLTEN values themselves are computed exactly, the gap is testable: for random decoherence configurations in both codes and a range of p and L, compare the loop-encirclement criterion with the directly computed 7-plaquette TEN and report false-positive/false-negative rates. If the rule has exceptions, the homology-based interpretation and the thresholds must be qualified accordingly.","section":"Sec. III.C, Fig. 8"},{"comment":"The thresholds p_QL and p_UF are reported as point values (e.g., p_QL=0.22 for n=1 and 96; p_UF=0.29 and 0.54) without error bars or finite-size analysis. The event-based ensemble estimator (peak of Δ(t)=C_1(t+1)-C_1(t)) is nonstandard, and no distribution of t_max or sample-to-sample spread is shown. Because the paper uses the difference between p_QL and p_UF to argue that QLTEN 'reacts to decoherence faster than the 1-form symmetry and the non-contractible logical orders,' statistical uncertainties and system-size dependence are needed. Please provide bootstrap or similar errors, the t_max distribution, and a scaling test for at least two additional system sizes.","section":"Sec. IV.A, IV.D, Fig. 13"},{"comment":"The qualitative contrast between the color code (QLTEN insensitive to n-trial EP) and the toric code (clearly n-dependent) is central to the conclusion that a universal microscopic description is challenging. However, the two simulations use different system sizes (color code (12,8), toric code (12,12)) and different maximal n (96 versus 120). Please confirm the contrast at matched system sizes and n, or explain quantitatively why the mismatch cannot affect the comparison. This is particularly relevant to the 'precocious QLTEN' claim in Sec. V, which relies on the timing of QLTEN-cluster growth relative to logical-order disappearance.","section":"Sec. V, Figs. 15-19"}],"minor_comments":[{"comment":"The proposed operator with arbitrary λ is not used anywhere in the paper. If it is only an outlook remark, say so explicitly; otherwise its presence introduces an unused free parameter and distracts from the main results.","section":"Sec. III.C, Eq. (21)"},{"comment":"Typos: 'holomogical' should be 'homological'; 'extra-ordinary' is nonstandard. The manuscript would also benefit from a careful pass for minor grammatical errors.","section":"Sec. VI"},{"comment":"Several variance/largest-cluster panels would be easier to read if the legend explicitly identified the subsystem size or n value inside the panel; currently some identification is only in the caption. Also state in captions whether error bars represent standard error or standard deviation.","section":"Figs. 5, 12, 13"},{"comment":"The disorder parameter D_X(Γ) is defined with ρ_D in both the numerator and denominator; this is clear enough, but for reproducibility please specify the string shape and boundary conditions more explicitly than 'details are specified in the practical calculation.'","section":"Sec. II.C.1, Eq. (13)"},{"comment":"The statement 'available from the authors on reasonable request' is acceptable, but for a numerical paper of this type, providing the stabilizer-formalism code (or a minimal example) would strengthen reproducibility. This is a suggestion, not a requirement.","section":"Data Availability"}],"recommendation":"major_revision","confidential_remarks":"The paper contains exact numerics and a plausible new local observable, but the main interpretive claim (QLTEN as a quasi-closed-loop cohomology) is explicitly heuristic and the thresholds are unquantified. I would not reject the manuscript, but I would ask for a numerical validation of the loop-encirclement rule and error bars on p_QL/p_UF before accepting the percolation-based conclusions. The authors' reliance on their own prior papers for the stabilizer-rank formula is understandable, but the novelty of QLTEN relative to [6] should be clarified in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nPunchline: the QLTEN idea is the real innovation here—a quasi-local evaluation of topological entanglement negativity on 7-plaquette hexagons gives a spatial picture of decoherence destroying topological order. The comparative result between the color code and the toric code under n-trial explosive percolation is also new, and the main contrast (color code insensitive, toric code sensitive) is supported not only by QLTEN clusters but by global TEN and logical-operator data. That is the paper's most robust finding.\n\nWhat's good: the stabilizer-formalism calculations are exact for the models, so the TEN and string-operator numbers are dependable as far as they go. The homology interpretation—TEN as first homology, 1-form disorder parameter as zeroth—is a clean way to understand why the two probes differ in the critical regime. The paper is also honest about its own limitations.\n\nSoft spots, in order: (1) The rule that a hexagon is a \"TC region\" when a quasi-closed loop of merged stabilizers encircles it is verified only for selected configurations; the paper calls it a heuristic reflection. This label is what the percolation thresholds p_QL and p_UF are built from, so the rule is load-bearing. I don't think it's wrong—the snapshots look plausible and the mean QLTEN is just the 7-plaquette TEN—but it needs a proof or a systematic numerical check. (2) The thresholds are quoted without error bars, and the FSS on TEN variance is admittedly unsatisfactory. (3) No code or data files are provided, making independent verification harder than it should be. (4) The TEN scaling law is carried from the authors' own prior papers; that's legitimate, but an independent benchmark would help.\n\nWho it's for: researchers working on decoherence in topological codes, mixed-state topological order, and quantum error mitigation. The QLTEN as an early-warning signal for logical qubit loss is suggestive and worth testing.\n\nIt deserves a serious referee. Send it to peer review with a request to either prove the labeling rule or validate it numerically, and to add error bars and code availability.\n\nBest","headline":"QLTEN is a genuinely useful new probe and the color-code/toric-code contrast under explosive percolation is real, but the labeling rule that defines QLTEN regions is heuristic and needs proof before the extracted thresholds are trusted.","tokens_in":20860,"tokens_out":3924,"would_cite":false,"duration_ms":53660,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Decoherence-induced topological phase transitions are percolation events, and a new quasi-local negativity (QLTEN) makes this visible at the microscopic scale.","keywords":["topological phase transition","decoherence","topological entanglement negativity","color code","toric code","1-form symmetry","percolation","explosive percolation"],"falsifier":"Enumerate all decoherence patterns on a small patch of the triangular lattice, and for each 7-plaquette hexagon in each pattern compute both the exact QLTEN value and whether a (quasi-)closed loop of merged S^Z stabilizers encircles it; find one pattern with QLTEN=1 but no encircling loop, or the reverse, and the heuristic rule is refuted. A coarser check: in many decoherence realizations, compare the percolation threshold read from QLTEN clusters with the threshold read from the exact global TEN; if the two transitions drift apart, QLTEN is not faithfully representing the global topological t","tokens_in":19852,"feed_emoji":"🧩","tokens_out":8898,"duration_ms":115405,"temperature":0.7,"pith_summary":"The paper tries to establish a microscopic, spatially resolved picture of decoherence-driven topological phase transitions: as random noise eats away a stabilizer state, regions of the new topological order grow, merge, and percolate, much like droplets in an Ising magnet. Its central object is QLTEN, a quasi-local topological entanglement negativity computed on 7-plaquette hexagonal subsystems, whose snapshots show γ_N=1 'toric-code regions' nucleating inside the γ_N=2 color-code state. The authors read TEN as a first-homology quantity—a (quasi-)closed loop of merged stabilizer generators—and the 1-form-symmetry disorder parameter as a zeroth-homology quantity—a connected cluster—which explains why the two indicators behave differently in the critical regime. Under biased 'explosive' decoherence, the color-code transition is insensitive to the bias while the toric-code transition is delayed; the authors conclude that a universal microscopic description is challenging even for closely related codes. They flag that the QLTEN identification is a heuristic checked on selected configurations and that a satisfactory finite-size scaling of the TEN variance was not achieved.","feed_headline":"Topological phase transitions under noise are percolation events","feed_subtitle":"A quasi-local entanglement probe tracks the growth of new topological order—and shows two codes respond differently.","key_machinery":"QLTEN: the topological entanglement negativity computed locally on adjacent 7-plaquette hexagonal subsystems (A,B,C) via γ_N = −N_A − N_B − N_C − N_ABC + N_AB + N_BC + N_AC; γ_N=1 marks a toric-code-like region, γ_N=0 a Higgs region. The homological reading: TEN answers to (quasi-)closed loops of merged S^Z stabilizers (first simplicial homology), while the 1-form disorder parameter D_X(Γ) answers to connected clusters of merged stabilizers (zeroth homology). The explosive-percolation decoherence schedule selects, at each step, the bond whose attached clusters have the smallest size product, so large decohered clusters are suppressed.","core_discovery":"On the paper's own terms: the decoherence-induced transition from the color code to the toric code is a percolation transition in the space of anyon-proliferation events. Each red-link XX-decoherence merges two red plaquette stabilizers, so the state's evolution is a growing bond network on a triangular lattice; when a (quasi-)closed loop of merged S^Z stabilizers surrounds a 7-plaquette hexagon, the local TEN drops from 2 to 1, marking a 'TC region.' QLTEN snapshots reveal these regions, and their largest cluster percolates at p≈0.22, earlier than the 1-form disorder parameter's critical point p_s≈0.31. In the toric code under Z-decoherence, QLTEN shows γ_N=0 Higgs regions, and the death of","pith_inferences":["Because QLTEN uses a fixed 7-plaquette resolution, its cluster statistics (thresholds, exponents) may drift as the subsystem size changes; a natural test is to repeat the percolation analysis with larger and smaller hexagonal subsystems and check convergence.","The color code's insensitivity to explosive percolation hints that its transition is driven by local anyon-pair creation rather than by the growth of global decohered clusters; this could be tested in other stabilizer codes, such as the surface code with different boundary conditions.","If QLTEN is a genuine percolation order parameter, its cluster-size distribution should obey standard percolation scaling; comparing its cluster-size exponent with bond percolation on the triangular lattice would be a quantitative check.","The paper leaves open whether the second-homology loop operator restores a universal description; a direct test is to measure its expectation value in both codes under the same explosive-percolation schedule and see whether it tracks a common threshold."],"forward_implications":["QLTEN configurations can be assembled from the stabilizer syndrome on a classical computer, making the percolation of TC or Higgs regions observable as an early warning that logical qubits are about to fail.","The non-coincidence of the 1-form critical point (p_s≈0.31, ν≈1.2, close to bond percolation) and the TEN-variance peak (p≈0.22) is explained by their different homological character: cluster vs loop.","In the toric code, the moment the largest QLTEN 'Higgs' cluster spans the system is the moment the logical qubit disappears; QLTEN is precocious relative to the non-contractible-loop order.","Biased decoherence results imply that fault-tolerance thresholds and noise-bias strategies are code-specific and will not transfer between the color code and the toric code.","The operator ∏_{⟨i,j⟩∈loop}(λ+X_i X_j) is introduced as a second-homology probe for the emergence of the Z2 gauge theory in the color-code-to-toric-code transition."],"fun_headline_variants":["Percolation drives noise-induced topological transitions","Entanglement probe shows percolation in phase shifts","Two codes, two paths: percolation in topological transitions","Quasi-local probe tracks topological order growth by percolation","Noise triggers topological shifts via percolation, probe finds"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that a 7-plaquette hexagon has γ_N=1 (the toric-code value) exactly when a (quasi-)closed loop of merged S^Z stabilizers encircles it; the paper checks this rule for selected configurations and explicitly says general cases are difficult to analyze analytically, so if other configurations violate it, QLTEN's snapshots mislabel the local topological order and the conclusions built on those labels weaken.","fun_headline_variants_meta":{"raw":{"variants":["Percolation drives noise-induced topological transitions","Entanglement probe shows percolation in phase shifts","Two codes, two paths: percolation in topological transitions","Quasi-local probe tracks topological order growth by percolation","Noise triggers topological shifts via percolation, probe finds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1251,"prompt_tokens":786,"completion_tokens":465,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":385}},"tokens_in":530,"tokens_out":465,"duration_ms":6907,"temperature":1.0,"reasoning_tokens":385,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T03:17:18.348678+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all decoherence patterns on a small patch of the triangular lattice, and for each 7-plaquette hexagon in each pattern compute both the exact QLTEN value and whether a (quasi-)closed loop of merged S^Z stabilizers encircles it; find one pattern with QLTEN=1 but no encircling loop, or the reverse, and the heuristic rule is refuted. A coarser check: in many decoherence realizations, compare the percolation threshold read from QLTEN clusters with the threshold read from the exact global TEN; if the two transitions drift apart, QLTEN is not faithfully representing the global topological t","supporting_citations":[],"review_version":2}