{"id":"9077ce99-2b78-4b73-a4ac-cd73440c72ce","arxiv_id":"2608.11319","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A numerically exact hopping model shows that quantum Darwinism redundancy curves depend on whether one traces over particles or over lattice sites, with fermionic environments sometimes showing a more pronounced redundancy plateau than bosonic or site-averaged ones.","lead":"The authors simulate a one-dimensional lattice in which a quantum impurity broadcasts its position into a surrounding gas of indistinguishable fermions or hard-core bosons, and they define how to measure the information stored in fractions of that gas. The paper matters because it gives quantum Darwinism, a leading theory of how classicality emerges, a concrete route toward ultracold-atom experiments.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Particle-based QMI rescales, not cancels, C(N,p); the fermionic redundancy advantage in Figs. 3 and 5 may be a normalization artifact.","rationale":"The reader's weakest assumption pointed to the pRDM normalization, and my analysis agrees that this is the most load-bearing issue, but I sharpen it: the C(N,p) factors do not merely cancel; they leave a multiplicative factor C on the entropy difference. This converts a small difference in the normalized QMI (fermions ≈0.945 vs bosons ≈0.913 at p=1 for Set R-LR) into the large apparent separation in Fig. 3. The paper's site-based statistics-independence result appears sound, because tracing over sites in the occupation-number basis gives identical reduced density matrices for fermions and hard-core bosons when the Fock-space coefficients coincide; the configuration-space exact diagonalization is a genuine strength. The decoherence analysis and the parameter taxonomy are also informative. However, the central Darwinism claim for particles relies on Eq. (14), and that equation as written is not the standard QMI of the normalized p-particle states. It can be negative for simple product states, and its scale depends on C. The paper neither proves the properties it uses for this quantity nor defends the unnormalized convention as the operationally relevant one for quantum Darwinism. This warrants keeping the verdict conditional: the authors should either re-present the particle-based curves with normalized pRDMs or explicitly justify why the C-rescaled quantity is the correct QD measure. I therefore recommend no change to the reader's CONDITIONAL verdict, but the requested normalization check is the decisive next step.","tokens_in":29090,"tokens_out":27447,"duration_ms":297020,"concrete_test":"Recompute Eqs. (12)-(14) after normalizing both particle-reduced density matrices: use \\tilde D_{I,1...p} = D_{I,1...p}/C(N,p) and \\tilde D_{1...p} = D_{1...p}/C(N,p), and plot I_norm = S(ρ_I) + S(\\tilde D_{1...p}) − S(\\tilde D_{I,1...p}) for fermions and hard-core bosons for Sets R and R-LR at the same t_QMI values used in Fig. 3. If the fermion-boson gap persists in I_norm and the bosonic curve remains non-plateau-like, the unnormalized convention is not the driver of the claimed effect; if both species approach I_norm≈S(ρ_I)=1 at small fractions, or the gap shrinks below the relevant scale, the headline fermionic redundancy advantage is an artifact of the missing C(N,p) normalization. As a separate diagnostic, evaluate Eq.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central fermionic-vs-bosonic redundancy claim is carried by the particle-based QMI in Eq. (14): I_part = S(ρ_I) + S(D_{1...p}) − S(D_{I,1...p}). From Eqs. (12)-(13), D_{I,1...p} and D_{1...p} are partial traces over N−p particles of a normalized pure state, with Tr D_{I,1...p} = Tr D_{1...p} = C(N,p). Writing D = C σ with Tr σ = 1, the von Neumann entropy of the unnormalized matrix is S(D) = C S(σ) − C log2 C. The additive terms −C log2 C do cancel in S(D_{1...p}) − S(D_{I,1...p}), but the physically relevant difference of normalized entropies is multiplied by C: I_part = S(ρ_I) + C [S(σ_{1...p}) − S(σ_{I,1...p})] = C I_norm + (1−C) S(ρ_I), where I_norm uses normalized pRDMs. Thus the combinatorial factor does not simply cancel; it rescales the entropy difference. For a product state with S(ρ_I)=1 and C=8, this gives I_part = 1−8 = −7, so Eq. (14) is not guaranteed nonnegative and is not a standard mutual information. More importantly, the claimed large fermionic plateau in Figs. 3 and 5 is largely this C-fold amplification. From Table V (p=1, C=8), the fermionic I_part≈0.56 corresponds to I_norm≈0.945, while the bosonic I_part≈0.30 corresponds to I_norm≈0.913: both species are close to the ideal value S(ρ_I)=1 in the normalized measure. The paper does not prove that this rescaled quantity is the quantum-Darwinism-relevant measure or that it satisfies the QMI properties it implicitly uses (nonnegativity, subadditivity, mirror symmetry). Without such a justification, the headline fermionic redundancy advantage is not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a numerically exact one-dimensional lattice model in which a static impurity in a spatial superposition interacts with an environment of indistinguishable fermions or hard-core bosons, and studies decoherence and quantum Darwinism by exact diagonalization. The central methodological contribution is a proposal for defining 'fractions of the environment' for indistinguishable particles: particle-based p-particle reduced density matrices (pRDMs) versus site-based reduced density matrices. The paper reports that the particle-based QMI depends on particle statistics, while the site-based QMI does not, and it claims that for certain parameter sets (R and R-LR) fermionic environments develop markedly more developed redundancy plateaus than bosonic or site-averaged ones. Those claims are carried by the particle-based QMI defined in Eq. (14) and by the numerical curves in Figs. 3 and 5.","tokens_in":29564,"tokens_out":11878,"duration_ms":118478,"significance":"If the claims held as stated, the paper would fill a genuine gap in the quantum-Darwinism literature by treating indistinguishable environments in a tunable, experimentally inspired many-body model, and it would clarify an important ambiguity in defining environment fractions. The exact-diagonalization implementation, the transparent parameter tables, and the structural result that site-based QMI is independent of particle statistics are definite strengths. However, the headline fermion-boson comparison relies on a particle-based QMI whose normalization is not specified and which is not a standard quantum mutual information as written. Because the reported statistics effect is amplified by the trace normalization factor, the quantitative content of the main claim is not established by the present analysis. The paper is likely salvageable, but it requires a substantial reanalysis of the central quantity.","major_comments":[{"comment":"The particle-based QMI is not well defined as written. The p-particle reduced density matrices defined in Eq. (12) have trace C(N,p), not 1. Inserting an unnormalized operator D = C σ into the von Neumann entropy gives S(D) = C S(σ) - C log2 C. In Eq. (14) the additive -C log2 C terms cancel between S(D_{1...p}) and S(D_{I,1...p}), but the remaining entropy difference is multiplied by C, so I_part = S(ρ_I) + C [I_norm - S(ρ_I)], where I_norm is the QMI computed with trace-normalized pRDMs. For C=8 and a state with S(ρ_I)=1 and I_norm=0, Eq. (14) gives -7, so the quantity is not nonnegative and is not a standard mutual information. This is not a pedantic point: in Table V, the R-LR fermionic values S(D_{I,1})=4.00 and S(D_1)=3.56 correspond to normalized entropies 3.50 and 3.445, a difference of 0.055, whereas the unnormalized difference is 0.44. The plotted difference between fermions and bosons in Figs. 3 and 5 is therefore substantially a rescaling artifact of the missing normalization. The authors should define I_part with trace-normalized pRDMs, redo Figs. 3 and 5 (and Appendix B) with that definition, and state whether the claimed fermionic redundancy advantage survives.","section":"Section III A, Eqs. (12)-(14); Table V; Figs. 3 and 5"},{"comment":"The redundancy curves are evaluated at hand-picked times, and one of the two selection criteria is itself a plateau diagnostic. The text states that the time is chosen so that D(t)≈0 at a local minimum and 'where the QMI has its smallest slope at F=1/2' within the evaluated interval. Minimizing the slope at F=1/2 directly selects a flat, plateau-like curve, so the subsequent interpretation of those curves as evidence of quantum Darwinism is partly circular. Because the criterion is applied separately for each parameter set, the comparison among Sets NE, S, and R, and the long-range versions, may reflect different dynamical instants rather than different redundancy capacities. I ask the authors to adopt a fixed, time-independent selection rule (for example, the first time after τ_QSL with D(t) below a threshold), to show curves at several decohered times, or to average I_part over a decohered time window, and to confirm that the fermion-boson difference is robust under that protocol.","section":"Section IV B 1; Table IV; Appendix C"},{"comment":"The operational meaning of the particle-based fraction is asserted but not derived, and the claimed QMI properties are not proven. In standard quantum Darwinism, a fraction is a subset of physical constituents accessible to an observer; for indistinguishable particles, the pRDM average is a plausible mathematical generalization, but the manuscript does not connect it to a concrete measurement scenario or coarse-graining. In addition, the statements that I_part obeys the mirror symmetry I_part(p)+I_part(N_E-p)=2 and equals 1 at p=N_E/2 are presented without proof. These properties are not automatic for pRDMs of indistinguishable particles, and for the unnormalized quantity they do not follow from the standard properties of mutual information. The authors should either prove these relations for the normalized quantity or explicitly present them as part of the chosen convention; without this, the reader cannot distinguish a property of the quantum state from a property of the averaging prescription.","section":"Section III A, Eq. (14)"}],"minor_comments":[{"comment":"The definition of the projection operator appears to contain a typo: '\\hat P_{JK} = \\hat C^\\dagger_J |0\\rangle\\langle 0| \\hat C_J S_K' should presumably read '\\hat C_J \\hat C_K' or similar. Please clarify the action of this operator.","section":"Section III A, Eq. (13)"},{"comment":"In the paragraph discussing long-range sets, the text says 'Sets NE-LR (W_EE=-1, W_IE=-1, i_max=7) and R-LR (W_EE=-1, W_IE=-1, i_max=7)', but Table II gives R-LR as W_IE=-4, W_EE=0, i_max=7. This is a typo, but it is confusing in a section whose conclusions depend on the parameter values.","section":"Section IV B 1"},{"comment":"The caption says the table lists entropies 'for each of the six long-range parameter sets', but the table contains three parameter sets (NE-LR, S-LR, R-LR), each with fermionic and bosonic entries. Please reword the caption to avoid implying there are six distinct Hamiltonians.","section":"Appendix B, Table V"},{"comment":"The caption would be easier to use if the line styles or colors for the six parameter sets were explicitly identified. The text refers to individual curves, but the reader currently has to infer the legend.","section":"Fig. 3"},{"comment":"The average in Eq. (20) is written as an average over all X with a given cardinality, but X was introduced as an ordered tuple. Please clarify whether the average is over unordered subsets of sites, which appears to be the intended meaning.","section":"Section III B, Eq. (20)"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the model and the site-based analysis are sound and likely publishable after revision; the main risk is that the headline claim of a fermionic redundancy advantage is an artifact of using unnormalized pRDMs in Eq. (14). I recommend requesting a revised version in which the particle-based QMI is redefined with normalized pRDMs, the corrected curves are shown, and a fixed time-selection protocol is adopted. If the authors find that the normalized curves collapse the fermion-boson distinction, the paper's conclusions will need to be substantially rewritten."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First QD study I know that takes indistinguishable-particle environments seriously, and the particle-trace versus site-trace distinction is a genuinely useful contribution. The headline fermion-vs-boson redundancy gap, though, is largely an unnormalized-entropy artifact, and the paper never notices.\n\nThe good parts first. The model is simple, experimentally motivated (cold atoms, static impurity, hard-core bosons vs fermions), and exactly diagonalized; the recovery of equilibration, revivals, and scrambling across parameter sets is clean. The site-trace statistics-independence is proven properly—the fermionic and bosonic site RDMs are unitarily equivalent with identical spectra—so that result stands. The particle-trace mirror-symmetry identities follow from pure-state Schmidt structure, though stated too casually. The engagement with the averaged-QMI literature (Chisholm et al.) is honest, and the citation pattern looks fair and current.\n\nThe soft spot is load-bearing. Eq. 14 plugs unnormalized pRDMs with trace C(N,p) into the standard QMI formula. The combinatorial factor does not cancel; it rescales: I_part = C·I_norm + (1−C)·S(ρ_I). Two consequences. First, the quantity is not a mutual information: on their own decohered pure states, if the impurity is maximally mixed but a single particle carries no position information, I_part(1) = 1−8 = −7. Nonnegativity is never checked. Second, the dramatic fermion advantage in Fig. 3 is mostly this amplification. From their own Table V at p=1, the fermionic 0.56 vs bosonic 0.30 corresponds to normalized values 0.945 vs 0.913—a real but small gap. Worse, the 'scrambled' Set S, read as non-redundant (I_part ≈ 0.11), has I_norm ≈ 0.89, near-ideal. The convention changes the qualitative reading of the curves, not just their height, and the paper never defends the unnormalized choice as the QD-relevant measure.\n\nMinor issues: redundancy curves sit at single hand-picked times (transparent, some robustness in App. C); the site-averaged curve rests on three points; no code or data deposited.\n\nThis deserves a serious referee. The framework is new, the site-trace result is solid, and the soft spot is addressable: normalize the pRDMs or defend the unnormalized convention operationally, then replot. The fermion advantage will likely survive in sign, not at the advertised magnitude. For people in QD, objectivity, or cold-atom simulators, a useful paper to cite—with care.","headline":"First QD study to take indistinguishable-particle environments seriously, with a solid site-trace result—but the headline fermion advantage largely comes from an unnormalized entropy convention the paper never justifies.","tokens_in":30051,"tokens_out":26542,"would_cite":true,"duration_ms":236105,"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":"In an environment of indistinguishable particles, the quantum mutual information depends on how a fraction is defined, and fermions can reach a redundancy plateau more readily than bosons.","keywords":["quantum Darwinism","decoherence","indistinguishable particles","quantum mutual information","redundancy plateau","fermions","hard-core bosons","ultracold atoms"],"falsifier":"Numerically evaluate the entropies $S(\\hat{D}_{I,1...p})$ and $S(\\hat{D}_{1...p})$ from explicitly normalised p-particle reduced density matrices (dividing by $\\binom{N_E}{p}$) and recompute the particle-based QMI curves; if the reported mirror symmetry $I_{\\mathrm{part}}(F)+I_{\\mathrm{part}}(1-F)=2$ breaks or the plateau heights shift, the fermionic redundancy advantage is an artefact of the unnormalised definition. A second check is to recompute the R and R-LR curves at generic post-decoherence times rather than the hand-selected times in Table IV and see whether the fermion-boson difference persists.","tokens_in":28916,"feed_emoji":"⚛️","tokens_out":9742,"duration_ms":82890,"temperature":0.7,"pith_summary":"The paper introduces a numerically exact one-dimensional lattice model in which an impurity, initially in a superposition of two end positions, is measured by an environment of indistinguishable hopping particles. It establishes how to define a fraction of such an environment for quantum-Darwinism purposes: trace out a fixed number of particles to obtain p-particle reduced density matrices, or trace out a set of sites and average over all site subsets of a given size. The paper's central claim is that this choice controls the results: the particle-trace quantum mutual information depends on whether the particles are fermions or hard-core bosons, whereas the site-trace mutual information does not. For parameter sets with strong impurity-environment attraction and no environment-environment interaction, fermionic environments develop a clearly more developed redundancy plateau than hard-core bosonic or site-averaged environments. This matters because it gives an experimentally accessible route, such as ultracold atoms in optical lattices, to test how indistinguishability shapes decoherence and the quantum-to-classical transition.","feed_headline":"Fermion environments can beat bosons at redundant measurement storage","feed_subtitle":"Particle-trace vs site-trace mutual information split the quantum-Darwinism curves; site averaging hides the plateau.","key_machinery":"The central object is the quantum mutual information $I[I:F]$ computed with two inequivalent partial traces. The particle-based trace yields p-particle reduced density matrices $\\hat{D}_{I,1...p}$ (including the impurity) and $\\hat{D}_{1...p}$ (after tracing out the impurity), with $I_{\\mathrm{part}}[I:\\hat{D}_{1...p}] = S(\\hat{\\rho}_I)+S(\\hat{D}_{1...p})-S(\\hat{D}_{I,1...p})$, and the fraction is $F=p/N_E$. The site-based trace yields Fock-space reduced density matrices $\\hat{\\rho}_{I,X}$ and $\\hat{\\rho}_X$ for a chosen subset $X$ of sites, averaged over all subsets of cardinality $|X|$, with fraction $F=|X|/M_S$. The argument turns on the spectra of these reduced states: for site traces the fermionic and bosonic blocks are unitarily equivalent, so $I_{\\mathrm{site}}$ is statistics-independent; for particle traces the creation- and annihilation-operator algebra makes the spectra different. The named mechanism that explains the fermionic advantage is the Fermi edge in the spectrum of $\\hat{D}_{I,1}$: the left- and right-conditioned natural orbitals become saturated, flattening the eigenvalue distribution only slightly compared with $\\hat{D}_1$, which increases the particle-based mutual information.","core_discovery":"On the paper's own terms, the discovery is that indistinguishability is not a side detail in quantum Darwinism but changes what the central observable means. For particle-based partial traces, the reduced density matrices $\\hat{D}_{I,1...p}$ and $\\hat{D}_{1...p}$ inherit the exchange symmetry of the environment, so fermionic and bosonic mutual information curves $I^{\\mathrm{F}}_{\\mathrm{part}}$ and $I^{\\mathrm{B}}_{\\mathrm{part}}$ differ. For site-based partial traces, the fixed-particle-number blocks of the reduced state are equal for fermions and bosons up to a global sign that is removed by unitary equivalence of their spectra, so $I_{\\mathrm{site}}$ is identical for both species. In the parameter sets called R and R-LR (strong impurity-environment attraction, no environment-environment interaction), the fermionic particle-trace QMI approaches the ideal quantum-Darwinism plateau, while the hard-core bosonic curve and the average over all site subsets stay closer to linear. The paper attributes this to a Fermi edge in the impurity-conditioned one-particle reduced density matrix: fermionic anti-bunching saturates the natural orbitals of the left and right branches, and the resulting entropy difference $S(\\hat{D}_{I,1})-S(\\hat{D}_1)$ is smaller for fermions than for bosons.","pith_inferences":["Beyond the paper, the same particle-versus-site ambiguity should affect other objectivity quantifiers, such as spectrum broadcast structures or strong quantum-Darwinism redundancy, whenever the environment is indistinguishable; any definition that labels particles will need the same normalisation and operational justification.","The Fermi-edge mechanism suggests a finite-size prediction the paper does not make: as $N_E$ grows, the fermionic plateau should remain limited by natural-orbital saturation, so the fermion-boson gap should shrink once the hard-core blocking effect dominates over statistics.","A direct testable extension is to replace the hand-selected QMI times with an ensemble average over many post-decoherence times; if the fermionic plateau is a robust feature, it should survive that average, and if not, the advantage is time-selection dependent.","The site-averaging result connects to known fraction-averaging ambiguities for distinguishable environments; the paper's construction effectively provides the indistinguishable-particle version of that averaging, which could be used to define a species-independent measure of objectivity."],"forward_implications":["For indistinguishable-particle environments, a single QMI-versus-fraction curve is not meaningful without specifying whether fractions are particles or sites and which statistics the particles obey; the two definitions can disagree qualitatively.","Fermionic environments with strong impurity coupling and no environment-environment interaction can redundantly encode the impurity pointer position even though hard-core exclusion prevents particles from fully gathering near the impurity; the plateau is close to ideal for the R and R-LR parameter sets.","Hard-core bosonic environments store pointer information less redundantly than fermionic ones in the same parameter regimes, because the impurity disturbs bosonic bunching in natural-orbital space and flattens the eigenvalue distribution.","Site-based averaging over all subsets washes out the information stored near the impurities, so whether a redundancy plateau appears depends on which sites an observer can access; corner subsets show plateaus while central subsets show the opposite.","The particle-statistics difference is in principle observable in ultracold-atom simulators, where fermionic and bosonic species with the same lattice parameters can be compared directly."],"supporting_citations":[{"why":"Defines quantum Darwinism and the redundancy criterion for classical information storage that the paper's QMI plateaus are testing.","marker":"[38]"},{"why":"Establishes the quantum mutual information as the quantitative measure of how much pointer information a fraction of the environment carries.","marker":"[42]"},{"why":"Shows that environment-environment interactions make redundant information storage decay, the baseline against which the paper's interacting-environment redundancy results are compared.","marker":"[43]"},{"why":"Characterises two-body Hamiltonians that produce pointer states and singly-branching structure; the paper relaxes these conditions and assesses partial redundancy.","marker":"[49]"},{"why":"Classifies two-body Hamiltonians for quantum Darwinism and underpins the paper's pointer-basis guarantee and the expected harm of intra-environment interactions.","marker":"[51]"},{"why":"Shows that averaging over environment fractions matters for objectivity quantifiers; the paper's site-averaging is the indistinguishable-particle analogue.","marker":"[52]"},{"why":"Provides Slater-determinant versus permanent representation of fermionic and bosonic many-body states, which is what makes particle-traced reduced densities statistics-dependent.","marker":"[76]"},{"why":"Defines p-particle reduced density matrices used for the particle-trace mutual information.","marker":"[82]"}],"fun_headline_variants":["Fermion environments boost redundancy in quantum Darwinism","Indistinguishable particles change quantum measurement information spread","Fermionic anti-bunching sharpens quantum Darwinism plateau","Particle statistics alter redundancy in broadcast quantum info","Fermion vs boson: redundancy in quantum measurement models"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The particle-based redundancy curves are well defined only if the unnormalised p-particle reduced density matrices may be inserted into the standard mutual-information formula with the binomial normalisation factors cancelling, and only if averaging over particle subsets is accepted as the operational meaning of a Darwinism fraction; the paper uses both without proving them.","fun_headline_variants_meta":{"raw":{"variants":["Fermion environments boost redundancy in quantum Darwinism","Indistinguishable particles change quantum measurement information spread","Fermionic anti-bunching sharpens quantum Darwinism plateau","Particle statistics alter redundancy in broadcast quantum info","Fermion vs boson: redundancy in quantum measurement models"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000577,"raw_usage":{"total_tokens":2792,"prompt_tokens":1085,"completion_tokens":1707,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":701,"completion_tokens_details":{"reasoning_tokens":1628}},"tokens_in":701,"tokens_out":1707,"duration_ms":10875,"temperature":1.0,"reasoning_tokens":1628,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:15:33.717358+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically evaluate the entropies $S(\\hat{D}_{I,1...p})$ and $S(\\hat{D}_{1...p})$ from explicitly normalised p-particle reduced density matrices (dividing by $\\binom{N_E}{p}$) and recompute the particle-based QMI curves; if the reported mirror symmetry $I_{\\mathrm{part}}(F)+I_{\\mathrm{part}}(1-F)=2$ breaks or the plateau heights shift, the fermionic redundancy advantage is an artefact of the unnormalised definition. A second check is to recompute the R and R-LR curves at generic post-decoherence times rather than the hand-selected times in Table IV and see whether the fermion-boson difference persists.","supporting_citations":[{"cited_title":"Horodecki, J","cited_arxiv_id":null,"evidence_quote":"Establishes the quantum mutual information as the quantitative measure of how much pointer information a fraction of the environment carries."},{"cited_title":"Korbicz, Roads to objectivity: quantum darwin- ism, spectrum broadcast structures, and strong quantum darwinism–a review, Quantum5, 571 (2021)","cited_arxiv_id":null,"evidence_quote":"Shows that environment-environment interactions make redundant information storage decay, the baseline against which the paper's interacting-environment redundancy results are compared."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Characterises two-body Hamiltonians that produce pointer states and singly-branching structure; the paper relaxes these conditions and assesses partial redundancy."},{"cited_title":"Touil, B","cited_arxiv_id":null,"evidence_quote":"Classifies two-body Hamiltonians for quantum Darwinism and underpins the paper's pointer-basis guarantee and the expected harm of intra-environment interactions."},{"cited_title":"Duruisseau, A","cited_arxiv_id":null,"evidence_quote":"Shows that averaging over environment fractions matters for objectivity quantifiers; the paper's site-averaging is the indistinguishable-particle analogue."},{"cited_title":"Esslinger, Fermi-hubbard physics with atoms in an optical lattice, Annu","cited_arxiv_id":null,"evidence_quote":"Provides Slater-determinant versus permanent representation of fermionic and bosonic many-body states, which is what makes particle-traced reduced densities statistics-dependent."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines p-particle reduced density matrices used for the particle-trace mutual information."}],"review_version":1}