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REVIEW 3 major objections 3 minor 58 references

Quantum information in Riemannian spaces

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper defines an observer-independent quantum phase-space entropy on arbitrary Riemannian spaces and derives a curved-space entropic uncertainty bound.

desk verdict 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. read the letter →

arxiv 2412.02979 v4 pith:YXDK5AWL submitted 2024-12-04 quant-ph hep-th

Pablo G. Camara This is my paper · ORCID
classification quant-phhep-th
keywords quantumphase-spaceentropyWignerfunctionRiemannianmanifoldsdifferentialBialynicki-Birula-Mycielskiinequalitycurvedspacetimequasiprobabilityanti-deSitterspace
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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].

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

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.

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 (3)
  1. [Section IV, Eq. (20)] 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.
  2. [Section VII, Eq. (29)] 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.
  3. [Section VI B, after Eq. (23)] 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.
minor comments (3)
  1. [Section III and Section IV] 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.
  2. [Section V, after Eq. (22)] 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.
  3. [Section VII, Eq. (31)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main entropy construction is a definition and the BBM bound is a corollary of an independent external theorem.

full rationale

The derivation chain is self-contained and non-circular. Equation (20) defines the Riemannian Wigner function in the lambda-wavefunction representation and is explicitly matched to the external construction of Gneiting et al. [24]; equation (22) is then a definition of phase-space entropy, not a fitted or inferred quantity. The flat and AdS2 oscillator entropies are computed by closed-form/residue integration and quadrature, with no parameter adjusted to match a target value. The generalized BBM inequality (29) follows by applying Beckner's flat-space inequality to the Fourier pair (lambda-wavefunction, momentum wavefunction) fixed by eq. (19); the metric factors in H_X and H_P cancel in the sum, so eq. (29) is a direct corollary of flat BBM rather than an independent curved-space theorem. This limits the novelty of that bound, but it is not circular: the proof invokes an external theorem and does not presuppose eq. (29). Footnote [35] and the chart-dependence of eq. (20) raise well-definedness and rigor concerns, but those are correctness issues, not circularity. There are no load-bearing self-citations.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central construction introduces no fitted free parameters; the Planck-length normalization cancels in the final entropies. The key axioms are standard tools (exponential charts, Beckner's inequality) plus domain assumptions: DeWitt operator ordering, single-valued λ-wavefunctions, and equivalence to the Gneiting et al. Wigner function. No new entities are postulated.

assumptions (6)
  • standard math Exponential charts cover the manifold up to a zero-measure cut locus, so integrals over X can be computed over a chart Dx.
    Used in Section II to define HX via eq. (4); relies on Chavel [26].
  • domain assumption The canonical momentum operator in curved space is defined with the DeWitt ordering in eq. (15).
    Section IV, eq. (15), citing [30,31]. This ordering choice affects the λ-wavefunctions and the Wigner function.
  • standard math Beckner's sharp Hausdorff-Young inequality applies to λ-wavefunctions supported on Dx.
    Section VII, eq. (29). The inequality is applied to ordinary Fourier transforms of λ-wavefunctions.
  • ad hoc to paper λ-wavefunctions must be single-valued, imposing the quantization condition eq. (25).
    Section VI B, after eq. (24). This restricts the model parameters mκ relative to R and j.
  • domain assumption The Wigner function eq. (20) is equivalent to the Gneiting et al. [24] construction and is the correct canonical-invariant generalization.
    Section IV, text after eq. (20). The paper states the equivalence explicitly during revision.
  • domain assumption A positive-definite spatial metric on a constant-time slice represents the curved background; Lorentzian and relativistic effects are deferred.
    Sections VI and VIII. The computations use spatial slices of Minkowski and AdS2.

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Pith. "Pith review of Quantum information in Riemannian spaces." pith.science (2026). https://pith.science/paper/YXDK5AWL

@misc{pith2026241202979,
  author       = {Pith},
  title        = {Pith review of: Quantum information in Riemannian spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YXDK5AWL}},
  note         = {Machine review of arXiv:2412.02979}
}
read the original abstract

We present a diffeomorphism-invariant formulation of differential entropy for Riemannian spaces, providing a fine-grained, coordinate-independent notion of quantum information for continuous variables in physical space. To this end, we consider the generalization of the Wigner quasiprobability density function to arbitrary Riemannian manifolds and analytically continue Shannon's differential entropy to account for contributions from intermediate virtual quantum states. We illustrate the framework by computing the quantum phase-space entropy of harmonic oscillator energy eigenstates in both Minkowski and anti-de Sitter geometries. Furthermore, we derive a generalized entropic uncertainty relation, extending the Bialynicki-Birula and Mycielski inequality to curved backgrounds. By bridging concepts from information theory, differential geometry, and quantum physics, our work provides a systematic approach to studying continuous-variable quantum information in curved spaces.

Figures

Figures reproduced from arXiv: 2412.02979 by the authors.

Figure 1
Figure 1. FIG. 1. Entropy of a two-state quasirandom variable as a [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Phase-space entropy (red circles) and the sum of the [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Left: Integration contour around the four poles of or [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Phase-space entropy (red circles) and the sum of the [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

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