{"id":"cbe9573f-549f-46da-9365-d3141dca77ca","arxiv_id":"2505.02125","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The second Rényi conditional mutual information and its Markov length classify average SPT, trivial, paramagnetic, and SWSSB mixed-state phases and detect the transitions between them in two 1D decohered spin chains.","lead":"This paper tests whether a quantum information measure called the second Rényi conditional mutual information can detect phases and phase transitions of noisy quantum states. It introduces a numerical way to compute this measure efficiently and shows it works on two one-dimensional spin models under decoherence.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (6) assumes an unproved state-independent proportionality between von Neumann and Rényi-2 CMI; without it, exponential decay of the second-Rényi CMI cannot be tied to Petz-map recoverability or to a mixed-state 'gap'.","rationale":"The reader's weakest-assumption analysis identifies exactly the same load-bearing point: the proportionality in Eq. (6) is the bridge between the computationally accessible Rényi-2 CMI and the Petz-map recoverability argument of Ref. [33]. My stress-test confirms that this bridge is not merely unproved; it is not a generic identity. The Rényi-2 quantity I^(2) defined by replacing von Neumann entropies with Rényi-2 entropies in the CMI formula does not inherit strong subadditivity, so it can be zero or negative where the von Neumann CMI is positive. Thus a positive state-independent β cannot hold universally. The paper's use of Eq. (6) in Eq. (7) is therefore a real soft spot in the central conceptual claim that the second-Rényi Markov length measures a mixed-state 'gap' and determines phase equivalence. This does not invalidate the numerical method or the observed peak/saturation behavior as possible diagnostics, but it does mean the interpretation attached to ξ_M^(2) is conditional on a direct check of the proportionality for the specific models studied. Since the Reader already flagged this assumption and issued a CONDITIONAL verdict, my independent review does not move the verdict; it remains CONDITIONAL pending the concrete comparison of I and I^(2).","tokens_in":14599,"tokens_out":10493,"duration_ms":140769,"concrete_test":"Directly compute the von Neumann CMI I(A:C|B) and the Rényi-2 CMI I^(2)(A:C|B) for the same decohered states and partitions used in Figs. 3 and 5, e.g., at (p_z,h_x) = (0.1,0.78), (0.1,1.0), (0.1,1.22) and (p_zz,h_x) = (0.11,1), (0.19,1), (0.28,1), for several system sizes L. Evaluate both quantities from the same MPS (S_X from entanglement spectra, S_X^(2) from Tr ρ_X^2) and check whether I / I^(2) is a single positive constant across all six parameter points and sizes within error bars. If the ratio varies by more than a moderate tolerance or fails to exist, Eq. (6) is refuted for these models and the recoverability/phase-equivalence interpretation in Sec. IIB cannot be used.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing assumption is Eq. (6): I(A:C|B) = β I^(2)(A:C|B) with a positive factor β, attributed to Refs. [39–41]. This relation is used in Eq. (7) to convert exponential decay of the Rényi-2 CMI into a Petz-map recovery-error bound, and hence to justify calling ξ_M^(2) a mixed-state 'gap' and to import the phase-equivalence framework of Ref. [33]. But I^(2), defined via Eq. (4), is not a conditional mutual information satisfying strong subadditivity; it can be zero or negative in simple states, so a universal positive β cannot exist. The paper itself only says 'we expect' the relation and cites numerical studies on different models, not a proof. If β is state-dependent or changes sign across the phase diagram, exponential decay of I^(2) no longer implies recoverability, and the central claim that the second-Rényi Markov length classifies non-trivial mixed-state phases and gaps is not established. The numerical diagnostics may still be useful phenomenologically, but the conceptual load is carried by Eq. (6).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes the second Rényi conditional mutual information (CMI), I^(2)(A:C|B), and its associated Markov length ξ_M^(2) as diagnostic quantities for non-trivial mixed-state phases and mixed-state phase transitions in one-dimensional spin systems. The authors introduce a doubled-Hilbert-space numerical scheme, based on applying depolarizing channels to compute reduced density-matrix purities via matrix product states. They apply this method to two models: (i) the cluster model in a transverse field under odd-site Z decoherence, where they report exponential decay of I^(2) away from h_x ≈ 1, a growing Markov length near the ASPT-to-trivial transition, and finite Markov lengths in both phases; and (ii) the transverse-field Ising model under ZZ and X decoherence, where they report growth of ξ_M^(2) approaching the SWSSB transition and an r-independent CMI saturating at ln 2 in the deep SWSSB phase. Appendix C gives an exact stabilizer calculation showing that the ln 2 value is obtained for a prototypical SWSSB state. The central claim is that the second Rényi CMI and its Markov length are good measures for classifying non-trivial mixed states and locating mixed-state phase transitions.","tokens_in":14840,"tokens_out":11091,"duration_ms":130281,"significance":"If the main claims are established, the paper would provide a numerically tractable proxy for the von Neumann CMI/Markov-length framework of Ref. [33] and a new diagnostic for strong-to-weak spontaneous symmetry breaking. The doubled-Hilbert-space computational scheme is simple and likely useful for future studies of decohered states. The ln 2 saturation in the SWSSB phase is a concrete, falsifiable prediction, and the exact stabilizer calculation in Appendix C is a genuine strength. However, the paper's conceptual claim that ξ_M^(2) is a mixed-state 'gap' and the phase-equivalence interpretation depend on an unproved proportionality between von Neumann CMI and second Rényi CMI, and the numerical evidence currently lacks error bars and finite-size control. The work is therefore a promising but not yet fully established contribution.","major_comments":[{"comment":"The proportionality I(A:C|B) = β I^(2)(A:C|B) with a positive factor β is introduced as an expectation, and Eq. (7) then uses it to convert exponential decay of the second Rényi CMI into a Petz-map recovery bound. This is load-bearing for the interpretation of ξ_M^(2) as a mixed-state 'gap' and for importing the phase-equivalence framework of Ref. [33]. The relation is not proven for the models studied, and the cited Refs. [39–41] are numerical and analytical studies of different settings. Because the second Rényi CMI does not in general inherit the information-theoretic properties of the von Neumann CMI (nonnegativity, strong subadditivity, data processing), a universal state-independent positive β is a nontrivial assumption. If β is state-dependent or if the relation fails, exponential decay of I^(2) does not imply recoverability, and the central conceptual claim is not established. I ask the authors to either prove or directly test Eq. (6) on the actual decohered states considered here, for example by computing the von Neumann CMI alongside I^(2) at the same parameters, or to explicitly restrict the claims to a phenomenological diagnostic and remove the recoverability/gap interpretation.","section":"Section II A, Eq. (6)"},{"comment":"The Markov-length estimates are obtained by fitting I^(2)(r) = e^{-c0 r} + c1 using a small number of r values, with no error bars, no stated r range, no fit residuals, and no extrapolation in system size. Because the system size is tied to r through |C| = r, finite-size effects from the complementary region C can masquerade as exponential decay or as a constant offset. The central distinction between a finite Markov length and an infinite one is exactly what the fits are used to establish, so the absence of quantitative fit quality and of any finite-size scaling analysis is a serious gap. For example, at p_zz = 0.28 the reported ξ_M^(2) ≈ 3.825 is still finite, yet the text concludes that the Markov length becomes infinite throughout the SWSSB phase; that conclusion rests on an extrapolation to p_zz → 1/2 for which no r-dependence data are shown. I request that the authors report full parameters (L, r values, bond dimensions, truncation errors), provide error bars or at least show the fit curves together with the data, and include a finite-size scaling analysis at the purported transition points.","section":"Section IV A and Section IV B, fitting of ξ_M^(2)"},{"comment":"The statement that the second Rényi CMI takes the universal value ln 2, and hence that the Markov length is infinite in the whole SWSSB phase, extrapolates from data at p_zz ≲ 0.28 plus an exact calculation for a single stabilizer state. The stabilizer state in Appendix C has only the global stabilizer S = {∏_j X_j}, and while the ln 2 value is derived correctly in that limit, the claim that this value is universal across the entire SWSSB phase is not proven. The numerical saturation in Fig. 5(a) is supportive, but the r-dependence in the deep SWSSB region is not displayed. I recommend either presenting r-dependence data at larger p_zz values (e.g., p_zz = 0.4, 0.45, 0.5) or softening the universality claim to a prediction consistent with the data.","section":"Section IV B and Appendix C, SWSSB saturation claim"}],"minor_comments":[{"comment":"In the p_z = 0.3 paragraph of Appendix B, the extracted Markov lengths are reported 'at pzz = 0.11, 0.19 and 0.28', but the context is the cluster model and the correct parameters should be h_x = 0.78, 1.0, 1.22. This copy-paste from Section IV B should be corrected.","section":"Appendix B"},{"comment":"The heading 'Second Rényi CMI for Jzz→0 and pzz→∞' is inconsistent with the definition 0 ≤ p_zz ≤ 1/2 elsewhere in the paper; it should presumably read p_zz → 1/2.","section":"Appendix C heading"},{"comment":"The introductory paragraph refers to 'Section VIII' for the summary and conclusion, but the paper only has five numbered sections plus appendices; the reference should be corrected.","section":"Section I"},{"comment":"The Choi operators in Eqs. (18) and (19) include complex-conjugated identity and Pauli operators on the upper chain (e.g., Ê_ZZ contains to Z^∗_{j,u}); since the Pauli matrices are Hermitian this is harmless, but it may confuse readers and should be explained or simplified.","section":"Equations (18) and (19)"},{"comment":"The main-text figures do not specify the system sizes L, the values of r used in the fits, or the bond dimensions used for the mixed-state MPS. Reporting these in the captions or in the text would substantially improve reproducibility, especially because estimates of ξ_M^(2) are the central quantitative output.","section":"Figure captions and numerical settings"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely topic and the numerical method is potentially useful, but the conceptual claim regarding recoverability and the 'mixed-state gap' is currently carried by an unproved proportionality between von Neumann and Rényi-2 CMI. I would encourage the editor to ask for a revision that either validates Eq. (6) on the studied models or explicitly reframes the paper as a phenomenological diagnostic study. Strengthening the numerics with error bars and finite-size scaling is essential before publication. The ln 2 saturation in the SWSSB phase, backed by the stabilizer calculation, is a promising result that could be highlighted more cautiously."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, read this if you care about mixed-state phase diagnostics. The paper's real contribution is numerical: an efficient MPS scheme to compute the second Rényi CMI in the doubled Hilbert space, applied to two decohered 1D models. The cluster-model ASPT transition and TFIM under ZZ+X decoherence both show the expected behavior: exponential decay away from transitions, growing Markov length near them, and saturation at ln2 in the deep SWSSB phase. The Appendix C stabilizer calculation is a nice formal check that the ln2 value is not a fitting artifact. That is genuine evidence and should be credited.\n\nWhere it gets soft is the conceptual bridge. The whole interpretation of the second-Rényi Markov length as a mixed-state 'gap' and the import of the phase-equivalence framework from Ref. [33] go through Eq. (6), I = β I^(2) with positive state-independent β. The paper says 'we expect' and cites numerical studies on different models. But I^(2) is not a conditional mutual information satisfying strong subadditivity; it can be negative in simple states. A universal positive β cannot exist. If β is state-dependent or changes sign, exponential decay of I^(2) no longer implies a Petz-map recovery bound, and the central claim that this Markov length classifies non-trivial mixed-state phases is not established. That is load-bearing, not a footnote.\n\nTwo smaller issues: the fits have no error bars and only three points per curve; the divergence of ξ is inferred from a single point at hx=1 with ξ≈10, not from a scaling collapse. Finite-size scaling is absent, and no code/data are shipped. These are fixable but should be addressed.\n\nBottom line: the numerical diagnostics are probably useful phenomenologically, and the paper is honestly written—it does state 'we expect' rather than hiding the assumption. But the conceptual interpretation is overreach until Eq. (6) is either proved for the relevant states or at least tested directly by computing both I and I^(2) on the same states. That test is within reach of the same MPS toolbox. I would send this to a serious referee, with the expectation of major revision rather than acceptance as is. It is not a desk reject; the method is novel and the field will care.","headline":"Useful numerical method for second-Rényi CMI, but the conceptual 'gap' interpretation rests on an unproved proportionality that needs direct testing.","tokens_in":15368,"tokens_out":1839,"would_cite":true,"duration_ms":20896,"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":"The second Rényi conditional mutual information is a mixed-state 'gap' measure: its Markov length stays short in average-SPT and trivial mixed phases, peaks at their transition, and becomes infinite in SWSSB, where it saturates at ln 2.","keywords":["Rényi conditional mutual information","Markov length","mixed-state phases","average symmetry-protected topological order","strong-to-weak spontaneous symmetry breaking","doubled Hilbert space","matrix product states","decoherence"],"falsifier":"Compute the von Neumann conditional mutual information and the second Rényi CMI on the same decohered states—e.g., the cluster model at $p_z=0.1$ for $h_x = 0.78, 1.0, 1.22$, using the doubled-Hilbert-space matrix-product-state method with converged bond dimension—and test whether the ratio $I/I^{(2)}$ stays constant as $h_x$, $p_{zz}$, and $r$ vary. If the ratio drifts, or if an explicit recovery map built from the reduced state of $A\\cup B$ fails to reproduce an error that decays with rate $1/\\xi^{(2)}_M$, the identification of the Rényi Markov length with a mixed-state gap is falsified.","tokens_in":14408,"feed_emoji":"📏","tokens_out":21873,"duration_ms":177368,"temperature":0.7,"pith_summary":"The paper seeks to establish that the second Rényi conditional mutual information of a tripartite one-dimensional chain, and the Markov length extracted from its exponential decay, can serve as the mixed-state analogue of an energy gap: a diagnostic that classifies noisy quantum phases and signals the transitions between them. This matters because nontrivial mixed states such as average symmetry-protected topological (ASPT) states and strong-to-weak spontaneous symmetry breaking (SWSSB) states have no pure-state Hamiltonian description, so they currently lack a workable principle for 'gapped' versus 'gapless' classification. The authors propose an efficient numerical scheme—computing the second Rényi CMI as matrix-product-state norms in the doubled Hilbert space—and apply it to two decohered chains: a cluster model under odd-site Z noise, and a transverse-field Ising model under ZZ and X noise. Numerically, the Markov length stays short inside the ASPT and trivial mixed phases, grows sharply at the ASPT transition, and becomes infinite in the SWSSB phase, where the CMI saturates at the universal value $\\ln 2$. If the paper is right, a single information-theoretic length scale organizes the mixed-state phase diagram of these systems and provides a computable, channel-based notion of mixed-state phase equivalence.","feed_headline":"One length scale classifies decohered phases and flags transitions","feed_subtitle":"Noisy 1D states reveal average-SPT and SWSSB phases via the decay length of Rényi-2 CMI.","key_machinery":"The engine of the argument is the second Rényi conditional mutual information, $I^{(2)}(A:C|B)(r) = I^{(2)}(A,BC) - I^{(2)}(A,B)$, assembled from second Rényi entropies $S^{(2)}_X = -\\log \\operatorname{Tr}(\\rho_X^2)$; the second Rényi Markov length $\\xi^{(2)}_M$ is defined by the decay $I^{(2)}(A:C|B)(r) \\sim e^{-r/\\xi^{(2)}_M}$. Computationally, the paper vectorizes the density matrix into a supervector in the doubled Hilbert space and implements partial trace as a maximal depolarization operator $\\hat{D}_{\\bar X}$ obeying $\\hat{D}_{\\bar X}|\\rho\\rangle\\rangle = |I_{\\bar X}/d_{\\bar X} \\otimes \\rho_X\\rangle\\rangle$, so every second Rényi entropy reduces to the logarithm of a matrix-product-state norm that filtering methods evaluate directly. Conceptually, the load-bearing link is the quoted proportionality $I(A:C|B) = \\beta I^{(2)}(A:C|B)$, which carries the recovery-map bound of Ref. [33] into the Rényi-2 setting and gives $1/\\xi^{(2)}_M$ its meaning as a gap controlling recoverability and mixed-state phase equivalence.","core_discovery":"The paper's central claim is that the second Rényi conditional mutual information $I^{(2)}(A:C|B)(r)$, built from second Rényi entropies on a tripartition of a 1D spin chain, decays exponentially in the buffer size $r$ with a rate that defines the second Rényi Markov length $\\xi^{(2)}_M$, and that $1/\\xi^{(2)}_M$ behaves as a gap for mixed states: small inside gapped mixed phases, sharply peaked at mixed-state transitions, and effectively infinite inside the SWSSB phase. For the cluster model under odd-site $Z$ decoherence, the authors find $\\xi^{(2)}_M \\approx 2.8\\text{–}2.9$ inside both the ASPT and trivial mixed phases and a pronounced peak (about 10 at $p_z=0.1$) at $h_x\\approx 1$, the ASPT-to-trivial transition. For the transverse-field Ising model under $ZZ$ and $X$ decoherence, $\\xi^{(2)}_M$ grows from about 1.6 toward the SWSSB transition, and in the deep SWSSB phase the CMI loses its $r$-dependence and takes the constant value $\\ln 2$, which the paper verifies in a stabilizer limit; hence the Markov length is infinite throughout the SWSSB phase. The stated conclusion is that the second Rényi CMI and its Markov length are good measures to classify the regime of a nontrivial mixed state and to locate mixed-state phase transitions between them, mirroring the von Neumann CMI while being numerically far more tractable.","pith_inferences":["Because the paper never evaluates the von Neumann CMI on the same states, an immediate check of the proportionality $I = \\beta I^{(2)}$ is still open: if the ratio drifts with $r$, $h_x$, or $p_{zz}$, then $\\xi^{(2)}_M$ is a state-dependent proxy rather than a faithful gap, and the phase-equivalence statements inherit that caveat.","The growth of $\\xi^{(2)}_M$ from about 2.8 to about 10 at $h_x \\approx 1$ within a finite system suggests a critical scaling region; fitting $I^{(2)}(A:C|B)(r)$ to a power law $r^{-\\alpha}$ at the transition would extract a Rényi-2 critical exponent and could connect the ASPT transition to known measurement-induced or percolation criticality.","The $\\ln 2$ plateau is the Rényi-2 avatar of the topological entropy of the SWSSB state; extending the same norm-of-supervector technology to negativity or to 1-form-symmetric channels might expose intrinsic mixed-state topological order through the same computational route.","A practical extension would be to apply the method to monitored quantum circuits or to two-dimensional strips, where the matrix-product-state ansatz still works but the tripartition geometry changes, testing whether the Markov-length criterion survives beyond the quasi-1D setting."],"forward_implications":["The ASPT-to-trivial transition in the decohered cluster model is detectable as a sharp peak in $\\xi^{(2)}_M$ at $h_x \\approx 1$ for every decoherence strength tested ($p_z = 0.1, 0.2, 0.3$), so the second Rényi Markov length serves as a numerical order parameter for that transition.","Inside the ASPT and trivial mixed phases the finite, short Markov length implies that the decohered states admit efficient recovery maps, so states within each regime belong to a common mixed-state phase in the sense of Ref. [33].","In the SWSSB phase the exponential decay of the second Rényi CMI disappears, the Markov length is infinite, and the local recovery bound no longer closes with $r$; the plateau at $\\ln 2$ is the information-theoretic fingerprint of the phase.","The doubled-Hilbert-space matrix-product-state scheme is channel-agnostic, so the same computation applies to any local decoherence channel on a 1D spin chain, including the $ZZ$-only and $X$-only channels that make up the Ising-model noise studied here."],"supporting_citations":[{"why":"Defines the Markov length from exponential decay of the conditional mutual information and the recovery-map criterion for mixed-state phase equivalence that this paper extends to the Rényi-2 setting.","marker":"[33]"},{"why":"Provide the numerical evidence that the von Neumann CMI is proportional to its second Rényi version, the relation the gap interpretation relies on.","marker":"[39, 40]"},{"why":"Supplies the analytic argument for the proportionality between the CMI and its second Rényi counterpart.","marker":"[41]"},{"why":"Introduces the second Rényi conditional mutual information as the object whose Markov length this paper studies.","marker":"[31]"},{"why":"Predicts the universal $\\ln 2$ value of the CMI inside strong-to-weak spontaneous symmetry breaking phases, which the paper's numerics reproduce and extend.","marker":"[24]"},{"why":"Establishes the paramagnetic-mixed/SWSSB phase diagram of the decohered transverse-field Ising model and the filtering method that the present calculation reuses.","marker":"[22]"},{"why":"Provides the average-SPT phase characterization and the doubled-Hilbert-space representation of decoherence channels used for the cluster model.","marker":"[15]"},{"why":"Supplies the depolarization identity that converts partial traces into supervector norms, the basis of the efficient Rényi-2 CMI computation.","marker":"[44]"},{"why":"Provide the matrix-product-state software with which all numerical simulations in the paper are carried out.","marker":"[37, 38]"}],"fun_headline_variants":["Rényi-2 CMI yields a Markov length classifying mixed-state phases","A single decay length from Rényi-2 CMI maps mixed-state order","Markov length from Rényi-2 CMI flags mixed-phase transitions","Rényi-2 CMI length: a gap probe for decohered 1D states","Second Rényi CMI defines a phase-classifying length for mixed states"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the second Rényi conditional mutual information tracks the ordinary one up to a constant positive factor; if that factor varies from state to state, a short Rényi Markov length need not mean the noisy state can be recovered from its surroundings, and the phase classification built on it would slip.","fun_headline_variants_meta":{"raw":{"variants":["Rényi-2 CMI yields a Markov length classifying mixed-state phases","A single decay length from Rényi-2 CMI maps mixed-state order","Markov length from Rényi-2 CMI flags mixed-phase transitions","Rényi-2 CMI length: a gap probe for decohered 1D states","Second Rényi CMI defines a phase-classifying length for mixed states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001017,"raw_usage":{"total_tokens":4395,"prompt_tokens":1149,"completion_tokens":3246,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":765,"completion_tokens_details":{"reasoning_tokens":3149}},"tokens_in":765,"tokens_out":3246,"duration_ms":22322,"temperature":1.0,"reasoning_tokens":3149,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T01:01:06.996376+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the von Neumann conditional mutual information and the second Rényi CMI on the same decohered states—e.g., the cluster model at $p_z=0.1$ for $h_x = 0.78, 1.0, 1.22$, using the doubled-Hilbert-space matrix-product-state method with converged bond dimension—and test whether the ratio $I/I^{(2)}$ stays constant as $h_x$, $p_{zz}$, and $r$ vary. If the ratio drifts, or if an explicit recovery map built from the reduced state of $A\\cup B$ fails to reproduce an error that decays with rate $1/\\xi^{(2)}_M$, the identification of the Rényi Markov length with a mixed-state gap is falsified.","supporting_citations":[],"review_version":1}