{"id":"a41e4f30-65dc-4881-b364-be9f9e9c1d7e","arxiv_id":"2506.18475","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The Rényi-2 generalized Shannon mutual information of the critical transverse-field Ising model reproduces the Ising central charge across a broad range of measurement relaxation and local decoherence.","lead":"The paper introduces a Rényi-2 generalized Shannon mutual information that interpolates between two known entanglement measures, and numerically tests it on a critical quantum spin chain. It finds the extracted central charge stays near the Ising value across broad ranges of measurement relaxation and local decoherence.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed c2≈1 plateau rests on four-point fits with no residuals or error bars; adding or removing one subsystem size could shift c2 and shrink the robust region.","rationale":"The reader's weakest assumption already identifies the finite-size fitting reliability in Sec. V as the key risk, and my reading agrees. The paper's central claim is numerical, so the quality of the fits is the load-bearing element. The endpoint cases pm=0 and pm=1/2 connect to established results, but the broad interior plateau is new and is assembled from fits with only four LA values and no reported goodness-of-fit. The Z-basis dip in c2 shows that the extracted quantity is pm-sensitive, so the absence of error bars matters even more. I do not see an internal inconsistency in the definition of the R2GSMI or in the doubled-space calculation; the concern is that the numerical evidence as presented cannot distinguish a genuine plateau from a fitting artifact. This does not overturn the paper, but it keeps the confidence at conditional. The proposed check—refitting with all subsystem sizes and with a jackknife—would settle whether the plateau survives.","tokens_in":12122,"tokens_out":7173,"duration_ms":79424,"concrete_test":"Recompute c2 at a representative plateau point, e.g. (pm,py) = (0.3, 0.2), by fitting Eq. (17) to R2GSMI data for all subsystem sizes LA in {4,6,8,10,12,14,16,18,20,22,24,26,28}, and report the resulting c2 with a reduced chi-squared and a jackknife over subsystem sizes. Repeat the fit using only the four sizes used in the paper and then with LA = 8 or LA = 16 excluded. If the central value moves by more than 10% relative to the four-point fit, or if the reduced chi-squared for the full-range fit exceeds 3, the claimed c2≈1 plateau is not robust to the fitting procedure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the R2GSMI of the critical TFIM follows Eq. (17) with c2≈1 across a broad (pm,py) region. The most load-bearing premise is that the two-parameter fit of Eq. (17) is reliable at every point in that region. This premise is least secure in Sec. V.B: the entire (pm,py) map of Fig. 4 is generated from fits to only four subsystem sizes, LA = 8, 12, 14, 16, at L = 32, giving two degrees of freedom per fit. The paper states these are “sufficient accurate results” but reports no chi-squared, residuals, or error bars. The new content is the interior of the parameter plane, since the endpoint lines pm = 0 and pm = 1/2 are already supported by prior work [18, 23]; an unstable four-point fit could easily produce a spurious broad plateau. The nonmonotonic dip for the Z-basis in Fig. 2(b) further shows that c2 is sensitive to pm, so region-level claims need per-point goodness-of-fit statistics and stability checks. Without those, the robustness claim is numerically underdetermined.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a Rényi-2 generalized Shannon mutual information (R2GSMI) that interpolates between the Rényi-2 mutual information (pm = 0) and the Rényi-2 Shannon mutual information (pm = 1/2). The authors propose a doubled-Hilbert-space matrix product state scheme for computing the R2GSMI, apply it to the L = 32 critical transverse-field Ising chain, and fit the CFT scaling ansatz of Eq. (17) to extract the Rényi-2 central charge c2. They report that c2 stays close to 1 for both Z-basis and X-basis measurements as pm varies, and that a broad interior region of the (pm,py) plane remains near c2 = 1 under Y-decoherence, from which they conclude that Ising CFT properties are robust against local decoherence.","tokens_in":12560,"tokens_out":5004,"duration_ms":53728,"significance":"If the numerical evidence were conclusive, the paper would offer a useful new observable that connects two established information measures and provides a practical way to test central charge robustness under decoherence. The doubled-Hilbert-space method is clearly described, and the endpoint benchmarks pm = 0 and pm = 1/2 agree with earlier work. However, the central claim rests on two-parameter fits with only four subsystem sizes at a single system size, with no reported residuals, error bars, or goodness-of-fit statistics; the anomalous dip near pm ≈ 0.1 for the Z basis is also left unexplained. The result is plausible but currently underdetermined by the presented evidence.","major_comments":[{"comment":"The entire (pm, py) phase map, including the claimed broad c2 ≈ 1 plateau, is generated from two-parameter fits of Eq. (17) using only LA = 8, 12, 14, 16 at fixed L = 32. This leaves only two degrees of freedom per fit, and the paper provides no residuals, chi-squared values, bootstrap uncertainties, or stability checks. Since the interior of the parameter plane is the new content beyond the previously studied pm = 0 and pm = 1/2 lines, the authors should provide per-point goodness-of-fit diagnostics and demonstrate that the extracted c2 is stable under small changes in the fit range, such as omitting one of the four subsystem sizes.","section":"Section V.B, Eq. (17), Fig. 4"},{"comment":"The nonmonotonic dip in c2 near pm ≈ 0.1 for the Z basis is reported without explanation. If the fit ansatz of Eq. (17) is poor precisely in this region, then the claim that 'all cases are well-fitted' is not supported, and the robustness statement is weakened. The authors should quantify the fit quality for the points in the dip, for example by showing residuals or a chi-squared value, and should discuss whether the dip is a physical effect of the interpolation between R2MI and R2SMI or an artifact of the fitting procedure.","section":"Section V.A, Fig. 2(b)"},{"comment":"All central charge extractions are performed at a single system size, L = 32, with no finite-size scaling analysis. A universal central charge estimate normally requires either a check that results are stable with L or a controlled large-L extrapolation. Without such a check, the reported c2 ≈ 1 plateaus could reflect a finite-size crossover rather than the Ising CFT value, especially in the decohered regions where the deviation grows at larger py.","section":"Section V, numerical setup"},{"comment":"The paper states that the four subsystem sizes LA = 8, 12, 14, 16 give 'sufficient accurate results,' but no justification for this specific choice is given, and it is not consistent with the earlier statement that data for 8 ≤ LA ≤ 24 were used. The authors should either use the full available range or explain why the restricted range is appropriate for Eq. (17), and they should report the resulting fit uncertainties.","section":"Section V.B"}],"minor_comments":[{"comment":"There are several typographical and terminology inconsistencies, including 'R´enyi' for 'Rényi', 'R2RSMI' used interchangeably with 'R2GSMI', and 'docohred' instead of 'decohered'. The authors should carefully proofread the manuscript.","section":"Throughout"},{"comment":"The sentence 'In 0.2≤ pm≤ 0, 5' contains a clear typo; this should be corrected to a proper interval such as '0.2 ≤ pm ≤ 0.5'.","section":"Section V.A"},{"comment":"The color map is generated by cubic interpolation from discrete data points, which can create visually large plateaus and suppress the actual sampling density. Showing the data points explicitly or using a scatter plot would make the evidence for the plateau more transparent.","section":"Figure 4"},{"comment":"The phrase 'Stein-spring representation' should be 'Stinespring representation'.","section":"Section III.C"},{"comment":"The claim that the R2GSMI 'can offer more experimentally accessible alternative' to entanglement entropy would benefit from a concrete experimental protocol or reference; as written, the accessibility claim is not quantified.","section":"Abstract and Introduction"},{"comment":"Reference [18] reports c2 = 1.02 for the XY model, which is mentioned only in the text; the numerical comparison to that value is useful and could be highlighted as an internal benchmark.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the journal's scope and the proposed quantity is a reasonable extension of known mutual information measures. The main concern is numerical: the central robustness claim rests on fits with very few data points and no error quantification. I would be willing to reconsider after the authors add per-fit diagnostics and stability checks, at minimum for the representative points along the pm axis and the interior of the (pm, py) plane."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper introduces R2GSMI, an interpolating quantity between Rényi-2 mutual information (p_m=0) and Rényi-2 Shannon mutual information (p_m=1/2). That interpolation is new, and the doubled-Hilbert-space MPS method used to compute it is appropriate and cleanly presented. The design makes physical sense: a partially dephasing channel is a natural way to relax the projective measurement assumption behind Shannon entropy, and the link to measurement noise is a useful framing for quantum simulators.\n\nThe numerical content is also worthwhile. The X-basis result—c2≈1 across the full range of p_m—is the strongest part of the paper, and the p_m=0 line reproduces known results for decohered critical states, which is a good sanity check. The authors also cite the earlier conjecture c_n=2c honestly and do not overstate their analytical contribution. The bibliography is balanced; the self-citations are to method papers and are not padding.\n\nWhere the paper is soft is exactly where the reader puts the finger: the central claim, Fig. 4, is a map of c2 extracted from fits to only four subsystem sizes, L_A=8,12,14,16, at fixed L=32. That is two degrees of freedom per fit, and the paper reports no residuals, no goodness-of-fit, and no error bars. The sentence that these are “sufficient accurate results” is an assertion, not a demonstration. Because the Z-basis data in Fig. 2(b) show a dip around p_m≈0.1, we know c2 is sensitive to p_m in some region; without per-point fit quality and stability checks, the broad plateau could be an artifact of the limited fit window. The lack of released code or data compounds this: “available upon reasonable request” is not a reproducibility statement. I would also like to see some finite-size dependence, since L=32 is a single moderate size.\n\nThe circularity concern is real but not fatal. Fitting the CFT ansatz is standard practice, and the comparison to c=1/2 is an external benchmark, not a manufactured target. The problem is not the ansatz; it is the evidence that the fits are reliable at every point in the map.\n\nOverall: this is a serious, honest numerical paper with a new quantity and a plausible robustness result. It deserves a referee, not a desk rejection, but the referee should require error bars or residual plots, stability checks under adding/removing fit points, and ideally code or data. I would not cite it in its current form, but the revised version could be useful for people working on CFT probes in open quantum systems.","headline":"A genuinely new interpolating mutual information, but the central robustness claim rests on four-point fits with no error bars or residuals, so treat the plateau as plausible rather than established.","tokens_in":12927,"tokens_out":1915,"would_cite":false,"duration_ms":21890,"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 paper introduces a Rényi-2 generalized Shannon mutual information and shows the critical Ising chain keeps its conformal-field-theory central charge under both relaxed measurements and local decoherence.","keywords":["Rényi-2 generalized Shannon mutual information","critical transverse-field Ising model","conformal field theory scaling","central charge","local decoherence","mixed states","doubled Hilbert space","matrix product states"],"falsifier":"Repeat the $(p_m,p_y)$ scan at $L=64$ and $L=128$ with the same fit window: if the extracted $c_2$ drifts with system size or the boundaries of the $c_2\\simeq 1$ plateau move, the robustness is a finite-size effect; alternatively, compute the residuals of the Eq. (17) fit as a function of $L_A$—systematic curvature near $p_m\\approx 0.1$ would show that the dip in $c_2$ reflects a failure of the scaling ansatz rather than a physical change in the state.","tokens_in":11935,"feed_emoji":"⚛️","tokens_out":9996,"duration_ms":84653,"temperature":0.7,"pith_summary":"The paper introduces a Rényi-2 generalized Shannon mutual information (R2GSMI) that continuously interpolates between the Rényi-2 mutual information ($p_m=0$) and the Rényi-2 Shannon mutual information ($p_m=1/2$) by replacing perfect projective measurements with local decoherence of strength $p_m$. For the critical transverse-field Ising model, it shows numerically that this quantity obeys the conformal-field-theory scaling law $I^{(2)}(A,B,p_m)=\\frac{c_2}{4}\\ln\\!\\big(\\frac{L}{\\pi}\\sin\\frac{\\pi L_A}{L}\\big)+b_2$ with $c_2=1$, the Ising value, over a broad region of the parameter plane. Under a global $Y$-decoherence of strength $p_y$ applied to the whole system, the extracted $c_2$ stays close to 1 across a wide $(p_m,p_y)$ region, indicating that the Ising CFT properties are robust against both measurement relaxation and local decoherence. If the claim holds, a noisy or partially decohering measurement protocol can still certify the Ising central charge without requiring perfect projective measurements.","feed_headline":"Central charge stays at 1 through noisy measurements and decoherence","feed_subtitle":"A generalized Shannon mutual information interpolates between two standard measures and still sees the Ising CFT.","key_machinery":"The central object is the Rényi-2 generalized Shannon mutual information $I^{(2)}(A,B,p_m)=S^{(2)}_{A,M}(p_m)+S^{(2)}_{B,M}(p_m)-S^{(2)}_{A\\cup B,M}(p_m)$, built from a subsystem Rényi-2 entropy $S^{(2)}_{A,M}(p_m)=-\\log\\operatorname{Tr}_A[(\\rho^{M,A}_{A,p_m})^2]$ where the pure state is first partially decohered by a local channel of strength $p_m$ in the measurement basis $\\hat{M}$. At $p_m=1/2$ this channel acts as a non-selective projective measurement and the quantity reduces to the known Rényi-2 Shannon mutual information; at $p_m=0$ it reduces to the Rényi-2 mutual information. The numerical machinery is the doubled Hilbert space (Choi) representation, in which the density matrix becomes a supervector on a ladder, and a maximal depolarization channel on subsystem $B$ projects out $B$ so that the norm of the supervector gives $\\operatorname{Tr}_A[(\\rho^M_A)^2]$. The fitted scaling law $I^{(2)}=\\frac{c_2}{4}\\ln\\!\\big(\\frac{L}{\\pi}\\sin\\frac{\\pi L_A}{L}\\big)+b_2$ is the diagnostic that converts the numerical data into a central-charge estimate.","core_discovery":"The central claim is that the CFT scaling of the R2GSMI and its Rényi-2 central charge $c_2$ are robust properties of the critical TFIM ground state, not artifacts of a particular measurement limit. For the $\\hat{M}=X$ conformal basis, $c_2=1$ for every $p_m$ from 0 to 1/2, meaning that relaxing the projective measurement all the way to the Shannon limit leaves the extracted central charge unchanged. For the $\\hat{M}=Z$ basis, $c_2=1$ both at $p_m=0$ (the Rényi-2 mutual information, where the exact Ising result is known) and for $p_m\\ge 0.2$, with a continuous dip near $p_m\\approx 0.1$; the paper takes this as a partial but not complete breakdown of robustness in that basis. When the whole system is subjected to local $Y$-decoherence, the fitted $c_2$ remains near 1 over a broad region of the $(p_m,p_y)$ plane, deviating only for strong decoherence $p_y\\gtrsim 0.3$ and especially at $p_y=1/2$. The paper concludes that the Ising CFT properties observed through both the conventional Rényi-2 Shannon mutual information and Rényi-2 mutual information survive in this generalized mutual information under local decoherence.","pith_inferences":["Editorial inference: the dip in $c_2$ near $p_m\\approx 0.1$ for the $Z$ basis may reflect a crossover between measurement-dominated and entanglement-dominated information, but a finite-size artifact is equally plausible; a system-size scan at $L=64,128$ would distinguish the two.","Editorial inference: if the same interpolation is applied to a bosonic CFT such as the XXZ chain (a direction the paper flags as future work), a similar robustness plateau would suggest that weak local decoherence generically preserves CFT central charges, not just the Ising one.","Editorial inference: the $p_m$-tuning offers an experimental handle: a measurement apparatus with controlled noise could still certify the central charge as long as its effective $p_m$ stays inside the robust plateau, turning measurement noise from a nuisance into a probe.","Editorial inference: replacing $\\hat{M}=Z$ or $X$ by other local operators in the R2GSE definition may produce a phase diagram of robustness that tracks whether the chosen basis is a conformal boundary condition; this is testable with the same numerical scheme."],"forward_implications":["Because the R2GSMI connects the Rényi-2 mutual information and the Rényi-2 Shannon mutual information as limits, any experimental setup that realizes partial decoherence of strength $p_m$ can interpolate between the two quantities and should still detect $c_2=1$ for the Ising critical point.","In the $\\hat{M}=X$ basis the extracted central charge is $c_2=1$ for all $p_m$, so the CFT fingerprint is immune to relaxing the measurement in that basis.","In the $\\hat{M}=Z$ basis the robustness has a window: $c_2\\simeq 1$ for $p_m\\ge 0.2$ and at $p_m=0$, with a dip near $p_m\\approx 0.1$, so the plateau region defines where noisy Z-basis measurements remain reliable.","For a globally $Y$-decohered critical state, $c_2\\simeq 1$ persists up to $p_y\\sim 0.3$ in the $(p_m,p_y)$ plane, indicating that local environmental decoherence does not immediately destroy the Ising CFT signature.","The doubled-space depolarization scheme provides a practical route to compute such generalized mutual informations for mixed states, not only for the pure ground state treated here."],"supporting_citations":[{"why":"provides the exact conformal-field-theory scaling law and the Ising central charge $c=1/2$ that the numerical values of $c_2$ are compared against.","marker":"[7]"},{"why":"introduces the universal Shannon mutual information scaling of critical chains and motivates the experimentally accessible observable that the R2GSMI generalizes.","marker":"[15]"},{"why":"defines the Rényi-2 Shannon mutual information and states the conjecture $c_n=c\\,n/(n-1)$ that fixes the benchmark $c_2=1$ for the Ising CFT.","marker":"[16]"},{"why":"reports the earlier numerical estimate $c_2\\simeq 1.02$ for the $Z$ basis in a free-fermion CFT, the baseline against which the present $p_m=0$, $p_y=0$ point is compared.","marker":"[18]"},{"why":"studied the Rényi-2 mutual information of Y-decohered critical states and supplies the $p_m=0$ line of the $(p_m,p_y)$ plane that the present results reproduce.","marker":"[23]"},{"why":"supplies the tensor-network routines used to prepare the critical ground state of the transverse-field Ising model.","marker":"[26,27]"},{"why":"gives the maximal depolarization channel construction used to trace out subsystem B in the doubled Hilbert space calculation of the R2GSE.","marker":"[35]"},{"why":"identifies the Z-basis projection as a conformal boundary condition of the Ising CFT, the property invoked to interpret the Z-basis $p_m$ dependence.","marker":"[41]"}],"fun_headline_variants":["Central charge stays 1 in generalized mutual info under decoherence","Rényi-Shannon mutual info robust to decoherence, retains Ising CFT","New mutual info measure keeps central charge 1 amid decoherence","Decohered critical system still shows central charge c=1","Generalized Rényi-Shannon info insensitive to local decoherence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the sine-form CFT scaling law with constant $c_2$ and $b_2$ holds at the numerically accessible size $L=32$ for every $(p_m,p_y)$ considered, so that fitting only four subsystem sizes ($L_A=8,12,14,16$) in the decohered case yields the true $c_2$; if the ansatz or the fit window fails at those sizes, the claimed robustness is not established.","fun_headline_variants_meta":{"raw":{"variants":["Central charge stays 1 in generalized mutual info under decoherence","Rényi-Shannon mutual info robust to decoherence, retains Ising CFT","New mutual info measure keeps central charge 1 amid decoherence","Decohered critical system still shows central charge c=1","Generalized Rényi-Shannon info insensitive to local decoherence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000245,"raw_usage":{"total_tokens":1629,"prompt_tokens":1131,"completion_tokens":498,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":747,"completion_tokens_details":{"reasoning_tokens":402}},"tokens_in":747,"tokens_out":498,"duration_ms":4874,"temperature":1.0,"reasoning_tokens":402,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:47:29.507194+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the $(p_m,p_y)$ scan at $L=64$ and $L=128$ with the same fit window: if the extracted $c_2$ drifts with system size or the boundaries of the $c_2\\simeq 1$ plateau move, the robustness is a finite-size effect; alternatively, compute the residuals of the Eq. (17) fit as a function of $L_A$—systematic curvature near $p_m\\approx 0.1$ would show that the dip in $c_2$ reflects a failure of the scaling ansatz rather than a physical change in the state.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the universal Shannon mutual information scaling of critical chains and motivates the experimentally accessible observable that the R2GSMI generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"reports the earlier numerical estimate $c_2\\simeq 1.02$ for the $Z$ basis in a free-fermion CFT, the baseline against which the present $p_m=0$, $p_y=0$ point is compared."}],"review_version":2}