{"id":"f20f28a8-176a-4ccc-901c-4a87cbb9327a","arxiv_id":"2506.02970","paper_version":3,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Weakly noisy quantum dynamics makes small-support integrals of motion appear as the slowest-decaying operators, which can be used to identify exact and approximate conservation laws.","lead":"This paper shows that in quantum systems with weak noise, conserved quantities that act on a few sites decay much more slowly than ordinary operators, so the slowest-decaying modes of the noisy dynamics reveal those conservation laws. The authors test this on several spin-chain models and propose using it as a tool to find hidden conserved quantities, including approximate ones.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Mori-gap convergence in Sec. IV is the weak point: if perpendicular-space hydrodynamic modes or other IOMs cause slow J1(t) decay, the eigenoperator–IOM correspondence can fail without leaving any numerical trace in the L≤9 data.","rationale":"The reader's weakest_assumption is exactly the Mori-gap convergence in Sec. IV, Eqs. (16)–(22): the mutual orthogonality of J1/J2 to all other slow modes and the finiteness of E_g^<. I agree this is the load-bearing concern. My independent reading of Appendix A strengthens it: the variance bound (A7) does not establish a small commutator for slow eigenoperators absent an unproven 'reasonable distribution' assumption, so the converse direction (slow eigenvalue implies approximate IOM) is also unproven. The paper's own Sec. IV Third comment concedes the Mori gap 'might not exist at all' after projecting out hydrodynamic modes, which is an internal concession. The numerics in Fig. 5(a,b) (a single MFIM point, L=9) and the spectrum plots in Figs. 2–4 cannot exclude a small but non-vanishing V_perp, and the correspondence for approximate IOMs in Secs. IIIC–D relies on the same unproven assumption. Thus the verdict CONDITIONAL is appropriate; I am not recommending REJECT because the exact-IOM numerics are convincing and the theory is plausible, and no internal inconsistency has been demonstrated. The concrete test above would settle whether the Mori-gap assumption holds in a controlled setting.","tokens_in":22715,"tokens_out":1965,"duration_ms":20861,"concrete_test":"Numerically test the Mori-gap assumption in a system where the perpendicular subspace is forced to be non-generic. Take the MFIM parameters of Fig. 5 (J=1, hx=2, hz=0.05), but with the parallel subspace enlarged to include the exact and approximate IOMs {A_m, B_m, H} (up to size ~4), and compute the projected Lindbladian L_perp on the orthogonal subspace. If the slowest decay rate of L_perp is O(γ) for L=7,9,11 (i.e., E_g^< ~ γ, not sqrt(v_B γ)), then Eq. (22) gives ||V_perp|| ~ ϵ1/γ, which can be O(1) and would invalidate the correspondence for approximate IOMs. Alternatively, run exact time evolution of J1(t) for an approximate IOM in this enlarged parallel space and check whether ||J1(t)|| decays significantly faster than e^{γ t} over a time window large enough to converge the integral in Eq. (20); if the integral does not converge, the Mori-gap assumption fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim needs the perpendicular component J1(t) in Sec. IV, Eqs. (16)–(22), to decay quickly enough that ||V_perp|| ≈ ϵ1/E_g^< is small. The authors assume a Mori gap E_g^< ~ min(γL, sqrt(v_B γ)), relying on J1(t) growing ballistically and decaying faster than e^{γ_V t}. This is load-bearing because the eigenoperator–IOM correspondence, and the proposed IOM-discovery method, hinge on V_perp being small. The paper itself identifies the danger in Sec. IV, Third comment: if J1 or J2 overlap with other IOMs or hydrodynamic modes, the perpendicular dynamics need not be chaotic, and no convergence of Eq. (20) is guaranteed. The numerical checks in Fig. 5(a,b) probe a single MFIM point, and the L≤9 spectra cannot exclude slow modes localized in the perpendicular subspace. A second, sharper gap: Appendix A's variance bound (A7) does not exclude a large |[H,V]| for a slow mode, because the size-distribution variance is only bounded under a 'reasonable distribution' assumption not proven from the Lindblad equation. Thus the converse direction (slow eigenvalue implies approximate IOM) is also weaker than claimed. The authors themselves concede in Sec. IV that the Mori gap 'might not exist at all' once hydrodynamic modes are projected out; if that concession is realized, the perturbation theory collapses and the central correspondence becomes a finite-size, small-γ artifact. This is an internal-consistency risk in the perturbative argument, not a disagreement with external consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper considers the Heisenberg-picture Lindbladian dynamics of a many-body spin chain under weak local depolarizing noise. Its main claim is that integrals of motion (IOMs) with small operator size decay more slowly than generic operators, so that the low-lying eigenoperators of the Lindbladian superoperator L-dagger have large overlap with the small-sized IOMs of the underlying Hamiltonian. The authors support this with exact diagonalization of several models (MFIM, TFIM, Heisenberg chains) and with a perturbative Mori-based argument. They also use the low-lying eigenoperators to detect approximate IOMs in the mixed-field Ising model and in Heisenberg chains, and propose a 'chaoticity measure' based on the overlap of the first few slow modes with powers of the Hamiltonian.","tokens_in":23153,"tokens_out":5807,"duration_ms":66769,"significance":"The numerical core is clean, and the exact statement that depolarizing noise makes an operator decay at rate gamma times its Pauli size (Eq. (5)) is a useful starting point. If the correspondence is correct, the paper offers a practical spectral tool for identifying exact and approximate local conservation laws, of interest for prethermalization and for quantum-simulation experiments. The strength of the claim, however, rests on two unproven assumptions: fast decay of the perpendicular component J1(t) in Section IV and the 'reasonable distribution' variance bound in Appendix A. Until these are either justified or stated as standing assumptions, the theoretical framing is conditional rather than established.","major_comments":[{"comment":"The estimate ||V_perp|| <~ |epsilon_1| integral e^{gamma_V t} ||J_1(t)|| dt in Eq. (20) is the load-bearing step of the perturbative justification, and it depends critically on the assumed stretched-exponential or ballistic decay in Eq. (21). The authors explicitly concede in the first comment of Section IV that after projecting out exact IOMs and hydrodynamic modes H_k, the Mori gap 'might not exist at all.' Since J_1 is defined only by orthogonality to Q, it may still overlap with other IOMs or with hydrodynamic modes, in which case ||J_1(t)|| may decay only algebraically or saturate, making the integral in Eq. (20) diverge and the expansion V = Q + V_perp uncontrolled. The numerical evidence in Fig. 5(a,b) is a single MFIM point at L=9 and cannot rule out slow modes living in the perpendicular subspace. This does not invalidate the numerical spectra, but it means the perturbative argument does not yet establish the eigenoperator-IOM correspondence in general.","section":"Section IV, Eqs. (16)-(22)"},{"comment":"The converse direction, that a slow eigenvalue implies an approximate IOM, uses Eq. (A7), ||[H,V]||^2 = gamma^2 ( <S^2>_V - <S>_V^2 ). The paper claims that for 'reasonable distributions' the variance is O(1) times <S>_V^2, but this is an assumption and is not derived from the Lindblad equation. A slow eigenoperator with a narrow distribution of Pauli sizes, for example one dominated by a single size, could have small variance while still having a large commutator with H. In addition, the derivation assumes V is Hermitian, which is justified only for real eigenvalues; the low-lying spectrum is not shown to be real, and complex-conjugate pairs would require a separate argument. Thus the claimed converse, stated in Section III as 'under mild conditions, ||[H,Q]|| = O(1) x gamma', is weaker than stated.","section":"Appendix A, Eq. (A7)"},{"comment":"The identification of approximate IOMs in the Heisenberg and mixed-field Ising examples is partly circular: the operators are discovered by reading them off the low-lying eigenoperators and then the same eigenoperators are cited as confirmation that these operators are approximate IOMs. For the newly proposed approximate IOMs, such as S_x^2 + S_y^2 in the XXZ case, the paper should report independent diagnostics, for example the norm of [H,Q] normalized by ||Q|| or the decay of Q(t) under unitary dynamics, rather than only the overlap with the eigenoperator used to find them.","section":"Section III.C-D"}],"minor_comments":[{"comment":"The phrase 'under mild conditions [51]' is too vague; the reader should be pointed to the precise assumptions in Appendix A, namely real eigenvalue and the Pauli-size variance bound.","section":"Section III"},{"comment":"The color-scale limits for the overlap matrices are not stated; adding explicit color bars or numeric values would make the near-unity and near-zero overlaps easier to assess.","section":"Fig. 2(c) and Fig. 3"},{"comment":"The quantities E_g^< and E_g^> are used before being defined; they should be introduced as the inverse decay timescales of ||J_1(t)|| and (J_2|J_1(t)), respectively.","section":"Section IV"},{"comment":"Calling the expression in Eq. (23) the decay rate is potentially misleading, since it is only the leading correction to gamma_Q; the full decay rate also contains the first-order term gamma_Q.","section":"Eq. (23)"},{"comment":"The term 'Loschmidt echo' is used informally for the Frobenius norm decay; please align the terminology with Eq. (5), which defines the norm decay rate rather than a Loschmidt echo amplitude.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a good fit for the journal and the numerics are valuable. The main issue is that the abstract and Section III state the eigenoperator-IOM correspondence in an unconditional form, while Section IV itself concedes that the Mori gap might not exist once hydrodynamic modes are included. I would support publication after the authors either prove the needed conditions, make them explicit standing assumptions with clear caveats in the abstract, or add numerical evidence probing the perpendicular subspace, for example the decay of J_1(t) in a model with hydrodynamic modes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a useful paper. The qualitative point that conserved operators decay more slowly under local noise was already in the air, but the systematic eigenoperator–IOM correspondence tested across MFIM, TFIM, and Heisenberg chains is new, and the proposed method for detecting approximate IOMs is a genuine step forward. The numerics are clean: the overlap matrices in Figs. 2–3 are convincing, and the recovery of known IOMs gives independent grounding. Appendix E, where the authors derive the complete set of Majorana-bilinear IOMs for the TFIM, is a nice analytic contribution. Data are on GitHub, which helps.\n\nThe soft spots are real but mostly where the authors themselves point. The converse direction—slow eigenvalue implies approximate IOM—relies on Appendix A's real-eigenvalue assumption and on the size-variance bound in Eq. (A7), which is asserted for 'reasonable distributions' rather than proven from the Lindblad equation. Since the superoperator is non-Hermitian, real eigenvalues are not automatic, and this gap matters. The bigger issue, as the stress-test notes, is the convergence of the perturbative integral in Eq. (20): the bound on ||V_perp|| needs J1(t) to grow ballistically and then decay fast, and if the perpendicular subspace contains hydrodynamic or other slow modes, the Mori gap may not exist. The authors concede this in Sec. IV, saying the gap 'might not exist at all' after projecting out hydrodynamic modes. The L<=9 spectra cannot exclude that scenario. This is a genuine limitation, but it is stated openly and the paper does not oversell the rigorous status of the perturbation theory.\n\nI do not think the circularity concern in the approximate-IOM discovery is fatal. The method is validated against known IOMs first; reading off new candidates from eigenoperators and then reusing them as confirmation is somewhat self-referential, but the candidates are plausible and backed by independent literature (the dressed particle number of Wurtz–Polkovnikov, the Sx^2+Sy^2 near the XXX point, the domain-wall swap at large Delta). Independent commutator-norm checks would strengthen this, and that is a fair referee request, not a fatal flaw.\n\nWho is this for? Researchers working on open quantum systems, operator growth, and conservation-law detection in many-body models. It deserves a serious referee. I would send it to peer review with requests to (a) prove or weaken the real-eigenvalue claim, (b) test the size-variance bound numerically or analytically, (c) add independent commutator-norm checks for new approximate IOMs, and (d) release runnable code with a commit hash. None of these are desk-reject grounds; the central numerical correspondence is solid and the limitations are honestly flagged.","headline":"A clean numerical demonstration that low-lying Lindbladian eigenoperators overlap with integrals of motion, with an honest perturbative framework whose converse direction rests on unproven assumptions; worth refereeing seriously.","tokens_in":23539,"tokens_out":1965,"would_cite":true,"duration_ms":25040,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Weak dissipation turns a Hamiltonian's conserved operators into its slowest-decaying modes, so diagonalizing the Lindbladian recovers exact and approximate integrals of motion.","keywords":["Lindblad equation","integrals of motion","slow modes","operator growth","dissipative quantum dynamics","exact diagonalization","spin chains","approximate conservation laws"],"falsifier":"Measure the slowest-decaying operators in a weakly depolarized spin chain, for example the transverse-field Ising model at $J=1$, $h=0.5$, by exact diagonalization at $L=10$–$12$ or by randomized measurements on a quantum simulator, and compare the reconstructed slow modes with the Majorana bilinear operators of Eqs. (9)–(11); if the reconstructed operators have little overlap with the known IOMs, or if the first few decay rates do not track the IOM sizes, the correspondence is ruled out.","tokens_in":22508,"feed_emoji":"⏳","tokens_out":9381,"duration_ms":103069,"temperature":0.7,"pith_summary":"This paper claims that in a many-body quantum system subjected to weak local noise, every integral of motion (IOM) with small support—an operator that commutes with the Hamiltonian and acts on few sites—becomes a slow mode of the dissipative dynamics. Because a generic operator grows in size under chaotic unitary evolution while an IOM does not, the IOM's norm decays exponentially slowly while a generic operator decays superexponentially; consequently, the slowest-decaying eigenoperators of the Lindbladian superoperator are almost exactly linear combinations of the Hamiltonian's IOMs. The paper verifies this correspondence by exact diagonalization for the mixed-field, transverse-field, and Heisenberg spin chains, and supports it with a perturbative argument based on a finite spectral gap in the subspace orthogonal to the IOMs. If correct, the result gives a spectral method to identify exact and approximate conservation laws—including prethermal symmetries—from weakly dissipative dynamics, on a classical computer or in a quantum simulator.","feed_headline":"Conserved operators become the slowest modes under weak noise","feed_subtitle":"Diagonalizing a weakly dissipated spin chain recovers its exact and approximate integrals of motion.","key_machinery":"The central object is the spectrum of the Heisenberg-picture Lindbladian together with the decomposition of operators into Pauli strings, whose decay rates under depolarizing noise are proportional to their sizes. The load-bearing mechanism is the split $L^\\dagger[Q] = -\\gamma_Q Q + \\varepsilon_1 J_1$, which separates a candidate conserved operator $Q$ from its orthogonal fast component $J_1$; the spectral gap of the orthogonal subspace is estimated as $\\min(\\gamma L,\\sqrt{v_B\\gamma})$, which makes the perturbative eigenoperator $V = Q + V_\\perp$ converge with $\\lVert V_\\perp\\rVert \\lesssim \\varepsilon_1/E_g^<$ small. This is what converts the intuitive growth-versus-no-growth picture into the quantitative claim that low-lying eigenoperators are the IOMs.","core_discovery":"In the Heisenberg-picture Lindbladian with weak local depolarizing noise, the decay rate of an operator is approximately the noise strength times its operator size. Under chaotic unitary dynamics a generic operator spreads ballistically, so its norm decays superexponentially, while a conserved operator keeps its support fixed and decays only as $e^{-\\gamma S t}$. The paper's central assertion is therefore that IOMs with $O(1)$ support are slow modes, and that the low-lying eigenoperators of the Lindbladian—those with decay rate $O(1)\\times\\gamma$—have large overlap with the exact and approximate IOMs of the underlying Hamiltonian. This is demonstrated for the mixed-field Ising model, where the first three eigenoperators are combinations of the identity, the energy, and its square; for the transverse-field Ising model, where the first five are Majorana bilinear IOMs; and for Heisenberg chains, where up to twenty-five eigenoperators are accounted for by known IOMs and additional slow modes reveal approximate conservation laws such as the spin current and $S_x^2+S_y^2$. The paper further argues, by a perturbation theory around each IOM $Q$, that an operator satisfying $L^\\dagger[Q] = -\\gamma_Q Q + \\varepsilon_1 J_1$ becomes an eigenoperator when the non-conserved component $J_1(t)$ decays fast enough, with the relevant gap scaling as $\\min(\\gamma L,\\sqrt{v_B\\gamma})$ rather than the exponentially small many-body level spacing.","pith_inferences":["Inference: the correspondence suggests a search strategy the paper does not itself implement—vary the noise strength across the crossover where the commutator-induced decay balances the noise-induced decay and track which operators enter the slow subspace, thereby estimating the commutator norms of hidden approximate IOMs.","Inference: if the gap argument extends to higher-dimensional lattices, weak local dissipation could turn any quantum simulator into a spectrometer for conserved densities, including multipole or subsystem conservation laws whose operators have small support.","Inference: the decomposition into a conserved parallel part and a fast orthogonal part can serve as a variational ansatz: given a candidate IOM, minimize the norm of $L^\\dagger[Q]+\\gamma_Q Q$ to find the best approximate conservation law; the paper defines exactly this residual but does not run such an optimization."],"forward_implications":["Diagonalizing a weakly dissipative Lindbladian yields exact IOMs: for the transverse-field Ising model the first five eigenoperators are the Majorana bilinear conserved operators, and for Heisenberg chains the low-lying eigenoperators are spanned by total spin, energy, and the higher conserved operator $K$.","Approximate conservation laws appear as slow modes alongside exact ones: the dressed particle number in the mixed-field Ising model at $h_x=h_z=0.4$, the spin current at anisotropy $\\Delta=0.1$, and $S_x^2+S_y^2$ at $\\Delta=1.5$ all show up among the slowest eigenoperators.","The first few eigenoperators define a measure of chaoticity: a model is fully chaotic only if its slow modes are linear combinations of powers of the Hamiltonian; mapping this over the mixed-field Ising model parameters shows that large regions with a weak transverse field are not fully chaotic despite lacking exact local IOMs.","Because the decay rate of an approximate IOM contains a noise-independent commutator contribution alongside a term linear in $\\gamma$, tuning the noise strength scans which approximate symmetries are visible: stronger noise favors small approximate IOMs, while weaker noise tends to hide them behind exact ones.","The same physical picture extends to non-depolarizing local noise up to caveats: for dephasing-decay noise on Heisenberg chains, additional slow modes such as projectors onto polarized states appear, so the correspondence between low-lying eigenoperators and IOMs holds with qualification."],"supporting_citations":[{"why":"Supplies the weak-dissipation Liouvillian spectrum and the gap structure that the perturbation theory in Section IV builds on.","marker":"[24]"},{"why":"Gives the $E_g^< \\sim \\min(\\gamma L,\\sqrt{v_B\\gamma})$ scaling for spectral gaps in the weak-dissipation limit, used to show the perturbative correction $V_\\perp$ stays small.","marker":"[25]"},{"why":"Provides the operator-growth picture in open quantum systems, where ballistic growth causes superexponential decay of generic operators under weak noise.","marker":"[34]"},{"why":"Experimental precedent where slowly decaying Pauli strings identify a Majorana zero mode, motivating the idea that IOMs appear as slow modes.","marker":"[13]"},{"why":"The universal operator growth hypothesis used to justify the ballistic size growth $S(O(t)) \\sim v_B t$ for generic operators.","marker":"[43]"},{"why":"Establishes the absence of local conserved quantities in the generic mixed-field Ising model, so the expected IOMs are the identity, energy, and powers of energy.","marker":"[55]"},{"why":"Lists the exact integrals of motion of Heisenberg chains, used to match the low-lying eigenoperators to known conserved operators.","marker":"[68]"},{"why":"Constructs the dressed particle-number operator in the mixed-field Ising model that the paper detects among the slow modes.","marker":"[69]"}],"fun_headline_variants":["Conserved operators are the slowest modes under weak dissipation","Weak dissipation reveals integrals of motion via operator decay","Integrals of motion are the slow modes of weakly open systems","Operator size sets lifetime: conserved operators decay slowest","Slow Lindbladian eigenoperators are hidden integrals of motion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that the non-conserved part of each IOM grows and decays quickly enough that the perpendicular component $V_\\perp$ stays small; this requires that the subspace orthogonal to the known IOMs contains no other slow or hydrodynamic modes, a condition the authors concede may fail.","fun_headline_variants_meta":{"raw":{"variants":["Conserved operators are the slowest modes under weak dissipation","Weak dissipation reveals integrals of motion via operator decay","Integrals of motion are the slow modes of weakly open systems","Operator size sets lifetime: conserved operators decay slowest","Slow Lindbladian eigenoperators are hidden integrals of motion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00084,"raw_usage":{"total_tokens":3677,"prompt_tokens":980,"completion_tokens":2697,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":596,"completion_tokens_details":{"reasoning_tokens":2617}},"tokens_in":596,"tokens_out":2697,"duration_ms":20831,"temperature":1.0,"reasoning_tokens":2617,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:12:29.172580+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the slowest-decaying operators in a weakly depolarized spin chain, for example the transverse-field Ising model at $J=1$, $h=0.5$, by exact diagonalization at $L=10$–$12$ or by randomized measurements on a quantum simulator, and compare the reconstructed slow modes with the Majorana bilinear operators of Eqs. (9)–(11); if the reconstructed operators have little overlap with the known IOMs, or if the first few decay rates do not track the IOM sizes, the correspondence is ruled out.","supporting_citations":[{"cited_title":"Liouvillian-gap analysis of open quantum many-body systems in the weak dissipation limit,","cited_arxiv_id":null,"evidence_quote":"Supplies the weak-dissipation Liouvillian spectrum and the gap structure that the perturbation theory in Section IV builds on."},{"cited_title":"Operator Growth in Open QuantumSystems,","cited_arxiv_id":null,"evidence_quote":"Provides the operator-growth picture in open quantum systems, where ballistic growth causes superexponential decay of generic operators under weak noise."},{"cited_title":"Proof of absence of local conserved quantities in the mixed-field Ising chain,","cited_arxiv_id":null,"evidence_quote":"Establishes the absence of local conserved quantities in the generic mixed-field Ising model, so the expected IOMs are the identity, energy, and powers of energy."},{"cited_title":"QUANTUM INTE- GRALS OF MOTION FOR THE HEISENBERG SPIN CHAIN,","cited_arxiv_id":null,"evidence_quote":"Lists the exact integrals of motion of Heisenberg chains, used to match the low-lying eigenoperators to known conserved operators."}],"review_version":1}