{"id":"0df8439d-c7cb-48db-a3e0-a7dba059800c","arxiv_id":"2412.02979","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A coordinate-invariant phase-space entropy for quantum states on Riemannian manifolds is defined and computed for oscillator states in flat and AdS2 spacetimes.","lead":"This paper defines a quantum phase-space entropy that is unchanged by coordinate changes on curved spaces, and computes it for a harmonic oscillator in flat space and in anti-de Sitter space. The work is a candidate bridge between quantum information and geometry, though most of the mathematical ingredients come from earlier papers.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (20) defines the curved Wigner function by an affine convolution in an exponential chart; chart-independence and equivalence to Gneiting et al. are not proven, so the invariant entropy eq. (22) may not be well-defined on arbitrary Riemannian spaces.","rationale":"The reader identifies exactly the load-bearing assumption: eq. (20) must define a chart-independent Wigner function on an arbitrary Riemannian manifold before the entropy eq. (22) can have the claimed observer-independent meaning. My reading sharpens that concern: the formula uses affine shifts in an exponential chart, and in curved space those are not geodesic midpoint translations. The paper does not prove that both shifted points remain in the chart domain, nor that the construction is invariant under changing the chart or base point. The claimed equivalence to Gneiting et al. [24] is asserted after the fact and is not derived; the paper's own footnote [35] admits the absence of a noncommutative Radon-Nikodym analogue. These are correctness risks, not just missing exposition, because the AdS2 computation uses one global coordinate and would be reproduced identically even if W were chart-dependent. The concern is testable numerically or by direct comparison to Ref. [24], so it does not force rejection; it supports the reader's conditional verdict. I agree with the reader rather than proposing a different weakest assumption.","tokens_in":19573,"tokens_out":14995,"duration_ms":161216,"concrete_test":"Implement eq. (20) on a compact curved manifold (e.g. S^2) for a smooth state localized away from the cut locus. Evaluate W at the same physical phase-space point using two different exponential charts, and also compare with the midpoint construction of Gneiting et al. If the densities, or the integrated H_{X,P}, differ at O(1) level, eq. (22) is not observer-independent. A second control: recompute the AdS2 ground state eq. (26) after a nonlinear coordinate change u = f(x); if H_{X,P} changes, the claimed diffeomorphism invariance fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central object is eq. (20). It evaluates ψ_λ_x at x ± x'/2, where x and x' are vectors in the exponential chart at x, i.e. at points exp_x(x ± x'/2). For eq. (22) to be an observer-independent entropy, two conditions must hold: (i) for a.e. x' both points lie in the chart domain D_x, including when the chord crosses the cut locus; and (ii) the value of W is invariant under changing the chart/base point under arbitrary diffeomorphisms, not just isometries. Neither is established. Coordinate translations inside an exponential chart do not coincide with geodesic midpoint constructions in curved space: exp_x(v + w/2) and exp_x(v - w/2) are symmetric about exp_x(v) only in the flat case. On S^2 or AdS this fails for generic charts. The paper's only support is the statement that eq. (20) is 'mathematically equivalent' to Gneiting et al. [24], discovered during revision, without derivation or numerical cross-check. Footnote [35] additionally concedes that no noncommutative Radon-Nikodym analogue is supplied, so the classical measure-theoretic justification of §II does not automatically transfer. If W is chart-dependent or undefined on a non-negligible set, H_{X,P} in eq. (22) changes with the observer, and the AdS example, computed in one global coordinate x, does not demonstrate general invariance.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a diffeomorphism-invariant formulation of differential entropy for quantum states on Riemannian spaces. It defines an invariant position entropy H_X from the Riemannian density, introduces a phase-space entropy H_{X,P} based on a Wigner function on curved manifolds (Eq. (20)) and the log-absolute-value entropy functional (Eq. (22)), and claims that this entropy is invariant under symplectic transformations and reduces to Gibbs entropy in the classical limit. The framework is illustrated by computing the phase-space entropy of harmonic oscillator energy eigenstates in flat space and in AdS2, and a generalized Bialynicki-Birula-Mycielski inequality H_X+H_P ≥ D(1-log 2) is stated for arbitrary Riemannian spaces. An appendix contains the flat-space Laguerre integral and a residue computation for the AdS2 Wigner integral.","tokens_in":19823,"tokens_out":13513,"duration_ms":123214,"significance":"If the Wigner function in Eq. (20) were rigorously well-defined and chart-independent on arbitrary Riemannian manifolds, the resulting invariant phase-space entropy would be a genuinely useful coordinate-free information measure for continuous-variable quantum systems, with potential applications to quantum information in curved spacetime. The paper deserves credit for including explicit analytic computations: the flat-space entropy formula follows from an appendix calculation, the AdS2 j=1 entropy is reduced to a numerical integral quoted to 10^{-9} accuracy, and no parameters are fitted. At the same time, the advertised curved-space generalization of the BBM inequality is shown here to be a coordinate-invariant restatement of the flat-space result, and the well-definedness of the central Wigner construction is not established beyond the claim of equivalence to Gneiting et al.","major_comments":[{"comment":"The central object is not shown to be well-defined on arbitrary Riemannian spaces. The integrand evaluates the λ-wavefunction at x ± x'/2, that is, at exp_x(x ± x'/2), but the integration is over the entire exponential chart D_x; no argument establishes that x ± x'/2 lies in D_x for the relevant range of x' nor that the result is independent of the chosen chart and base point under diffeomorphisms. The assertion that Eq. (20) is 'mathematically equivalent' to Gneiting et al. [24] is given without derivation or numerical cross-check, and footnote [35] explicitly concedes that no noncommutative Radon-Nikodym analogue is supplied. Since Eq. (22) inherits this construction, the claimed observer-independence of H_{X,P} is not established; the AdS2 computation in one global coordinate chart does not demonstrate general invariance.","section":"Section IV, Eq. (20)"},{"comment":"The generalized BBM inequality is a restatement of the flat-space inequality rather than a curved-space generalization. Combining Eqs. (5) and (12) gives H_X+H_P = H_x+H_p - D log h in any chart, with the metric-dependent terms canceling identically. Therefore Eq. (29) is equivalent to the usual BBM bound applied to the λ-wavefunction and its ordinary Fourier transform; it contains no curvature-dependent term and would hold for any Riemannian metric. The paper should either present this as an invariance result or identify a genuinely geometric step beyond [25]; as written, the phrase 'extending the BBM inequality to curved backgrounds' overstates the content.","section":"Section VII, Eq. (29)"},{"comment":"The momentum operator used in the AdS2 Hamiltonian is inconsistent with the canonical momentum defined in Eq. (15). For D=1 with metric γ(x)dx^2, Eq. (15) gives P_x = -iħ(∂_x + γ'(x)/(4γ(x))) acting on position wavefunctions, whereas the text states P_x = -iħ[γ(x)]^{-1/2}∂_x. These operators differ by a first-order term that is not a pure gauge. The eigenproblem written in the text is therefore not the one defined by the paper's earlier canonical momentum, and it is unclear whether the wavefunction in Eq. (24) is an eigenstate of the Hamiltonian introduced in Section IV. If the calculation is intended to be done in the λ-wavefunction representation, this must be stated explicitly and the Hamiltonian transformed consistently.","section":"Section VI B, after Eq. (23)"}],"minor_comments":[{"comment":"The notation h and ħ are both used without an explicit statement that h = 2πħ; this makes the normalization conventions in Eqs. (7), (19), and (22) unnecessarily hard to verify.","section":"Section III and Section IV"},{"comment":"The claim that H_{X,P} is invariant under symplectic transformations of phase space needs a precise statement of the allowed class of transformations; Wigner functions are not covariantly transformed by arbitrary nonlinear canonical transformations in the standard flat-space construction.","section":"Section V, after Eq. (22)"},{"comment":"The conjectured bound in Eq. (31) is supported only by numerical evidence for the AdS2 ground state, as shown in Fig. 4; the text should state explicitly that this is numerical evidence for one family of states and not a proof.","section":"Section VII, Eq. (31)"}],"recommendation":"major_revision","confidential_remarks":"The main risk is that the paper's advertised curved-space results are partly restatements of flat-space statements: the BBM inequality is an invariant rewriting, and the Wigner construction relies on an unproven equivalence to Gneiting et al. The AdS2 entropy computation is valuable and appears carefully done, but the Hamiltonian inconsistency in Section VI B needs to be resolved before the example can be trusted. I recommend major revision rather than rejection because the central idea is defensible and the missing chart-independence proof may be within reach."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: this is a serious but uneven paper. The genuinely new material is concrete: an explicit flat-space Wigner-entropy formula for oscillator eigenstates (the appendix derivation looks right), an AdS2 ground-state entropy of log 2 − 1/2 for j = 1 via residues plus a numerical integral quoted to <10^-9, and a conjectured lower bound in eq. (31). The framework is honestly assembled from prior work the author names: Pennec's intrinsic entropy, the Gneiting–Fischer–Hornberger Wigner function (equivalence admitted), and the log|ρ| Wigner entropy of Laguna/Sagar and Cerf et al. No parameters are fitted and the citation pattern is fair.\n\nThe main soft spot is exactly the one the stress-test flags. Eq. (20) defines the curved Wigner function by affine shifts inside an exponential chart. The paper never proves that both x ± x'/2 lie in the chart domain for a.e. x', nor that the result is chart-independent under arbitrary diffeomorphisms. The text says the expression is equivalent to Gneiting et al. and leaves it at that; footnote [35] concedes the missing noncommutative Radon–Nikodym analogue. The AdS2 computation is done in one global coordinate, so it cannot demonstrate the invariance. That means the central phrase 'diffeomorphism-invariant entropy' is an assertion, not an established result.\n\nA second, smaller problem: the mixed-state entropy formula in Section V uses the chain rule on p_a ρ_a(x,p), but the Wigner function of a mixture is the convex combination Σ p_a ρ_a, not a joint distribution over ensemble label and phase space. The formula H_total = H_vN + Σ p_a H_(a) only holds for disjoint supports. This section needs reworking.\n\nThe generalized BBM inequality (29) is correct but is flat BBM applied to λ-wavefunctions; the curved content is in the definitions, not in a new bound. The conjectured inequality (31) is the more interesting item.\n\nBottom line: for readers working on continuous-variable quantum information in curved spaces or space-based QKD, this paper provides concrete reference computations and a framework worth arguing with. It deserves a serious referee. But the invariance claim needs a proof or a carefully qualified statement of its domain, and the mixture-entropy formula needs fixing. Send to review with expectation of major revision. I would not cite the invariant entropy as a definition until the well-definedness question is settled.","headline":"A serious but uneven paper: the explicit flat and AdS2 entropy computations are valuable, but the central coordinate-invariance claim is asserted rather than proven.","tokens_in":20401,"tokens_out":9489,"would_cite":false,"duration_ms":101231,"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":"This paper defines an observer-independent quantum phase-space entropy on arbitrary Riemannian spaces and derives a curved-space entropic uncertainty bound.","keywords":["quantum phase-space entropy","Wigner function","Riemannian manifolds","differential entropy","Bialynicki-Birula-Mycielski inequality","curved spacetime","quasiprobability","anti-de Sitter space"],"falsifier":"Express the same quantum state on a compact curved manifold, such as a sphere, in two overlapping exponential charts, compute $H_{X,P}$ from eq. (22) using each chart, and compare: unequal values would disprove the claimed diffeomorphism invariance. Also search any curved manifold for a state whose $\\lambda$-wavefunction has a well-defined Fourier transform and whose invariant entropies satisfy $H_X + H_P < D(1-\\log 2)$; such a state would refute the generalized Bialynicki-Birula--Mycielski inequality.","tokens_in":19308,"feed_emoji":"🌀","tokens_out":13350,"duration_ms":117827,"temperature":0.7,"pith_summary":"The paper tries to establish that quantum information can be made independent of the observer's coordinates even on curved space. It constructs a phase-space entropy $H_{X,P}$ for quantum states on any connected, geodesically complete Riemannian manifold, using a Wigner quasiprobability density adapted to the curved geometry and an analytic continuation of Shannon entropy to negative quasiprobabilities, $p\\log|p|$ in place of $p\\log p$. The entropy is claimed to be invariant under reparametrizations and symplectic transformations, to decompose as $H_X + H_{P|X}$, and to reduce to Gibbs entropy classically. The paper also derives a generalized Bialynicki-Birula--Mycielski inequality $H_X + H_P \\geq D(1-\\log 2)$ for arbitrary Riemannian spaces and illustrates the formalism on harmonic oscillator states in Minkowski and anti-de Sitter geometries. If correct, this gives continuous-variable quantum information a coordinate-free meaning tied to the geometry of physical space, with direct implications for quantum communication in curved or gravitational settings.","feed_headline":"Quantum entropy is invariant on curved space","feed_subtitle":"A Wigner-based phase-space entropy extends to Riemannian manifolds and produces a curved-space uncertainty bound.","key_machinery":"The load-bearing object is the $\\lambda$-wavefunction $\\psi^\\lambda_x(\\vec{x}) = l_P^{-D/2}[\\det g_x(\\vec{x})]^{1/4}\\psi_x(\\vec{x})$, a wavefunction normalized with respect to the local Lebesgue measure rather than the Riemannian volume. With this normalization, the curved-space Wigner function $\\rho_{x,p}(\\vec{x},\\vec{p}) = \\int [\\psi^\\lambda_x]^*(\\vec{x} - \\vec{x}'/2)\\,\\psi^\\lambda_x(\\vec{x} + \\vec{x}'/2)\\,e^{-i\\vec{p}\\cdot\\vec{x}'/\\hbar}\\,d^D\\vec{x}'$ has the same form in any exponential chart as in flat space. Entropy is then defined by replacing the probability $p$ in Shannon's formula with the absolute value $|p|$, which keeps the chain rule of conditional entropy and makes the integral finite for Wigner-negative states. The invariance is carried by the relations among the Riemannian, Lebesgue, and symplectic measures together with the exponential-chart construction, while the generalized uncertainty bound follows because the momentum wavefunction is the ordinary Fourier transform of the $\\lambda$-wavefunction.","core_discovery":"The central claim is that the entropy of a quantum state is not observer-relative once position and momentum degrees of freedom are both retained. Equation (22) defines $H_{X,P} = -h^{-D}\\int \\rho_{x,p}\\log|\\rho_{x,p}|\\,d^D x\\,d^D p$ with the curved-space Wigner function of eq. (20), and the paper argues that this object is invariant under diffeomorphisms of the Riemannian space and under symplectic transformations of phase space. Because the Wigner function is expressed through Lebesgue-normalized $\\lambda$-wavefunctions, it takes the same flat-space Fourier form in every exponential chart while the Riemannian, Lebesgue, and symplectic measures combine to make the marginals and the entropy coordinate-free. The paper verifies the construction on harmonic-oscillator eigenstates: in flat space the ground-state phase-space entropy is $1-\\log 2$ nats, and in AdS$_2$ it is $\\log 2 - 1/2$ nats for $j=1$, independent of all dimensionful parameters, with curvature generating Wigner negativity and position-momentum mutual information. It further claims the generalized bound $H_X + H_P \\geq D(1-\\log 2)$ for arbitrary Riemannian spaces and notes that the tighter inequality conjectured for Wigner-positive states in flat space can be violated in curved geometries.","pith_inferences":["Editorial inference: The same Fourier-transform argument that yields the generalized BBM bound should extend to manifolds with boundaries or conical singularities whenever a global $\\lambda$-wavefunction exists; computing $H_X + H_P$ on a cone would be a quick test of the scope.","Editorial inference: The choice $p\\log|p|$ is singled out by the chain rule, but the paper does not prove uniqueness; before the numerical values such as $0.307$ nats and $0.193$ nats are treated as absolute information contents, they should be compared with operational quantities extracted from quantum state tomography or weak measurements, since other analytic continuations would give different ","Editorial inference: If the position-momentum mutual information induced by curvature is real, it suggests an information-theoretic signature of gravity that could be probed in tabletop experiments realizing curved effective metrics, where the metric-dependent entropy difference should appear in tomographic reconstructions."],"forward_implications":["The phase-space entropy of a closed system is conserved under unitary evolution, and for a mixed state it splits as $H_{\\rm total} = H_{\\rm vN} + \\sum_a p_a H^{(a)}_{X,P}$, with the mutual information $I = H_{X,P} - \\sum_a p_a H^{(a)}_{X,P}$ quantifying how much of the ensemble information is carried by the microscopic position and momentum degrees of freedom.","Continuous-variable quantum key distribution and other quantum communication protocols can be analyzed with coordinate-independent entropic uncertainty relations, so gravitational or geometric effects on the channel can be incorporated through the generalized BBM bound.","Spatial curvature changes the trade-off between position and momentum information: in AdS$_2$ the ground state has lower phase-space entropy than in flat space because curvature induces mutual information between the two degrees of freedom, and varying the curvature radius transfers information between $H_X$ and $H_P$.","The phase-space entropies obtained for the oscillator states are independent of all dimensionful constants, including the Planck length, mass, and coupling, indicating a dimensionless, purely geometric contribution to the information content of those states.","The conjectured bound $H_{X,P} \\geq H_X + H_P + \\int \\rho_x \\log\\sqrt{\\det g}\\,d^D x$ is verified numerically for the AdS$_2$ harmonic-oscillator ground state at all values of $j$, making it a candidate curved-space replacement for the tighter flat-space inequality."],"supporting_citations":[{"why":"Supplies Shannon's entropy, whose analytic continuation to quasiprobabilities defines the phase-space entropy in eq. (22).","marker":"[1]"},{"why":"Introduces the diffeomorphism-invariant differential entropy on Riemannian manifolds that the paper lifts to phase space.","marker":"[5]"},{"why":"Provides the curved-configuration-space Wigner function to which the paper's eq. (20) is equivalent, anchoring the arbitrary-manifold construction.","marker":"[24]"},{"why":"Gives the flat-space BBM inequality whose Fourier-transform proof is extended to arbitrary Riemannian spaces.","marker":"[25]"},{"why":"Supplies the curved-space position and momentum operator representation and Hamiltonian used to define wavefunctions and lambda-wavefunctions.","marker":"[30]"},{"why":"Establishes the deformation-quantization Wigner formalism whose flat-space expression eq. (20) mirrors.","marker":"[32]"},{"why":"Give the harmonic-oscillator eigenfunctions on AdS2 used in the explicit entropy computations.","marker":"[39, 40]"},{"why":"Supplies the sharp Hausdorff-Young inequality that is the analytic ingredient in the BBM proof the paper generalizes.","marker":"[44]"},{"why":"Introduce the quantum Wigner entropy and the tighter inequality whose behavior in curved spaces the paper examines and finds can be violated.","marker":"[8, 9]"}],"fun_headline_variants":["Curved-space entropy defies observer bias","Wigner entropy goes coordinate-free in curved space","Quantum entropy made invariant on any Riemannian manifold","Curved-space uncertainty bound derived from Wigner entropy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the integral in eq. (20) is a well-defined Wigner function on any connected, geodesically complete Riemannian manifold: for every point $\\vec{x}$ and every integration variable $\\vec{x}'$, both $\\vec{x} \\pm \\vec{x}'/2$ must lie inside the same exponential chart, and the result must be independent of which chart is chosen; the paper does not prove this directly, deferring instead to equivalence with the curved-space Wigner function of [24].","fun_headline_variants_meta":{"raw":{"variants":["Curved-space entropy defies observer bias","Wigner entropy goes coordinate-free in curved space","Quantum entropy made invariant on any Riemannian manifold","Curved-space uncertainty bound derived from Wigner entropy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000127,"raw_usage":{"total_tokens":1125,"prompt_tokens":964,"completion_tokens":161,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":580,"completion_tokens_details":{"reasoning_tokens":102}},"tokens_in":580,"tokens_out":161,"duration_ms":2448,"temperature":1.0,"reasoning_tokens":102,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:54:19.150109+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Express the same quantum state on a compact curved manifold, such as a sphere, in two overlapping exponential charts, compute $H_{X,P}$ from eq. (22) using each chart, and compare: unequal values would disprove the claimed diffeomorphism invariance. Also search any curved manifold for a state whose $\\lambda$-wavefunction has a well-defined Fourier transform and whose invariant entropies satisfy $H_X + H_P < D(1-\\log 2)$; such a state would refute the generalized Bialynicki-Birula--Mycielski inequality.","supporting_citations":[{"cited_title":"Pennec, Intrinsic statistics on Riemannian mani- folds: Basic tools for geometric measurements, Journal of Mathematical Imaging and Vision 25, 127 (2006)","cited_arxiv_id":null,"evidence_quote":"Introduces the diffeomorphism-invariant differential entropy on Riemannian manifolds that the paper lifts to phase space."},{"cited_title":"Gneiting, T","cited_arxiv_id":null,"evidence_quote":"Provides the curved-configuration-space Wigner function to which the paper's eq. (20) is equivalent, anchoring the arbitrary-manifold construction."},{"cited_title":"Bia/suppress lynicki-Birula and J","cited_arxiv_id":null,"evidence_quote":"Gives the flat-space BBM inequality whose Fourier-transform proof is extended to arbitrary Riemannian spaces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the curved-space position and momentum operator representation and Hamiltonian used to define wavefunctions and lambda-wavefunctions."},{"cited_title":"Zachos, Deformation quantization: quantum mechan- ics lives and works in phase-space, International Journal of Modern Physics A 17, 297 (2002)","cited_arxiv_id":null,"evidence_quote":"Establishes the deformation-quantization Wigner formalism whose flat-space expression eq. (20) mirrors."},{"cited_title":"Beckner, Inequalities in Fourier analysis on Rn, Pro - ceedings of the National Academy of Sciences 72, 638 (1975)","cited_arxiv_id":null,"evidence_quote":"Supplies the sharp Hausdorff-Young inequality that is the analytic ingredient in the BBM proof the paper generalizes."}],"review_version":1}