{"id":"5370ba08-8e1a-4075-8dbb-3b2df70613a9","arxiv_id":"1908.07083","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"In a small fermionic lattice, observational entropy matches Boltzmann-like behavior at its extreme values, while entanglement entropy can fall to near zero by emptying a subsystem.","lead":"This paper compares two ways of defining entropy for isolated quantum systems, entanglement entropy and observational entropy. It finds that observational entropy tracks the classical Boltzmann entropy even in extreme states, while entanglement entropy does not.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main risk: Eq. (9) inherits the Pmax <= pi^2/16 bound from Ref. [47], but physical-lattice transfer is tested only up to L=30 with residual fluctuations; larger Pmax in the thermodynamic limit would erase the lower bound on SxE(min).","rationale":"Reader's weakest_assumption is the same as mine: the load-bearing link is the transfer of the Ref. [47] localization bound to physical eigenstates. I agree. The manuscript itself flags the risk in Sec. VI when it attributes residual Pmax fluctuations to non-random eigenvectors; that admission makes the extrapolation the weak spot rather than a hidden one. I considered whether the simplex-search procedure used for extrema is equally fragile, but Eq. (9) protects the main 'not near zero' conclusion even if a search missed the true minimum; hence the Pmax transfer is more central. The proposed L=40 run is feasible because only the relevant energy window is needed, and N/M^2 becomes more favorable (9880 vs 100), making it a direct test of whether the bound holds outside the random-vector and small-size regime. No change to the CONDITIONAL verdict is needed: the qualitative conclusion is plausible and supported, but not fully established.","tokens_in":19097,"tokens_out":9199,"duration_ms":93886,"concrete_test":"Run the same Pmax maximization for the interacting fermionic lattice at Np=3 with L=40 (Hilbert-space dimension 9880, M still 10), using a Krylov/Lanczos solver restricted to the beta=0.01 energy window, with 100 complex RPTSs, and compare the mean and spread of Pmax to the pi^2/16 prediction and to the same maximization over Gaussian random vectors of matched N and M. If the lattice Pmax is not at or below pi^2/16 within one standard deviation, or if its finite-size trend does not decrease toward the constant as L grows from 30 to 40, the transfer of the Ref. [47] bound to physical eigenstates, and with it Eq. (9), is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim is Eq. (9): SxE(loc) >= (1 - Pmax) Sth, so SxE(min) cannot fall below roughly half of Sth only if Pmax <= 1/2 (real RPTS) or <= pi^2/16 (complex RPTS). Those constants are proved in Ref. [47] for random energy eigenvectors and a weakly interacting gas, not for the interacting fermionic lattice used here. Section VI tests the transfer with Np=3 and L up to 30, where N/M^2 is only marginally large (e.g., L=30 gives N=C(30,3)=4060 and M=C(5,3)=10), and the figure shows systematic finite-size effects; the paper itself says 'the presence of some fluctuations is expected since our model is a real system with non-random energy eigenvectors.' The extrapolation to the thermodynamic limit is therefore carrying the argument. If the maximum localization probability of physical eigenstates is larger than pi^2/16, or worse, tends to 1 as L grows, the bound in Eq. (9) weakens or disappears, and the central contrast between SxE and Sent (a nonzero lower bound near half the maximum versus a minimum approaching zero) is no longer established. All of Sections VII and VIII rely on this link.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper compares two entropy measures for isolated quantum systems out of equilibrium: bipartite entanglement entropy and observational entropy with position-and-energy coarse graining. Using exact diagonalization of a one-dimensional lattice of interacting spinless fermions (N_p = 2 or 3, L up to 30, starting from random pure thermal states), the authors characterize the typical, maximal, and minimal values reached under unitary evolution. They report that observational entropy behaves qualitatively like classical Boltzmann entropy even at its extremes: the maximum is close to the uniform thermal state, and the minimum is bounded below at roughly half the thermodynamic entropy, achieved by localizing particles as much as possible. In contrast, entanglement entropy is minimized by emptying the subsystem and maximized by splitting particles between the two partitions, and neither extreme is typical. The quantitative anchor is Eq. (9), S_xE(loc) ≥ (1 - P_max) S_th, where P_max ≤ 1/2 (real pure thermal states) or π²/16 (complex pure thermal states) comes from the authors' earlier work [47].","tokens_in":19431,"tokens_out":6635,"duration_ms":69659,"significance":"If established, the paper's central distinction is significant: observational entropy would be a Boltzmann-like entropy that remains physically meaningful far from equilibrium and at extreme fluctuations, whereas bipartite entanglement entropy would not. This matters for discussions of thermalization, entropy production, and arrow-of-time questions in closed systems. The manuscript is transparent about its numerical setup, uses exact diagonalization with explicitly stated parameters, reports error bars from six initial states, and connects its results to analytic bounds. It does not, however, ship code or data, and its quantitative claims rely on a transfer of results from a companion paper [47] that is tested here only on very small systems.","major_comments":[{"comment":"The lower bound S_xE(loc) ≥ (1 - P_max) S_th is load-bearing for the paper's central claim that observational entropy cannot fall below about half its maximum, but the constants 1/2 and π²/16 are proved in Ref. [47] for random energy eigenvectors and a weakly interacting gas, not for the interacting fermionic lattice used here. The numerical transfer test in Fig. 8 uses N_p = 3 and L ≤ 30, where N/M² ≈ 40.6 at L = 30 (N = C(30,3) = 4060, M = C(5,3) = 10); this is a modest ratio, and the paper itself notes that 'the presence of some fluctuations is expected since our model is a real system with non-random energy eigenvectors.' Since a larger P_max in the thermodynamic limit would weaken or erase the lower bound, please add a finite-size scaling analysis of P_max as a function of N/M² at fixed β and Δx, and investigate the sensitivity of the limiting P_max to the interaction strengths V, V′ and to N_p. Without this, the claim that S_xE(min) cannot fall below roughly half of S_th is an extrapolation rather than an established result.","section":"Sec. VI, Eqs. (8)-(9)"},{"comment":"The reported global minima and maxima of Sent and SxE are obtained with a local simplex search over phases φ_E = Et, with no demonstration that the search is globally convergent. For nonconvex entropy landscapes this is a real risk: the values labeled 'Sent(min)', 'Sent(max)', and 'SxE(min)' could be local extrema, and the quantitative statements such as 'Sent(min) ≈ 0.05' or 'SxE(min) ≈ 63% of the maximum' could be artifacts of the starting point. Because the paper's central claims concern extreme values, please establish robustness of the extrema, for example by repeated optimization from many random phase initializations, or by an exhaustive phase grid for small L where the Hilbert-space dimension is small enough, and report the spread of the resulting extrema.","section":"Sec. III (simplex search algorithm)"},{"comment":"Equation (8) is stated without derivation and is attributed to Ref. [47], which shares three authors with this paper. Since Eq. (9) is the quantitative anchor for the main conclusion, the manuscript should either reproduce the derivation in an appendix or state precisely which conditions from Ref. [47] guarantee the formula, including the large-L assumption and the treatment of finite-size corrections. Note also that Eq. (8) is then plotted in Fig. 9 for system sizes as small as L = 8, even though it is introduced as a large-L expression; the expected finite-size error at these small L should be quantified or the plot restricted to the regime of validity.","section":"Sec. VII, Eq. (8)"}],"minor_comments":[{"comment":"The word 'compliment' appears where 'complement' is meant (e.g., 'between a subsystem and its compliment').","section":"Sec. I and Sec. III"},{"comment":"The condition on the Hilbert-space dimensions appears to be typeset as 'N ≥ M²'; it should be 'N ≫ M²' to match the condition required by the argument in Ref. [47].","section":"Sec. VI"},{"comment":"The paper takes t = t′ = 1.9 and V = V′ = 0.5 for all simulations, but does not discuss whether this parameter set is in the non-integrable regime or how the qualitative conclusions depend on the choice of couplings; a brief justification or a test with different couplings would make the numerics more convincing.","section":"Sec. II"},{"comment":"The description of the histogram construction says the system is evolved 'for a long time' and sampled 'at each small fixed time step,' but the actual time step and total duration are not stated; these details matter for assessing whether the sampled distribution is representative of the long-time behavior.","section":"Sec. III"},{"comment":"The presentation of macrostate sizes and the percentages in Fig. 10 is hard to follow; please clarify the definitions of Ω and the percentages in the caption or in the main text so that the claimed correspondence between macrostate size and observational entropy can be checked directly.","section":"Fig. 10 and associated text"}],"recommendation":"major_revision","confidential_remarks":"The paper leans heavily on Ref. [47], which is from the same group, and the present manuscript does not derive Eq. (8) or the P_max bounds; the editor may wish to verify that Ref. [47] is in press or accepted and that the present numerical confirmation is sufficiently independent. For a numerical paper, the absence of a code/data repository is a weakness, though not fatal if the parameters are as fully specified as they appear to be. The main concern is that the thermodynamic-limit claim in the central lower bound is not yet supported by the finite-size data; with additional scaling analysis and a more robust optimization study, the work could be made convincing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this paper is a careful numerical comparison of two entropy definitions in a small fermionic lattice, and it makes a real point — observational entropy tracks Boltzmann entropy even at its extreme values, while bipartite entanglement entropy doesn't. The new content is the first detailed mapping of extreme and typical entropy values versus system size and temperature for this model, and the clean physical picture that emerges: SxE is minimized by localizing particles, Sent by emptying a subsystem.\n\nThe paper does several things well. It uses random pure thermal states, computes histograms of fluctuations, and finds extrema with a simplex search, with error bars over 6 initial states. The figures are convincing that SxE(min) stays around half of Sth while Sent(min) drops to near zero as L grows. The contrast with Sent(max) being L-independent is explained well via an upper bound from Ref. [59]. The paper is also honest about numerical limitations.\n\nThe main soft spot is exactly what the stress-test flags: Eq. (9) inherits the Pmax ≤ pi^2/16 bound from Ref. [47] (three shared authors), and the transfer to this interacting lattice is only tested numerically for Np=3 and L up to 30, where the ratio N/M^2 is not huge. The paper itself notes remaining fluctuations. So the thermodynamic-limit claim that SxE(min) >= (1-pi^2/16) Sth is an extrapolation. If physical eigenstates allowed larger localization, the lower bound would erode. That doesn't break the qualitative picture for the sizes studied, but it does mean the central quantitative anchor is not fully established. A second, minor issue: the simplex search is local, not global, so the 'extreme' values may be local extrema; for small systems this could be checked by enumeration. Also, no code or data is provided, which is a reproducibility gap for a numerical paper.\n\nOverall, this is a solid, readable paper that deserves a serious referee. I'd send it to peer review, with requests for the numerical data/code, a more careful discussion of the Pmax transfer and its finite-size scaling, and a check of the extremum search on small cases. It's a useful contribution for anyone working on entropy in isolated systems, quantum thermodynamics, or foundations of statistical mechanics.","headline":"Solid numerical case that observational entropy, unlike entanglement entropy, behaves Boltzmann-like at its extremes, though the quantitative lower bound leans on a same-group analytic result that is only extrapolated to the thermodynamic limit.","tokens_in":19938,"tokens_out":3096,"would_cite":true,"duration_ms":31897,"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":"Observational entropy behaves like classical Boltzmann entropy even at its extreme values, while bipartite entanglement entropy does not.","keywords":["observational entropy","entanglement entropy","random pure thermal state","quantum thermalization","entropy fluctuations","localization probability","isolated quantum system","fermionic lattice"],"falsifier":"Directly maximize $P_{\\mathrm{max}}$ over unitary evolution for many complex random pure thermal states in the same fermionic lattice with $N_p=3$, $L=30$ or larger, and a localized region satisfying the three stated conditions; if the mean maximum localization probability clearly exceeds $\\pi^2/16$, the localization bound that produces the floor in Eq. (9) is false. Alternatively, prepare a cold-atom lattice in a thermal pure state, reconstruct $S_{xE}$ by coarse-grained counting, and look for a measured value below $(1-P_{\\mathrm{max}})S_{\\mathrm{th}}$.","tokens_in":18917,"feed_emoji":"⚛️","tokens_out":10039,"duration_ms":99154,"temperature":0.7,"pith_summary":"An isolated quantum system that starts in a random pure thermal state spends its long unitary evolution near many different states, and this paper asks how two entropy definitions behave at the extremes of that evolution. The central claim is that observational entropy $S_{xE}$—the entropy an observer assigns using only coarse-grained position and energy measurements—behaves like classical Boltzmann entropy even in its extreme values: its maximum is close to thermal equilibrium with particles spread uniformly, and its minimum, although achieved by packing particles into one spatial region, cannot fall below roughly half of that maximum. Bipartite entanglement entropy, in contrast, can drop almost to zero by emptying the observed subsystem into the rest of the lattice, and reaches its maximum for a non-typical, non-uniform configuration. This distinction matters because it determines which entropy can meaningfully describe isolated, far-from-equilibrium, long-lived quantum systems.","feed_headline":"Observational entropy tracks Boltzmann even at its extremes","feed_subtitle":"For isolated quantum systems, its minimum stays near half of maximum; entanglement entropy can dive to zero.","key_machinery":"The load-bearing object is the maximum localization probability $P_{\\mathrm{max}}$: the largest probability, reachable by evolving a random pure thermal state unitarily, that all particles are found in a chosen spatial region of size $\\Delta x$. A prior analytical result [47] bounds $P_{\\mathrm{max}}\\le 1/2$ for real initial states and $P_{\\mathrm{max}}\\le \\pi^2/16$ for complex ones, provided the localized subspace is much smaller than the full Hilbert space, the region is larger than the thermal wavelength, and the interactions are local; the paper checks these conditions numerically on its fermionic lattice. Through the identity $S_{xE}(\\mathrm{loc}) \\ge (1-P_{\\mathrm{max}})S_{\\mathrm{th}}$, this bound turns the physical impossibility of concentrating all particles into a small region into a universal floor on observational entropy. The contrasting object for entanglement entropy is the upper bound in Eq. (5), derived in [59], which depends only on particle number and partition sizes and explains why the maximal entanglement entropy stays constant as the bath grows.","core_discovery":"Using exact diagonalization of a one-dimensional fermionic lattice with $N_p=2$ or $3$ particles, the paper shows that over infinite-time unitary evolution starting from a random pure thermal state, the maximum and minimum of $S_{xE}$ are both close to its typical value and to the thermodynamic entropy $S_{\\mathrm{th}}$, while the minimum of the bipartite entanglement entropy $S_{\\mathrm{ent}}$ approaches zero and its maximum approaches a constant, $\\ln 6$ for the parameters studied, that is independent of the bath size and much larger than the average. The quantitative anchor is the maximum localization probability $P_{\\mathrm{max}}$: for initial real random pure thermal states $P_{\\mathrm{max}}\\le 1/2$, and for complex ones $P_{\\mathrm{max}}\\le \\pi^2/16$, so a localized state obeys $S_{xE}(\\mathrm{loc})\\ge (1-P_{\\mathrm{max}})S_{\\mathrm{th}}$ and the minimum observational entropy stays at a fixed fraction of the thermodynamic entropy. The paper verifies the localization bound numerically for its interacting lattice, identifies the localized state as the one minimizing $S_{xE}$ (but not $S_{\\mathrm{ent}}$), and concludes that observational entropy tracks Boltzmann entropy through the correspondence between macrostate size $\\Omega$ and entropy, whereas entanglement entropy does not.","pith_inferences":["If the localization bound is the true origin of the $S_{xE}$ floor, then the same floor should appear for other spatial coarse-grainings finer than the thermal wavelength; varying $\\Delta x$ and comparing measured minima with Eq. (9) would test this directly.","The dip in $S_{xE}(\\mathrm{min})$ near $\\beta\\approx 0.5$ suggests a temperature-controlled crossover in the ability to localize particles at the coarse-graining scale, which might be observable in cold-atom experiments as a change in the rarity of low-entropy configurations.","A closely related question not settled here is whether the sum of local von Neumann entropies, a multipartite entanglement measure, also obeys a $(1-P_{\\mathrm{max}})$-type floor; this is a direct extension of the paper's closing observation.","For cosmological questions such as the arrow of time, the relevant entropy should be one that is bounded below even under extreme fluctuations and remains defined for a closed system; observational entropy is a concrete candidate if its localization floor survives in a gravitational state space."],"forward_implications":["If $S_{xE}$ is Boltzmann-like, then a single coarse-grained measurement protocol gives a practical measure of how far an isolated system is from thermal equilibrium, including during rare large fluctuations.","Starting from a random pure thermal state, observational entropy cannot fluctuate below about half of the thermodynamic entropy, so spontaneous concentration of particles into a small region is limited in entropy cost; entanglement entropy has no such floor and can vanish.","The maximal entanglement entropy is not a symptom of a large macrostate: it is achieved by non-typical configurations with balanced particle numbers in the two partitions, and it stays constant as the bath size grows.","The gap between maximal and typical entanglement entropy grows with system size, so any claim that uses extreme entanglement values as a thermodynamic entropy proxy becomes worse for larger systems."],"supporting_citations":[{"why":"Supplies the analytical bound $P_{\\mathrm{max}}\\le 1/2$ for real states and $\\le \\pi^2/16$ for complex states, together with Eqs. (8)–(9), the quantitative anchor for the $S_{xE}$ minimum.","marker":"[47]"},{"why":"Defines observational entropy for multiple coarse-grainings and establishes its approach to thermodynamic entropy in thermalizing systems.","marker":"[17]"},{"why":"Derives the upper bound on maximal entanglement entropy used to explain why $S_{\\mathrm{ent}}(\\mathrm{max})$ is constant independent of bath size.","marker":"[59]"},{"why":"Introduces the thermal pure quantum (random pure thermal) states used as initial conditions in all numerical work.","marker":"[55]"},{"why":"Provides the simplex search algorithm used to find the extreme values of both entropies over long-time evolution.","marker":"[58]"},{"why":"Generalizes observational entropy to multiple coarse-grainings and provides the construction in which $S_{xE}$ is defined.","marker":"[16]"},{"why":"Supplies canonical typicality, the basis for identifying average entanglement entropy with the thermodynamic entropy of the subsystem at high temperature.","marker":"[44]"}],"fun_headline_variants":["Observational entropy tracks Boltzmann even at extremes","Entanglement entropy's minimum can drop to zero, observational cannot","For isolated quantum systems, observational entropy follows Boltzmann","One entropy mimics Boltzmann, the other doesn't"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that the localization bounds on $P_{\\mathrm{max}}$, proved for random-vector and weakly-interacting-gas eigenstates, transfer to the physical non-random eigenstates of the interacting fermionic lattice and to the thermodynamic limit, with the constants $1/2$ and $\\pi^2/16$ approached quickly enough.","fun_headline_variants_meta":{"raw":{"variants":["Observational entropy tracks Boltzmann even at extremes","Entanglement entropy's minimum can drop to zero, observational cannot","For isolated quantum systems, observational entropy follows Boltzmann","One entropy mimics Boltzmann, the other doesn't"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000279,"raw_usage":{"total_tokens":1690,"prompt_tokens":1009,"completion_tokens":681,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":625,"completion_tokens_details":{"reasoning_tokens":620}},"tokens_in":625,"tokens_out":681,"duration_ms":7659,"temperature":1.0,"reasoning_tokens":620,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:26:57.596534+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly maximize $P_{\\mathrm{max}}$ over unitary evolution for many complex random pure thermal states in the same fermionic lattice with $N_p=3$, $L=30$ or larger, and a localized region satisfying the three stated conditions; if the mean maximum localization probability clearly exceeds $\\pi^2/16$, the localization bound that produces the floor in Eq. (9) is false. Alternatively, prepare a cold-atom lattice in a thermal pure state, reconstruct $S_{xE}$ by coarse-grained counting, and look for a measured value below $(1-P_{\\mathrm{max}})S_{\\mathrm{th}}$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the analytical bound $P_{\\mathrm{max}}\\le 1/2$ for real states and $\\le \\pi^2/16$ for complex states, together with Eqs. (8)–(9), the quantitative anchor for the $S_{xE}$ minimum."},{"cited_title":"ˇSafr´ anek, J","cited_arxiv_id":null,"evidence_quote":"Defines observational entropy for multiple coarse-grainings and establishes its approach to thermodynamic entropy in thermalizing systems."},{"cited_title":"Faiez and D","cited_arxiv_id":null,"evidence_quote":"Derives the upper bound on maximal entanglement entropy used to explain why $S_{\\mathrm{ent}}(\\mathrm{max})$ is constant independent of bath size."},{"cited_title":"Sugiura and A","cited_arxiv_id":null,"evidence_quote":"Introduces the thermal pure quantum (random pure thermal) states used as initial conditions in all numerical work."},{"cited_title":"ˇSafr´ anek, J","cited_arxiv_id":null,"evidence_quote":"Generalizes observational entropy to multiple coarse-grainings and provides the construction in which $S_{xE}$ is defined."}],"review_version":1}