{"id":"feb60ead-94f2-45db-b668-e978454af2fb","arxiv_id":"2501.04526","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":7,"one_line_summary":"Non-Markovian dephasing and depolarizing noise can help GHZ and W states keep or revive entanglement, but the paper's predicted infinite-qubit survival for the one-versus-rest W cut is an extrapolation artifact.","lead":"This paper examines how non-Markovian noise, where the environment sends information back into the system, affects entanglement in multi-qubit GHZ and W states. It finds that such noise can make entanglement settle at non-zero values and can revive entanglement after it collapses, which matters because sustained multiparty entanglement is a core resource for quantum information tasks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The asymptotic W-state robustness claim is an extrapolation artifact: the exact 1-Rest logarithmic negativity under local dephasing is log2(1 + 2√(N−1)λ²/N), which tends to 0, not to the fitted 0.24 ebits.","rationale":"The reader's weakest-assumption analysis identifies precisely the load-bearing flaw: the exponential-plus-offset extrapolation assumes a nonzero asymptote without an exact basis. I independently confirm the exact dephased-W 1-Rest formula, which is decisive. The claim of 24% asymptotic robustness is not merely uncertain; it is contradicted by an exact calculation. The paper does contain correct finite-N numerics and a plausible highest-cut saturation, but the headline result — W states retain nonzero entanglement for any bipartition in the asymptotic limit under non-Markovian dephasing — is false. The depolarising revival section has a separate concern about whether the chosen rates γz(t)=α sin(t), γx=γy=0.1 correspond to a physically derived completely positive dynamics, but the dephasing calculation alone is sufficient to reject the central advertised claim. Therefore the reader's REJECT verdict should stand.","tokens_in":13840,"tokens_out":12802,"duration_ms":133939,"concrete_test":"Using Eq. (8) with s=2.47, compute λ(t)=exp[−2∫₀ᵗ γ(t′)dt′] at t=30; evaluate the exact expression E_N = log2(1 + 2√(N−1)λ²/N) for N=3..10 and compare with Fig. 4(c). If the exact values match the numerical data but E_N continues to decrease toward 0 at N=20, 50, 100 while the fitted ansatz saturates at b²=0.24, then the asymptotic claim is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim that W states retain non-zero entanglement across any bipartition as N→∞ relies on fitting E_N = a e^{−c(N−3)} + b² to numerical data for N=3..10 and reading off the offset b²≈0.24 (Section III, after Fig. 4(c), a=0.2329, b=0.4906, c=0.1559). But the 1-Rest logarithmic negativity of a locally dephased W state is exactly computable. The dephased state is ρ = (1−λ²)/N Π₁ + λ² |W_N⟩⟨W_N|, where λ = exp[−2∫γ(t′)dt′] is the single-qubit coherence factor, because every pair of single-excitation basis states differs in exactly two qubits. Partial transposition over one qubit gives trace norm 1 + 2√(N−1)λ²/N, hence E_N^{1-Rest} = log2(1 + 2√(N−1)λ²/N). For any fixed λ≤1, this tends to 0 as N→∞; even the decoherence-free case λ=1 gives a vanishing 1-Rest negativity. The finite-N data in Fig. 4(c) are compatible with this exact form, and the fitted offset b² is an artifact of extrapolating a slowly varying function over N=3..10. Thus the advertised 'any bipartition' asymptotic saturation fails for the 1-Rest cut. The highest-cut saturation may survive, but it does not rescue the broader claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the effect of local non-Markovian dephasing and depolarizing noise on the entanglement of N-qubit GHZ and W states. The authors numerically solve the time-local master equation for N=3 to 10, compute logarithmic negativity for the 1-Rest and highest-cut bipartitions, and extrapolate the N-dependence using fitted exponential forms. The main advertised results are that non-Markovian dephasing makes W-state entanglement saturate to a nonzero value as N grows (24% for the 1-Rest cut, about 49.7% for the highest cut), that W states are more robust than GHZ states, that there is an even-odd dichotomy in the highest-cut entanglement, and that non-Markovian depolarizing noise induces entanglement revival after collapse.","tokens_in":14203,"tokens_out":11431,"duration_ms":118647,"significance":"If the asymptotic claims were correct, the paper would establish a striking resource-preservation effect of non-Markovian noise and would be a useful contribution to the open-systems literature. The finite-N numerical work for N=3 to 10 appears internally consistent, and the identification of the optimal Ohmicity region (s between 2.3 and 2.5) is a useful observation. However, the central asymptotic W-state claim is directly contradicted by an exact calculation for the 1-Rest bipartition, so the main message of the paper is not reliable. The depolarizing-noise revival phenomenology and the finite-N comparisons may survive a revision, but the advertised asymptotic robustness does not.","major_comments":[{"comment":"The claimed asymptotic saturation E_N = 0.24 for the 1-Rest bipartition is an extrapolation artifact. For any local dephasing channel the evolved W state is rho(t) = lambda^2 |W_N><W_N| + (1-lambda^2)/N P_1, where lambda = exp[-integral_0^t gamma(t') dt'] and P_1 projects onto the single-excitation subspace. The 1-Rest logarithmic negativity is exactly E_N^{1-Rest} = log2(1 + 2 sqrt(N-1) lambda^2 / N). For a legitimate CPTP dephasing evolution 0 <= lambda <= 1, and for every fixed lambda this quantity tends to 0 as N goes to infinity; even the decoherence-free case lambda = 1 behaves as log2(1 + 2/sqrt(N)) and vanishes asymptotically. The data in Fig. 4(c) for N=3..10 are compatible with this slow decay, and the fitted offset b^2 = 0.4906^2 is not a physical limit. This directly invalidates the statement in the Conclusion that the W state retains a nonzero amount of entanglement for any bipartition in the asymptotic limit.","section":"Section III, 'Results for Dephasing noise', paragraph after Fig. 4(c)"},{"comment":"The asymptotic predictions are obtained by fitting E_N = a e^{-c(N-3)} + b^2 (or the reciprocal form for the odd-N highest cut) to eight data points at a single time t=30 and reading off b^2 as the asymptotic value. No model selection, uncertainty quantification, or validation on larger N is provided. This procedure is especially fragile here because the exact 1-Rest W formula decays only as a power law, so an exponential-plus-offset fit over N=3..10 will generically return a spurious positive offset. The extrapolation method is therefore not a reliable basis for any of the asymptotic claims in the paper.","section":"Section III, extrapolation procedure for GHZ and W states"},{"comment":"The concluding claim that the W state retains a nonzero amount of entanglement for any bipartition in the asymptotic limit is false for the 1-Rest cut, as shown by the exact formula above. In addition, for N>=4 the paper computes only two cuts (1-Rest and highest-cut); intermediate bipartitions such as 2 vs N-2 for N>=6 are not computed, so the 'any bipartition' wording is unsupported even independently of the exact contradiction.","section":"Section IV, Conclusion"}],"minor_comments":[{"comment":"The ratio omega_c/omega_0 never appears explicitly in the text, although the dimensionless time t = omega_0 t~ is used throughout; the authors should state the value of omega_c/omega_0 used in the numerical simulations.","section":"Section II, Eq. (8)"},{"comment":"The phrase 'single to (N-1) mode excitation' is used, but a multimode W state is never defined; Eq. (2) gives only the single-excitation W state, so the depolarising-section statement about multimode W states is ambiguous.","section":"Section III, depolarising-noise subsection"},{"comment":"The text says the evolution runs to t=100, yet all N-dependence data are reported at t=30; a convergence check in t for the largest N would clarify that the values plotted are indeed saturated values.","section":"Section III, 'characterising the final state entanglement'"},{"comment":"The extrapolation panels extend to N=60 while the data cover only N=3..10, and no error bars or goodness-of-fit measures are shown; this makes it difficult to assess the reliability of the fitted offsets.","section":"Fig. 4(c) and Fig. 4(d)"}],"recommendation":"reject","confidential_remarks":"The exact 1-Rest formula for the dephased W state is decisive and I verified it against the known N=3 pure-state value. The paper's main advertised result fails, and the error is load-bearing: the asymptotic W-state robustness claim cannot be repaired without removing the central claim and substantially changing the scope of the manuscript. Some finite-N observations and the depolarising revival phenomenology are salvageable, but they would constitute a different paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper makes a specific, testable claim: under non-Markovian dephasing, N-qubit W states retain a non-zero fraction of entanglement in the 1-Rest bipartition as N grows, and they quote 0.24 ebits. That claim is wrong, and the error is easy to pin down. For a locally dephased W state, every off-diagonal coherence decays by the same factor λ², and the 1-Rest logarithmic negativity is exactly log2(1+2√(N-1)λ²/N), which tends to 0 for any fixed λ. The 0.24 comes from fitting E_N = a e^{-c(N-3)} + b² to N=3..10 and reading off b². The fit is fine; the inference is not.\n\nWhat is genuinely new is the finite-N numerical study: the saturation curves for GHZ and W states for s>2, and the even-odd pattern in the balanced-cut W entanglement. The balanced-cut saturation at about 0.4977 ebits is consistent with an exact limiting value, and the even-odd dichotomy is a real finite-size effect in their data. Those parts are worth keeping.\n\nThe soft spots beyond the main error: the depolarizing revival results use γz(t)=α sin(t) with γx=γy=0.1 and never prove the evolution is completely positive. The paper cites CP conditions but does not verify them for the chosen rates, so the revivals may be artifacts of a non-physical map. Also, the choice s=2.47 is justified only by scanning to maximize retained entanglement; that is a free parameter, not a derivation.\n\nThe citation pattern is adequate, and the authors do not oversell their methods. The central claim is simply contradicted by the exact calculation, and the 'any bipartition' asymptotic statement in the text is false. This is a solid cautionary example for a reading group about extrapolation, but not a paper to cite or build on. A serious editor could still send it to referees, because the error is subtle and the topic is relevant; but the outcome should be rejection unless the authors remove the asymptotic claim and reframe the paper as a finite-N numerical study.\n\nRecommendation: send to review if you have a referee who can work through the exact formula; otherwise desk-reject. My vote: reject.","headline":"The paper's finite-N numerics are plausible, but the advertised asymptotic W-state robustness under dephasing is an extrapolation artifact contradicted by the exact 1-Rest negativity formula.","tokens_in":14790,"tokens_out":3981,"would_cite":false,"duration_ms":37262,"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":"Under noise with memory, W states retain entanglement as the number of qubits grows.","keywords":["multipartite entanglement","non-Markovian dynamics","dephasing channel","depolarising channel","W states","GHZ states","logarithmic negativity","entanglement revival"],"falsifier":"Take the $N$-qubit W state and apply local dephasing with coherence survival factor $p$; the exact 1-Rest logarithmic negativity is $\\log_2\\!(1+\\frac{2\\sqrt{N-1}\\,p}{N})$, which tends to zero for any fixed $p<1$ as $N\\to\\infty$. Evaluating this closed form for $N=10,20,50,100$ using the paper's dephasing factor would directly show whether the extrapolated $0.24$ floor is real or a finite-size fitting artifact.","tokens_in":13549,"feed_emoji":"⚛️","tokens_out":11811,"duration_ms":109687,"temperature":0.7,"pith_summary":"Entanglement is the working resource for distributed quantum tasks, but real qubits are hit by their environments. This paper asks whether multiparty entanglement can survive when each qubit is locally coupled to a reservoir that remembers its past, a non-Markovian channel, rather than a memoryless one. By solving the master equation for GHZ cat states and W states under local dephasing and depolarising noise, the authors find that memory effects change the answer: under non-Markovian dephasing, W-state entanglement saturates at a nonzero value in every bipartition as the number of qubits grows, while GHZ entanglement dies out around 16 qubits. Under non-Markovian depolarising noise, both state families show entanglement revival after collapse. If these claims hold, engineering or exploiting environmental memory is a way to protect multiparty entanglement.","feed_headline":"Quantum W states keep 24% entanglement under memory noise","feed_subtitle":"Cat states fade by 16 qubits; W states saturate and even revive after collapse.","key_machinery":"The load-bearing object is the time-local GKSL master equation with time-dependent decay rates, whose signs control whether the channel is Markovian or non-Markovian. For dephasing, the paper uses the zero-temperature Ohmic-bath rate $\\gamma_{T=0}(t,s)=\\omega_c[1+(\\omega_c t)^2]^{-s/2}\\Gamma[s]\\sin(s\\arctan(\\omega_c t))$: for $s>2$ the rate turns negative, producing information back-flow and non-Markovianity. The evolved state $\\rho(t)$ is obtained by integrating the master equation for each $N$, and entanglement is quantified by logarithmic negativity over the 1-Rest and highest-cut bipartitions. The $N$-dependence of the saturated negativity is then extrapolated with exponential-plus-constant fits, and the constant term is what yields the claimed asymptotic values.","core_discovery":"The central discovery, stated on the paper's own terms, is that information back-flow from a non-Markovian environment changes the asymptotic fate of multiparty entanglement. For local dephasing with an Ohmic reservoir at zero temperature, the time-dependent rate $\\gamma_{T=0}(t,s)$ becomes negative for Ohmicity $s>2$, and in that regime the time-saturated logarithmic negativity, an entanglement measure for mixed states, stays positive for W states when evaluated on 1-vs-rest and highest-cut bipartitions. Fitting results for $N=3$ through $N=10$ at $t=30$ to $E_N = a e^{-c(N-3)}+b^2$, the authors obtain $b^2=0.24$ for the 1-Rest cut and conclude that W states are 24% robust asymptotically; the highest-cut saturates at $E_N=0.4977$ for even $N$ and approaches $0.4962$ for odd $N$, an even-odd dichotomy. By contrast, GHZ states retain a nonzero value only up to about $N=16$ before it vanishes. Under local depolarising noise with a periodically sign-changing rate, both GHZ and W states collapse and then revive, with a second revival for $N\\ge7$ in the highest-cut case.","pith_inferences":["An exact calculation of the dephased single-mode W state's 1-Rest logarithmic negativity gives $\\log_2\\!(1+2\\sqrt{N-1}\\,p/N)$, which vanishes as $N\\to\\infty$; if that calculation applies to the paper's model, the reported $0.24$ asymptotic value is a finite-size fitting artifact rather than a true limit.","The even-odd saturation pattern suggests a design rule for dephasing-prone protocols: choose an even number of parties so the highest-cut entanglement floor is flat in $N$.","The revival windows in the depolarising channel suggest scheduling entanglement distribution or distillation during the high-entanglement intervals, turning non-Markovian memory into a timing resource."],"forward_implications":["For dephasing, W states are claimed to retain a nonzero amount of entanglement in every bipartition in the asymptotic limit, with the 1-Rest value saturating at $E_N=0.24$.","GHZ states under the same non-Markovian dephasing are claimed to lose their retained entanglement completely by about $N=16$.","For the highest-cut bipartition under dephasing, even-$N$ W states saturate at $E_N=0.4977$ independent of $N$, while odd-$N$ W states approach $E_N=0.4962$, giving an even-odd dichotomy.","Under non-Markovian depolarising noise, both GHZ and W states first lose their entanglement and then revive it, with a second revival appearing for $N\\ge7$ in the highest-cut case."],"supporting_citations":[{"why":"Defines the multiparty-entanglement robustness question the paper extends, providing the depolarising-noise baseline for GHZ states.","marker":"[37]"},{"why":"Supplies the exact zero-temperature Ohmic-bath dephasing rate used in Eq. (8), which makes the decay rate negative for s>2.","marker":"[46]"},{"why":"Provides the open-quantum-systems master-equation and thermal-bath framework behind the dephasing rate in Eq. (6).","marker":"[22]"},{"why":"Establishes that GHZ and W states are SLOCC-inequivalent, which motivates treating them separately under noise.","marker":"[35]"},{"why":"Defines information back-flow as a measure of non-Markovianity, the mechanism the paper credits for sustaining entanglement.","marker":"[30]"},{"why":"Supplies the entanglement-based non-Markovianity measure that supports the paper's use of memory effects.","marker":"[31]"},{"why":"Gives the P-divisibility conditions used to choose decay rates in the depolarising model.","marker":"[47]"},{"why":"Provides the divisibility criteria that certify when the chosen depolarising rates make the dynamics non-Markovian.","marker":"[48]"}],"fun_headline_variants":["Entanglement revives in multiparty states under non-Markovian noise","W states outlast GHZ in non-Markovian noise, saturate at 24%","Memory noise lets W states keep entanglement alive","Non-Markovian noise: W states saturate, GHZ states revive","Even-odd entanglement dichotomy in W states under memory noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's asymptotic claims rest on assuming that a curve fit through data for $N=3$ through $N=10$ at one time continues to hold for all larger $N$, so the fitted constant term is taken to be the permanent entanglement value.","fun_headline_variants_meta":{"raw":{"variants":["Entanglement revives in multiparty states under non-Markovian noise","W states outlast GHZ in non-Markovian noise, saturate at 24%","Memory noise lets W states keep entanglement alive","Non-Markovian noise: W states saturate, GHZ states revive","Even-odd entanglement dichotomy in W states under memory noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000609,"raw_usage":{"total_tokens":2878,"prompt_tokens":1027,"completion_tokens":1851,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":643,"completion_tokens_details":{"reasoning_tokens":1755}},"tokens_in":643,"tokens_out":1851,"duration_ms":13029,"temperature":1.0,"reasoning_tokens":1755,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:33:57.474165+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the $N$-qubit W state and apply local dephasing with coherence survival factor $p$; the exact 1-Rest logarithmic negativity is $\\log_2\\!(1+\\frac{2\\sqrt{N-1}\\,p}{N})$, which tends to zero for any fixed $p<1$ as $N\\to\\infty$. Evaluating this closed form for $N=10,20,50,100$ using the paper's dephasing factor would directly show whether the extrapolated $0.24$ floor is real or a finite-size fitting artifact.","supporting_citations":[{"cited_title":"Zeilinger, M","cited_arxiv_id":null,"evidence_quote":"Defines the multiparty-entanglement robustness question the paper extends, providing the depolarising-noise baseline for GHZ states."},{"cited_title":"Mansour and S","cited_arxiv_id":null,"evidence_quote":"Supplies the exact zero-temperature Ohmic-bath dephasing rate used in Eq. (8), which makes the decay rate negative for s>2."},{"cited_title":"Goswami, S","cited_arxiv_id":null,"evidence_quote":"Provides the open-quantum-systems master-equation and thermal-bath framework behind the dephasing rate in Eq. (6)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that GHZ and W states are SLOCC-inequivalent, which motivates treating them separately under noise."},{"cited_title":"Shabani and D","cited_arxiv_id":null,"evidence_quote":"Defines information back-flow as a measure of non-Markovianity, the mechanism the paper credits for sustaining entanglement."},{"cited_title":"Schmid, K","cited_arxiv_id":null,"evidence_quote":"Supplies the entanglement-based non-Markovianity measure that supports the paper's use of memory effects."},{"cited_title":"Peng, F.-L","cited_arxiv_id":null,"evidence_quote":"Gives the P-divisibility conditions used to choose decay rates in the depolarising model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the divisibility criteria that certify when the chosen depolarising rates make the dynamics non-Markovian."}],"review_version":1}