{"id":"52814342-0aad-4fa0-bd09-e14fb903033d","arxiv_id":"2501.09708","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Sandwiching a state by ρ_B^{-1/2} turns Belavkin-Staszewski quantum Markov chains into ordinary quantum Markov chains, yielding recovery maps and superexponential conditional-independence decay.","lead":"The paper proves that quantum conditional-independence states defined through the Belavkin-Staszewski relative entropy are exactly ordinary quantum Markov chains after a simple reweighting, and uses this to build a Petz-style recovery map and a structural classification. The payoff is the first family of states whose conditional mutual information decays superexponentially with the size of the conditioning system, a new benchmark for correlation decay.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6.1 rests on an unproved prefactor bound: the proof concludes with the tautology ∥ρ_B^{-1}∥∞ ≤ C∥ρ_B^{-1}∥∞, so the advertised superexponential CMI decay is not yet established.","rationale":"The reader's weakest-assumption analysis correctly identifies the proof of Theorem 6.1 as the main gap. The manuscript itself contains the explicit tautological inequality in Section 6.1, and the proof does not independently establish the prefactor bound. This is a concrete, checkable defect in the application section, not a manufactured concern. I do not see a comparable flaw in the main classification theorem: Theorem 3.3's directions are argued with explicit algebra, and the recovery-map equivalence in Corollary 3.9 is plausible even though the converse direction is stated tersely. Proposition 4.5's reverse inequalities require the commuting-marginal assumption [η_AB,η_BC]=0, but this is acknowledged in the paper and is not the source of the Theorem 6.1 gap, since Theorem 6.1 uses only the unconditional lower-bound direction. Because the reader already issued a CONDITIONAL verdict and the concern I find is the same one, no verdict adjustment is needed; the appropriate recommendation is to keep the paper conditional until the prefactor bound is supplied or the external result is explicitly quoted.","tokens_in":31728,"tokens_out":21283,"duration_ms":200745,"concrete_test":"Locate [7, Corollary 4.4] and check whether it actually proves an upper bound on ∥ρ_B^{-1}∥∞ for finite-range 1D Gibbs states at positive temperature, and whether the bound is at most exponential in |B|. If yes, insert that bound in place of the final tautological inequality in the proof of Theorem 6.1 and verify the prefactor. If the external bound is missing or weaker, compute ∥ρ_B^{-1}∥∞ numerically for a noncommuting chain such as the transverse-field Ising chain at fixed β, for |B| up to about 14, and compare its growth with the decay rate of ε(|B|) used in Theorem 6.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central correspondence Theorem 3.3 and the recovery-map Corollary 3.9 appear internally coherent, and I do not see a defect requiring a change to the core classification result. The load-bearing weak point is the proof of Theorem 6.1, which is the advertised application producing the first family of states with superexponentially decaying CMI. After a chain of identities for ρ_B^{-1}, the proof states: 'Therefore, by [7, Corollary 4.4], we conclude ∥(ρ_B)^{-1}∥∞ ≤ C∥(ρ_B)^{-1}∥∞.' This displayed inequality is tautological: the constant C is not shown to be finite in a useful way, and no explicit exponential (or other sufficiently mild) bound on ∥ρ_B^{-1}∥∞ is derived. The desired bound on Iη(A:C|B) contains the prefactor ∥ρ_B^{-1}∥∞^{1/2}; for the product with the superexponentially decaying ε(|B|) to remain superexponential, this prefactor must not grow faster than the decay of ε, and at minimum its growth must be controlled. As written, that control is not supplied by the proof; it is delegated to an external result whose exact content and applicability are not quoted. If [7, Corollary 4.4] does provide a suitable bound, the gap is a simple repair; if it does not, Theorem 6.1 is unsupported. This does not affect Theorem 3.3, but it does affect the paper's headline application.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces Belavkin-Staszewski quantum Markov chains (BS-QMCs), states with vanishing BS-conditional mutual information, and relates them to ordinary quantum Markov chains (QMCs) via the map η_ABC = d_B^{-1} ρ_B^{-1/2} ρ_ABC ρ_B^{-1/2}. The central result, Theorem 3.3, characterizes BS-QMCs as exactly those states whose associated η_ABC is a QMC with maximally mixed B marginal; it also yields a structural decomposition and a recovery map Φ_{B→AB} in Petz form with an additional unitary, including the converse direction that was left open in [14]. The paper extends the correspondence to general states and channels saturating the BS data-processing inequality (Section 5), studies approximate BS-QMCs (Propositions 4.5 and 4.6), gives examples of BS-QMCs with entangled AC marginals (Proposition 4.3), and applies the correspondence to quantum spin chains, claiming superexponential decay of the CMI of η_ABC for Gibbs states of local, finite-range, translation-invariant Hamiltonians (Theorem 6.1) and a bound on the quantity Δ_ρ relevant to spectral gaps (Proposition 6.2).","tokens_in":32059,"tokens_out":2502,"duration_ms":26972,"significance":"If the main results stand, this is a substantial contribution to the structural theory of quantum conditional independence. Theorem 3.3 gives a complete classification of BS-QMCs in terms of the Hayden–Jozsa–Petz–Winter QMC structure theorem, and Corollary 3.9 resolves the open recovery-map question from [14]. The recovery map Φ_{B→AB} is linear, completely positive, and resembles the Petz map, which is a notable step since the original BS recovery condition is not positive. The generalization to BS-triples and Petz-triples in Section 5, with explicit structural decompositions, is valuable and appears to be proven carefully. The paper also provides concrete constructive examples, including entangled AC marginals for BS-QMCs, which sharpen the contrast with QMCs. The advertised spin-chain application, if its prefactor bound is properly established, would provide the first family of states with non-vanishing CMI that decays superexponentially with |B|; currently that application is not fully supported by the proof as written, so the overall significance is conditional on repairing Section 6.1.","major_comments":[{"comment":"The proof does not establish the required prefactor growth. After the chain of identities for ρ_B^{-1}, the text states: 'Therefore, by [7, Corollary 4.4], we conclude ∥(ρ_B)^{-1}∥_∞ ≤ C∥(ρ_B)^{-1}∥_∞.' This displayed inequality is tautological: it provides no bound on ∥ρ_B^{-1}∥_∞ beyond itself, and the constant C is not shown to be finite or to scale in a controlled way. The final bound in Theorem 6.1 contains the factor ∥ρ_B^{-1}∥_∞^{1/2}, so to conclude superexponential decay the proof must show that this factor grows at most sub-superexponentially, typically at most exponentially in |B|. The cited [7, Corollary 4.4] is not quoted, and its hypotheses and conclusion are not checked against the present setting. This is a load-bearing gap: it affects the paper's headline application, even though it does not affect the internal coherence of Theorem 3.3.","section":"Section 6.1, proof of Theorem 6.1"},{"comment":"The bound on ∥ρ_{BC}^{-1/2}ρ_{ABC}ρ_{BC}^{-1/2}∥_∞ is delegated to '[4, Theorem IX.1.1]' and to '[7, Corollary 3.4(i)]' and an 'analogous proof' to '[7, Corollary 4.4]'. The latter is not carried out, and the text does not state the resulting constants. Since this term also enters the prefactor of the theorem, the proof would benefit from an explicit statement of the bounds being used, including the dependence on |B| and on the inverse-temperature, strength, and range parameters. This is a presentation gap in the same chain of reasoning as the previous comment, and it should be addressed together with it.","section":"Section 6.1, proof of Theorem 6.1, final paragraph"}],"minor_comments":[{"comment":"There are several typographical errors, including 'beging' (page 12), 'satruate' (page 24), and 'Gibss' (page 26). These should be corrected in a revision.","section":"General"},{"comment":"The proof of the lower bound invokes [6] for the inequality I_η(A:C|B) ≤ 2(log min{d_A,d_C}+1) ∥η_ABC - P(η_AB)∥_1^{1/2}. The cited reference is an IEEE conference paper that may not be readily available; quoting the precise inequality and its hypotheses would improve self-containedness.","section":"Section 4.2, Proposition 4.5"},{"comment":"The notation bI_ρ^{os}(A;C|B), bI_ρ^{ts}(A;C|B), and bI_ρ^{rev}(A;C|B) is introduced, but the superscripts are not explained in the text; the reader must infer that 'os', 'ts', and 'rev' stand for 'operator', 'trace', and 'reversed'. A short sentence defining the labels would clarify the definition.","section":"Section 2.2, Eq. (7)"},{"comment":"The statement of Proposition 4.3 says 'ρ_ABC is a BS-QMC and ρ_AC has negative partial transpose if and only if α ∈ [-1, 1-√3)' but the proof only shows the 'if' direction by explicit computation. The 'only if' part appears to follow from the minimum of the quadratic expression, but it would be helpful to state this explicitly.","section":"Section 4.1, Proposition 4.3"},{"comment":"In the proof of (iv) ⇒ (ii), the displayed chain of equalities uses the identity tr[T(ρ)T(σ)^{-1}T(ρ)] = tr[ρ T^*(T(σ)^{-1}T(ρ))]; this is correct, but it may not be immediately obvious to all readers that T^* is taken with respect to the Hilbert-Schmidt inner product. A brief reminder would improve readability.","section":"Section 5.2, Theorem 5.13"},{"comment":"The function f(ρ) is said to be 'explicitly defined in the proof', but the proof defines it as g(ρ) = g_1(ρ)^{-1}; the final statement would be clearer if the dependence on d_A, d_C, d_D and on the norm factors were displayed in the proposition itself.","section":"Section 6.2, Proposition 6.2"}],"recommendation":"major_revision","confidential_remarks":"The central equivalence and the recovery-map results appear sound and are the main contribution of the paper. The reason for major revision is the gap in the proof of Theorem 6.1, which is the advertised application. The authors should either derive the needed bound on ∥ρ_B^{-1}∥_∞ by quoting [7, Corollary 4.4] with its exact statement and verifying its hypotheses, or replace Theorem 6.1 with a weaker but fully proven statement. If the external result indeed provides the missing bound, the revision is straightforward; otherwise the claim of superexponential CMI decay is unsupported. I do not see circularity in the main classification result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe core result here is real. Theorem 3.3 establishes a clean correspondence: ρ_ABC is a BS-quantum Markov chain iff η_ABC = d_B^{-1} ρ_B^{-1/2} ρ_ABC ρ_B^{-1/2} is an ordinary quantum Markov chain with η_B = τ_B. That gives a structural decomposition directly, and Corollary 3.9 answers the open question from Gondolf et al. by proving the Φ recovery condition is an iff. The proof of the correspondence is short but airtight; the decomposition inherits the Hayden-Jozsa-Petz-Winter structure. The example family with entangled ρ_AC (Prop 4.3) is a nice addition, showing the BS-QMC set is genuinely larger and the correspondence is entanglement-breaking between A and C while the inverse map can create entanglement.\n\nThe generalization in Section 5 to BS-triples and Petz-triples is more than bookkeeping. Theorem 5.7 and Corollary 5.11 give a completely positive recovery map in Petz form with an extra unitary factor, and Theorem 5.13 gives structural decompositions for both directions. I read these proofs as careful and self-contained. The dependence on prior work by the same authors ([5], [7], [14]) is for external results, not for this paper's claims; no circularity.\n\nThe soft spot is Section 6.1. Theorem 6.1 claims superexponential decay of I_η for η associated to 1D Gibbs states. The proof tries to bound the prefactor ∥ρ_B^{-1}∥_∞ and ends with the displayed inequality ∥ρ_B^{-1}∥_∞ ≤ C∥ρ_B^{-1}∥_∞. That is tautological; the constant C absorbs the thing that needs bounding. Since the final bound contains ∥ρ_B^{-1}∥_∞^{1/2}, the prefactor must not grow faster than the superexponential ε(|B|). The proof delegates to [7, Cor 4.4], but what is needed is an explicit exponential (or other controlled) bound on ∥ρ_B^{-1}∥_∞, and that is not quoted. So the advertised first family of states with superexponential CMI decay is not established as written. If [7, Cor 4.4] supplies the needed bound, the fix is a one-line citation; if not, Theorem 6.1 needs a different argument. This does not touch Theorem 3.3 or Section 5.\n\nAlso minor: Proposition 4.5's converse requires [η_AB, η_BC]=0, which is a genuine restriction; the authors acknowledge it and leave the general converse open.\n\nThis paper is for researchers in quantum information and mathematical physics working on Markov chains, recovery maps, and correlation decay. Read and cite Theorem 3.3 and Section 5; be cautious with Theorem 6.1 until the prefactor is pinned down. I would send this to a serious referee; the central classification result deserves the time.","headline":"Core structural correspondence between BS-quantum Markov chains and ordinary quantum Markov chains is new and convincing; the advertised superexponential CMI decay application has a proof gap that needs a fix.","tokens_in":32521,"tokens_out":4714,"would_cite":true,"duration_ms":42534,"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":"This paper proves that the states with zero Belavkin–Staszewski conditional mutual information are exactly those that become ordinary quantum Markov chains after a rescaling by the inverse square root of the middle marginal, and uses this…","keywords":["Belavkin-Staszewski relative entropy","quantum Markov chains","conditional mutual information","recovery map","data-processing inequality","Gibbs states","superexponential decay","quantum spin chains"],"falsifier":"Compute $\\|(\\rho_B)^{-1}\\|_\\infty$ for the Gibbs states of a fixed 1D local, finite-range, translation-invariant Hamiltonian at fixed inverse temperature as $|B|$ grows; if the growth is faster than exponential, Theorem 6.1's claimed superexponential decay of $I_\\eta$ lacks its prefactor control. The displayed chain in Section 6.1 already shows the risk: after invoking [7, Corollary 4.4] the proof concludes $\\|(\\rho_B)^{-1}\\|_\\infty \\le C\\|(\\rho_B)^{-1}\\|_\\infty$, which is tautological and cannot by itself establish the exponential prefactor.","tokens_in":31567,"feed_emoji":"🔗","tokens_out":9903,"duration_ms":89777,"temperature":0.7,"pith_summary":"The paper characterizes Belavkin–Staszewski quantum Markov chains: states whose conditional mutual information, defined through the Belavkin–Staszewski relative entropy instead of the standard one, is zero. Its central result is that such states are exactly the ones that become ordinary quantum Markov chains after rescaling by the inverse square root of the middle marginal, so their structure is fully captured by the known structure theorem for quantum Markov chains. This gives a linear recovery map for the Belavkin–Staszewski entropy, an entanglement-breaking correspondence, and bounds relating approximate versions of the two notions. As an application, the paper constructs the first family of states whose non-vanishing conditional mutual information decays superexponentially with the size of the conditioning system.","feed_headline":"Rescale by the middle system: BS-quantum Markov chains become known ones","feed_subtitle":"Rescaling by the middle system maps them onto known Markov chains, giving recovery and superexponential decay.","key_machinery":"The load-bearing object is the rescaling map $\\eta(X) = d_B^{-1} \\rho_B^{-1/2} X \\rho_B^{-1/2}$, applied to $\\rho_{ABC}$; it converts the purely algebraic BS condition $\\rho_{ABC} = \\rho_{AB} \\rho_B^{-1} \\rho_{BC}$ into the ordinary quantum Markov chain condition with commuting marginals $\\eta_{AB}$ and $\\eta_{BC}$ and middle marginal $\\tau_B$. Its inverse on quantum Markov chains with $\\eta_B = \\tau_B$ is $\\omega(X_B) = d_B^2 X_B^{1/2} \\eta_{AB} \\eta_{BC} X_B^{1/2}$, which generates families of BS-quantum Markov chains. The recovery map $\\Phi_{B\\to AB}(X) = d_B \\rho_B^{1/2} \\eta_{AB}^{1/2} \\rho_B^{-1/2} X \\rho_B^{-1/2} \\eta_{AB}^{1/2} \\rho_B^{1/2}$, equivalently $\\rho_{AB}^{1/2} W_{AB} \\rho_B^{-1/2} X \\rho_B^{-1/2} W_{AB}^* \\rho_{AB}^{1/2}$ with $W_{AB}$ unitary, is the mechanism that makes recovery an if-and-only-if statement; the commuting-marginal identity $[\\eta_{AB},\\eta_{BC}]=0$ is what turns the non-Hermitian BS recovery condition into this Hermitian one.","core_discovery":"The paper establishes the equivalence stated in Theorem 3.3: for a tripartite state $\\rho_{ABC}$ with invertible $\\rho_B$, the rescaled state $\\eta_{ABC} = d_B^{-1} \\rho_B^{-1/2} \\rho_{ABC} \\rho_B^{-1/2}$ is a quantum Markov chain with middle marginal maximally mixed if and only if $\\rho_{ABC}$ is a Belavkin–Staszewski quantum Markov chain, i.e. iff $\\rho_{ABC} = \\rho_{AB} \\rho_B^{-1} \\rho_{BC}$ and the marginals $\\eta_{AB}$, $\\eta_{BC}$ commute. It follows (Corollary 3.9) that $\\rho_{ABC}$ is a BS-quantum Markov chain iff it is recovered from $\\rho_{BC}$ by the linear completely positive map $\\Phi_{B\\to AB}$ with a unitary factor, closing the converse left open in an earlier work. The authors extend this correspondence to arbitrary pairs of states and channels saturating the respective data-processing inequalities, prove structural decompositions in both settings, and show that on BS-quantum Markov chains the rescaling is entanglement-breaking between $A$ and $C$, even though BS-quantum Markov chains can have entangled $AC$ marginals.","pith_inferences":["If the exponential prefactor bound on $\\|\\rho_B^{-1}\\|_\\infty$ can be proved directly, the superexponential CMI decay of $\\eta_{ABC}$ could transfer to spectral-gap statements for Davies generators through the paper's bound on $\\Delta_\\rho$, connecting conditional independence to open-system dynamics.","The entanglement-breaking property of the rescaling restricted to BS-quantum Markov chains suggests quantifying the 'BS-ness' of a state by how much entanglement the rescaling removes; such a measure would separate BS-quantum Markov chains that are ordinary quantum Markov chains from those with entangled AC marginals.","A natural testable extension is to check the paper's open rotated-recovery question, whether $(\\Phi^{\\mathrm{rot}}_{B\\to AB}\\otimes\\mathrm{id}_C)(\\rho_{BC})=\\rho_{ABC}$ holds for exact BS-quantum Markov chains; a positive answer would give a second recovery map and sharpen the approximate bounds.","The correspondence between saturation triples for the two relative entropies may offer an operational interpretation: BS-saturating triples are images of relative-entropy-saturating triples under a state-dependent rescaling, potentially allowing channel-capacity arguments formulated for one entropy to be ported to the other."],"forward_implications":["Every BS-quantum Markov chain admits the block decomposition of Theorem 3.3(v), so the structural theory of ordinary quantum Markov chains applies verbatim to zero-BS-CMI states.","The recovery map $\\Phi_{B\\to AB}$ is certified: a state is a BS-quantum Markov chain if and only if $(\\Phi_{B\\to AB}\\otimes\\mathrm{id}_C)(\\rho_{BC})=\\rho_{ABC}$, resolving the converse left open in the prior work.","Approximate BS-quantum Markov chains are approximate quantum Markov chains: the reversed BS-CMI lower-bounds an eighth power of the CMI of $\\eta_{ABC}$, and a partial upper bound holds under commuting marginals.","Saturation of the BS data-processing inequality for general channels is equivalent to saturation of the standard relative-entropy data-processing inequality for the associated partial trace, yielding a linear completely positive recovery map.","For Gibbs states of local finite-range translation-invariant one-dimensional Hamiltonians at any positive temperature, the associated $\\eta_{ABC}$ has conditional mutual information decaying superexponentially in $|B|$, the first such example with non-vanishing CMI."],"supporting_citations":[{"why":"Supplies the quantum Markov chain structure theorem that the correspondence targets.","marker":"[16]"},{"why":"Defines the recovery map whose converse is proved and provides the Gibbs-state superexponential decay input.","marker":"[14]"},{"why":"Gives the Belavkin–Staszewski data-processing saturation condition used to identify BS-quantum Markov chains.","marker":"[5]"},{"why":"Provides the maximal-divergence equality conditions used for the structure theorem and recovery criteria.","marker":"[17]"},{"why":"Supplies the norm bound on inverse marginals invoked in the spin-chain application.","marker":"[7]"},{"why":"Gives the standard recovery map and quantum Markov chain condition used throughout.","marker":"[32]"},{"why":"Supplies the decomposition of states saturating the relative-entropy data-processing inequality used in the general correspondence.","marker":"[19]"},{"why":"Defines the correlation quantity whose fast decay implies a Davies generator spectral gap.","marker":"[25]"}],"fun_headline_variants":["Rescaling by the middle system maps BS-Markov onto known chains","BS-Markov chains become known ones via middle rescaling","Recovery map and superexponential decay from BS-Markov rescaling","Middle-system rescaling yields recovery and superexponential decay"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The application to Gibbs states needs the inverse of the middle marginal, $\\|\\rho_B^{-1}\\|_\\infty$, to grow at most exponentially with $|B|$; the paper's proof of that step invokes an external bound and, as written, collapses to the tautology $\\|\\rho_B^{-1}\\|_\\infty \\le C\\|\\rho_B^{-1}\\|_\\infty$.","fun_headline_variants_meta":{"raw":{"variants":["Rescaling by the middle system maps BS-Markov onto known chains","BS-Markov chains become known ones via middle rescaling","Recovery map and superexponential decay from BS-Markov rescaling","Middle-system rescaling yields recovery and superexponential decay"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000804,"raw_usage":{"total_tokens":3579,"prompt_tokens":1042,"completion_tokens":2537,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":658,"completion_tokens_details":{"reasoning_tokens":2460}},"tokens_in":658,"tokens_out":2537,"duration_ms":17893,"temperature":1.0,"reasoning_tokens":2460,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:47:05.044240+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\|(\\rho_B)^{-1}\\|_\\infty$ for the Gibbs states of a fixed 1D local, finite-range, translation-invariant Hamiltonian at fixed inverse temperature as $|B|$ grows; if the growth is faster than exponential, Theorem 6.1's claimed superexponential decay of $I_\\eta$ lacks its prefactor control. The displayed chain in Section 6.1 already shows the risk: after invoking [7, Corollary 4.4] the proof concludes $\\|(\\rho_B)^{-1}\\|_\\infty \\le C\\|(\\rho_B)^{-1}\\|_\\infty$, which is tautological and cannot by itself establish the exponential prefactor.","supporting_citations":[{"cited_title":"Hayden, R","cited_arxiv_id":null,"evidence_quote":"Supplies the quantum Markov chain structure theorem that the correspondence targets."},{"cited_title":"Conditional Independence of 1D Gibbs States with Applications to Efficient Learning","cited_arxiv_id":"2402.18500","evidence_quote":"Defines the recovery map whose converse is proved and provides the Gibbs-state superexponential decay input."},{"cited_title":"Bluhm and ´A","cited_arxiv_id":null,"evidence_quote":"Gives the Belavkin–Staszewski data-processing saturation condition used to identify BS-quantum Markov chains."},{"cited_title":"Hiai and M","cited_arxiv_id":null,"evidence_quote":"Provides the maximal-divergence equality conditions used for the structure theorem and recovery criteria."},{"cited_title":"Bluhm, ´A","cited_arxiv_id":null,"evidence_quote":"Supplies the norm bound on inverse marginals invoked in the spin-chain application."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the standard recovery map and quantum Markov chain condition used throughout."},{"cited_title":"Jenˇ cov´ a and D","cited_arxiv_id":null,"evidence_quote":"Supplies the decomposition of states saturating the relative-entropy data-processing inequality used in the general correspondence."}],"review_version":1}