{"id":"9ff371ad-802a-47b8-b4f8-af00273e7a2b","arxiv_id":"2505.16525","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The largest entanglement eigenvalue of a quantum chaotic kicked Ising chain follows a Weibull-type extreme value distribution rather than the random-matrix Tracy-Widom law, even as ETH is satisfied.","lead":"Using a kicked spin chain, a standard model of quantum chaos, the authors find that its entanglement spectrum does not follow all random matrix predictions. The largest Schmidt eigenvalue deviates from the expected Tracy-Widom law and is instead fit by a Weibull-type extreme value distribution, while thermalization holds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Weibull-type claim rests on a single 3-parameter GEV fit at L=18, while the paper's own moment and KL-divergence data show KFIM approaching Tracy–Widom, so the asymptotic extreme-value classification is not established.","rationale":"The reader's weakest-assumption identification is exactly the load-bearing issue: the Weibull-type claim is built on a single three-parameter fit at L=18 without finite-size scaling of xi or comparison against alternative distributions. My independent reading confirms this and adds a specific mechanism: Tracy–Widom data are in the Gumbel max-domain, so a flexible GEV fit can easily return a nonzero xi on finite samples even when the asymptotic law is Tracy–Widom. The paper's own metrics (R, DKL, moments in Table I) move toward Tracy–Widom with increasing L, so the data do not distinguish a true Weibull asymptotic class from a finite-size crossover. The deviation-from-Tracy-Widom part of the paper is supported by multiple consistent observations and by the contrast with COE and Wishart data, and the ETH/autocorrelation results provide independent evidence of chaos; those parts are not the problem. The verdict CONDITIONAL remains appropriate because the core empirical deviation claim is credible but the Weibull classification needs the proposed systematic validation before it can be stated in the abstract as the distribution. Since my concern matches the reader's rather than introducing a new one, no verdict adjustment is needed.","tokens_in":13838,"tokens_out":5604,"duration_ms":53874,"concrete_test":"Fit the same 3-parameter GEV (Eq. 5) to 40960 Lmax samples generated from the trace-normalized 2^9 x 2^9 Wishart ensemble (W9), using the same fitting window as Fig. 1(d), and record the fitted xi. Then fit the KFIM L=18 data with xi free and with xi fixed to 0 (Gumbel), and compute the log-likelihood difference or BIC for both W9 and KFIM. Finally, repeat the free-xi GEV fit for KFIM at L=12,14,16 from the existing data and plot xi(L) with error bars. If xi(W9) is comparable to 0.17, or if the Gumbel fit is not strongly rejected for KFIM, the Weibull classification is not established; if xi(L) drifts toward 0 (or toward the Tracy–Widom-compatible value) with L, the asymptotic law is likely Tracy–Widom rather than Weibull.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the largest Schmidt eigenvalue of KFIM follows a Weibull-type extreme value distribution depends on accepting the fitted shape parameter xi = 0.17 at L=18 as representative of the asymptotic law. This step is insecure for two reasons. First, the Fisher–Tippett–Gnedenko theorem applies to maxima of iid draws, whereas Schmidt eigenvalues of a random-matrix state are strongly correlated and have Tracy–Widom fluctuations, which lie in the Gumbel domain (xi = 0 under the paper's sign convention). The cited weak-correlation extension (Ref. [47]) is not checked for the KFIM entanglement spectrum. Second, with three free parameters (alpha, beta, xi), the GEV family can absorb finite-size and Tracy–Widom tails and return a nonzero xi even when the limiting law is Tracy–Widom. The paper does not report xi as a function of L, does not compare the GEV fit against the finite-size Wishart reference W9 used elsewhere in the paper, and does not perform a nested test of xi=0 versus xi=0.17. Moreover, the paper's own Table I shows KFIM mean, variance, and skewness moving monotonically toward Tracy–Widom values as L grows (mean from -0.207 to -0.475 toward -1.207, variance from 2.057 to 1.741 toward 1.608, skewness from 0.577 to 0.326 toward 0.293), and Fig. 2(c) shows DKL decreasing after L=14. This is equally consistent with a slow crossover to Tracy–Widom, as the body text itself cautions. The abstract states the Weibull conclusion more strongly than the body's statement that it is difficult to conclude whether Tracy–Widom is achieved asymptotically.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the kicked field Ising model (KFIM) at the self-dual point J=b=π/4, a paradigmatic many-body quantum chaotic Floquet spin-1/2 chain, and investigates whether all random-matrix-theory (RMT) properties hold for this system. The authors compute the reduced density matrix of eigenstates near phase ϕ=π/2, and analyze the statistics of the largest Schmidt eigenvalue. They report that, even at L=18 (subsystem dimension 2^9), the distribution of the centered and scaled largest Schmidt eigenvalue deviates from the Tracy–Widom distribution, and they fit a three-parameter generalized extreme value (GEV) distribution with shape parameter ξ=0.17±0.011, interpreting this as a Weibull-type extreme value law. They also demonstrate that the KFIM satisfies the diagonal and off-diagonal eigenstate thermalization hypothesis via scaling of matrix-element fluctuations of σ^z, and that the spin-spin autocorrelation function decays exponentially and saturates to a system-size-dependent value. The conclusion is that a quantum chaotic model can deviate from certain RMT statistics while still thermalizing.","tokens_in":14199,"tokens_out":6118,"duration_ms":49448,"significance":"If the core claim holds, the paper provides a useful counterexample to the common assumption that spectral RMT behavior in many-body quantum chaos implies all RMT statistical properties, and it complements earlier work on entanglement-statistics deviations in the KFIM. The numerical effort is substantial and carefully controlled: system sizes up to L=18 with ~10^4–10^6 eigenstates per size, comparison against finite-size Wishart and COE references, and explicit ETH scaling fits with quoted parameters. The deviation from Tracy–Widom is well documented through histograms, cumulants, and distance measures. However, the specific identification of a Weibull-type asymptotic law is not established: it rests on a single GEV fit at L=18, with no finite-size extrapolation of the shape parameter and no goodness-of-fit comparison against the finite-size Wishart distribution used elsewhere in the paper. The ETH and autocorrelation analyses are convincing and support the secondary claim that deviations in extreme-value statistics do not destroy thermalization. The paper's significance would be strengthened considerably if the extreme-value classification were placed on firmer statistical ground.","major_comments":[{"comment":"The Weibull-type classification rests entirely on a single three-parameter GEV fit at L=18, with fitted parameters α=7.77×10^-3, β=9.40×10^-5, ξ=0.17±0.011. The manuscript does not report the fitted shape parameter as a function of L, does not test the null hypothesis ξ=0 (Gumbel) against the fitted value, and does not compare the GEV fit with the finite-size Wishart reference W9 used elsewhere in the paper. Since the GEV family with three free parameters can absorb finite-size corrections and even mimic Tracy–Widom tails, the fitted ξ is not evidence for a distinct asymptotic extreme-value domain. This is load-bearing because the abstract asserts that the distribution 'follows the extreme value distribution of Weibull type.' I recommend adding a finite-size scaling analysis of (α, β, ξ) and applying the same GEV fit to the W9 and COE data as a control.","section":"Main text, paragraph after Eq. (5)"},{"comment":"The application of the Fisher–Tippett–Gnedenko theorem to the largest Schmidt eigenvalue is not justified without checking the relevant correlation conditions. The theorem concerns maxima of i.i.d. draws (or weakly dependent sequences satisfying mixing conditions); the largest eigenvalue of a Wishart matrix is known to follow the Tracy–Widom distribution, which lies in the Gumbel domain (ξ=0 under the manuscript's sign convention). The manuscript cites Ref. [47] for the weak-correlation extension, but does not verify that the KFIM entanglement spectrum satisfies those conditions. A concrete and easily implementable test would be to fit the same GEV form to the numerically obtained W9 distribution and to the COE distribution: if those fits also produce nonzero ξ, the interpretation of ξ=0.17 as a Weibull-type index would be untenable.","section":"Main text, paragraph introducing Eq. (5)"},{"comment":"The paper's own convergence diagnostics are equally consistent with a slow crossover to Tracy–Widom rather than a distinct Weibull limit. Table I shows the KFIM mean, variance, and skewness moving monotonically toward the Tracy–Widom values as L increases (mean from -0.2067 to -0.4746 toward -1.207; variance from 2.057 to 1.741 toward 1.608; skewness from 0.5774 to 0.3257 toward 0.293), and Fig. 2(c) shows D_KL decreasing after L=14. The body text acknowledges this ambiguity ('it is difficult to conclude concretely... if the Tracy–Widom distribution is really achieved asymptotically'), but the abstract and conclusion present the Weibull classification as the definite outcome. The manuscript should either provide a quantitative asymptotic analysis supporting the Weibull limit or temper the claim to state that the distribution is not Tracy–Widom at accessible sizes and that a GEV fit at L=18 yields a Weibull-type shape parameter.","section":"Table I and Fig. 2(c)"}],"minor_comments":[{"comment":"The text says Q(x) is the distribution obtained from experimental data and P(x) is the theoretical distribution, but the formula sums P(x) log(P(x)/Q(x)); the roles of P and Q should be clarified to avoid confusion about the sign and direction of the KL divergence.","section":"Eq. (6)"},{"comment":"The sentence 'Due to the unclear trend for the KFIM case it is difficult to difficult to conclude concretely' contains a duplicated phrase 'difficult to'.","section":"Main text, paragraph on KL divergence"},{"comment":"The text refers to 'Fig. 2(bd)' when citing panels of Fig. 2; this appears to be a typo for 'Fig. 2(b)-(d)' or similar.","section":"Fig. 2 caption and text"},{"comment":"The phrase 'trace out first L/2 spin' should be 'trace out the first L/2 spins' for grammatical correctness.","section":"Numerical details paragraph"},{"comment":"The phrase 'where the we take' contains an extra article; it should read 'where we take'.","section":"Paragraph after Eq. (7)"},{"comment":"References [19] and [20] appear to be the same paper (Pausch et al., 'Chaos and ergodicity across the energy spectrum of interacting bosons,' Phys. Rev. Lett. 126, 150601 (2021)) with only the arXiv metadata differing; one of the two entries should be removed or replaced.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope and the numerical data appear sound. The main concern is the gap between the abstract's categorical Weibull-type claim and the evidence actually presented: a single three-parameter GEV fit at L=18 with no finite-size scaling or control fits. The deviation from Tracy–Widom is well supported, so the paper is not in danger of rejection on that ground; however, the extreme-value classification needs additional work. If the authors cannot provide a convincing asymptotic analysis, they should substantially soften the claim. The ETH and autocorrelation sections are solid and add value independent of the extreme-value issue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The numerical evidence that the largest Schmidt eigenvalue of the KFIM deviates from the Tracy-Widom distribution is solid. Large sample sizes up to L=18, comparison against a numerically generated Wishart distribution of matching dimension, and the clear contrast with COE matrices all support the deviation claim. The ETH analysis is also credible: diagonal and off-diagonal scalings are close to the expected 2^{-L/2} behavior, and the autocorrelation saturates as expected. That is a genuine and useful result: a clean counterexample to the assumption that level repulsion implies all RMT features, complementing Ref. [32] on entropy statistics.\n\nThe soft spot is the Weibull claim. It rests on a single three-parameter GEV fit at L=18 with shape parameter xi=0.17. The paper does not track xi across system sizes, does not compare the GEV fit against the finite-size Wishart reference used elsewhere, and does not test whether xi=0 (Gumbel/Tracy-Widom-like tail) is rejected. The Fisher-Tippett-Gnedenko theorem applies to maxima of iid draws, and Schmidt eigenvalues of a random state are correlated; the cited weak-correlation extension is not verified for this system. Moreover, Table I shows mean, variance, and skewness moving monotonically toward Tracy-Widom values as L grows, and D_KL decreases after L=14. That is at least as consistent with a slow crossover to Tracy-Widom as with an asymptotic Weibull law. The body text honestly says it is difficult to conclude, but the abstract states the Weibull conclusion more firmly.\n\nSo my verdict: the deviation claim is well supported and worth publishing; the Weibull classification needs more work. I would send this to peer review, but ask the authors to add finite-size scaling of the GEV shape parameter, compare against the Wishart reference, and soften the abstract accordingly. The paper is for people working on entanglement statistics of many-body chaotic systems and RMT breakdown. I would bring it to a reading group, though I would not cite it myself until the asymptotic classification is resolved.","headline":"Convincing demonstration of Tracy-Widom failure in a kicked spin chain, but the Weibull-type claim is overreaching without finite-size scaling.","tokens_in":14747,"tokens_out":2617,"would_cite":false,"duration_ms":21350,"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 shows that the largest Schmidt eigenvalue (the top entanglement-spectrum value) of eigenstates of a kicked quantum chaotic spin-1/2 chain does not follow the Tracy-Widom distribution even at L=18; a Weibull-type extreme value…","keywords":["quantum chaos","kicked field Ising model","entanglement spectrum","Schmidt eigenvalues","Tracy-Widom distribution","extreme value statistics","eigenstate thermalization hypothesis","random matrix theory"],"falsifier":"Repeat the generalized extreme-value fit for the largest Schmidt eigenvalue at L=12, 14, 16, and 18 under the same centering and test the L=18 fit against the Tracy-Widom F1 distribution with a goodness-of-fit statistic; if xi moves systematically toward 0 as L grows, or if Tracy-Widom fits within error bars at L=18, the Weibull-type classification is a finite-size artifact rather than the asymptotic law.","tokens_in":13612,"feed_emoji":"🎲","tokens_out":12056,"duration_ms":94173,"temperature":0.7,"pith_summary":"This paper challenges the common assumption that a many-body quantum chaotic system reproduces every prediction of random matrix theory once its spectrum shows level repulsion. Working with the kicked field Ising model, a standard maximally chaotic spin-1/2 Floquet system, the authors show that the largest eigenvalue of the reduced density matrix (the leading Schmidt coefficient) of mid-spectrum eigenstates does not follow the Tracy-Widom distribution, even at L=18. Instead, after centering and rescaling, its distribution is described by the generalized extreme value equation with fitted shape parameter xi = 0.17 +/- 0.011, which the paper identifies with a Weibull-type (bounded tail) law. The same model still satisfies the eigenstate thermalization hypothesis in both diagonal and off-diagonal matrix elements, and its spin autocorrelation decays exponentially and saturates to a value that shrinks with system size, so the deviation does not spoil thermalization. If correct, this means standard Gaussian and circular ensembles are not enough to capture all eigenstate statistics of local chaotic many-body systems.","feed_headline":"Chaotic spin chain's top entanglement eigenvalue skips Tracy-Widom","feed_subtitle":"Even at 18 spins the entanglement-spectrum extreme follows a Weibull law, while thermalization still holds.","key_machinery":"The central object is the entanglement spectrum of a bipartitioned eigenstate: the Schmidt coefficients $\\{\\lambda_i\\}$ from the Schmidt decomposition of the state, with the maximum coefficient $\\lambda_{\\max}$ carrying the largest weight in entanglement properties. The machinery is the Fisher-Tippett-Gnedenko theorem, which states that the centered and rescaled maximum of many independent variables converges to one of three universal extreme-value families (Gumbel, Fr\\'echet, Weibull) selected by the tail of the underlying density; the paper fits the generalized extreme-value form of Eq. (5) to $\\lambda_{\\max}$ to classify the tail. Supporting machinery includes the Marchenko-Pastur law for the bulk density of Schmidt eigenvalues, the Wishart ensemble (with unit trace) as the random-matrix model for the reduced density matrix, the Tracy-Widom distribution as the predicted law for the largest eigenvalue of large Wishart matrices, and the ETH ansatz for matrix elements of local observables.","core_discovery":"On the paper's own terms, the discovery is that the maximum Schmidt coefficient $\\lambda_{\\max}$ of eigenstates of the self-dual kicked field Ising model, centered and rescaled using the Wishart/Tracy-Widom scaling of Eq. (4), remains far from the Tracy-Widom $F_1$ distribution at the largest accessible size $L=18$. Fitting the generalized extreme-value distribution of Eq. (5) to the raw $\\lambda_{\\max}$ data yields $\\alpha = 7.77\\times 10^{-3}$, $\\beta = 9.40\\times 10^{-5}$, and shape $\\xi = 0.17 \\pm 0.011$; in the parametrization used, positive $\\xi$ corresponds to the Weibull-type class of bounded-tail extremes. The paper contrasts this with the circular orthogonal ensemble, which converges quickly to the Tracy-Widom law as measured by the distance ratio $R$ and the Kullback-Leibler divergence, while the kicked chain's $R$ stays at $0.56$ and its KL divergence only begins to decrease after $L=14$. Separately, the paper establishes that the diagonal and off-diagonal ETH ansatz for the spin observable are satisfied, with fluctuations decaying as $2^{-L/2}$, and that the averaged spin autocorrelation function decays exponentially and saturates to a late-time value that decreases with $L$.","pith_inferences":["A decisive check would be to track the fitted GEV shape parameter $\\xi$ as a function of $L$; if $\\xi$ trends toward $0$ or toward Tracy-Widom moments once $L$ is increased beyond 18, the Weibull-type conclusion would be a finite-size crossover rather than the asymptotic law.","The same analysis applied to other chaotic models (e.g., random local Hamiltonians or dual-unitary circuits) would show whether bounded-tail extreme value statistics is a generic feature of local many-body chaos or specific to this kicked model.","The paper's ETH finding suggests that entanglement-spectrum extremes probe a different layer of eigenstate structure than thermalization; a natural extension is to check whether the non-Tracy-Widom tail leaves a detectable signature in other ETH-violating observables or in out-of-time-order correlators at larger sizes.","More data or better fits could test whether the 'Wishart with unit trace' null model itself needs amendment; a modified random matrix ensemble that encodes the tensor-product Hilbert space structure might reproduce the Weibull-type tail while keeping the Marchenko-Pastur bulk."],"forward_implications":["If the Weibull-type fit is the correct asymptotic law, then the largest entanglement-spectrum eigenvalue of local chaotic Floquet systems belongs to a different universality class from the Wishart/Tracy-Widom prediction.","The deviation is specific to the kicked chain: the same statistics for the circular orthogonal ensemble converge to Tracy-Widom, so the standard RMT null model works for unstructured random unitaries but not for this structured Hamiltonian.","Because the largest Schmidt eigenvalue dominates the entanglement entropy, its non-Tracy-Widom tail is the likely origin of previously reported deviations of entanglement-entropy statistics in this model.","The ETH results imply that this non-RMT feature does not derail thermalization: local observables still thermalize in the eigenstate sense, and correlation functions show the expected ergodic decay and saturation.","Random matrix ensembles beyond the Gaussian and circular classes may be needed to model many-body chaotic eigenstates; the paper offers the Weibull-type extreme value law as a fingerprint to be reproduced by such ensembles."],"supporting_citations":[{"why":"Identifies the self-dual kicked field Ising model as a minimal model of many-body quantum chaos, the system whose eigenstates are studied.","marker":"[27]"},{"why":"Reports deviations from random-matrix entanglement statistics in kicked spin-1/2 chains, the effect this paper traces to the largest Schmidt eigenvalue.","marker":"[32]"},{"why":"Supplies the polynomial filter diagonalization method used to obtain eigenstates for L >= 13, including the L=18 data.","marker":"[39]"},{"why":"Extends polynomial filtering to Floquet unitary operators with a Walsh-Hadamard strategy, the numerical backbone for the largest sizes.","marker":"[40]"},{"why":"Provides the Tracy-Widom law and centering/scaling constants for the largest eigenvalue of Wishart matrices, the RMT prediction being tested.","marker":"[43]"},{"why":"Establishes the Fisher-Tippett-Gnedenko theorem on the three universal limiting distributions of maxima, the basis for classifying the tail.","marker":"[44]"},{"why":"Supplies the extreme-value statistics framework for correlated variables that justifies applying GEV forms to the Schmidt eigenvalues.","marker":"[46]"},{"why":"Supports the validity of extreme-value limits under weak correlations, which the paper invokes when fitting Eq. (5).","marker":"[47]"},{"why":"Defines the fluctuation measure used to test the diagonal eigenstate thermalization hypothesis.","marker":"[49]"},{"why":"Gives the Wishart/Marchenko-Pastur and random matrix background that sets up the reference distribution for the entanglement spectrum.","marker":"[5]"}],"fun_headline_variants":["Entanglement extreme in chaotic spin chain follows Weibull, not Tracy-Widom","Kicked spin chain: max entanglement eigenvalue skips Tracy-Widom","Weibull extreme replaces Tracy-Widom in chaotic spin chain","Largest entanglement eigenvalue deviates from Tracy-Widom in kicked chain","Kicked spin chain: Weibull extreme, but ETH still holds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the L=18 sample is large enough that the three-parameter generalized extreme-value fit tells us the true asymptotic tail; if the fitted shape parameter moves with system size, the Weibull-type label would not be the asymptotic law.","fun_headline_variants_meta":{"raw":{"variants":["Entanglement extreme in chaotic spin chain follows Weibull, not Tracy-Widom","Kicked spin chain: max entanglement eigenvalue skips Tracy-Widom","Weibull extreme replaces Tracy-Widom in chaotic spin chain","Largest entanglement eigenvalue deviates from Tracy-Widom in kicked chain","Kicked spin chain: Weibull extreme, but ETH still holds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000567,"raw_usage":{"total_tokens":2722,"prompt_tokens":1017,"completion_tokens":1705,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":633,"completion_tokens_details":{"reasoning_tokens":1621}},"tokens_in":633,"tokens_out":1705,"duration_ms":9482,"temperature":1.0,"reasoning_tokens":1621,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:58:54.358051+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the generalized extreme-value fit for the largest Schmidt eigenvalue at L=12, 14, 16, and 18 under the same centering and test the L=18 fit against the Tracy-Widom F1 distribution with a goodness-of-fit statistic; if xi moves systematically toward 0 as L grows, or if Tracy-Widom fits within error bars at L=18, the Weibull-type classification is a finite-size artifact rather than the asymptotic law.","supporting_citations":[{"cited_title":"Deviations from random matrix entanglement statistics for kicked quantum chaotic spin-$1/2$ chains","cited_arxiv_id":"2405.07545","evidence_quote":"Reports deviations from random-matrix entanglement statistics in kicked spin-1/2 chains, the effect this paper traces to the largest Schmidt eigenvalue."},{"cited_title":"Polynomial filter diagonalization of large Floquet unitary operators","cited_arxiv_id":"2102.05054","evidence_quote":"Extends polynomial filtering to Floquet unitary operators with a Walsh-Hadamard strategy, the numerical backbone for the largest sizes."},{"cited_title":"Random covariance matrices: Universality of local statistics of eigenvalues","cited_arxiv_id":"0912.0966","evidence_quote":"Provides the Tracy-Widom law and centering/scaling constants for the largest eigenvalue of Wishart matrices, the RMT prediction being tested."},{"cited_title":"On the distribution of the largest eigenvalueinprincipalcomponentsanalysis,","cited_arxiv_id":null,"evidence_quote":"Establishes the Fisher-Tippett-Gnedenko theorem on the three universal limiting distributions of maxima, the basis for classifying the tail."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the extreme-value statistics framework for correlated variables that justifies applying GEV forms to the Schmidt eigenvalues."},{"cited_title":"Extreme value statistics of correlated random variables: a pedagogical review","cited_arxiv_id":"1910.10667","evidence_quote":"Supports the validity of extreme-value limits under weak correlations, which the paper invokes when fitting Eq. (5)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Wishart/Marchenko-Pastur and random matrix background that sets up the reference distribution for the entanglement spectrum."}],"review_version":1}