Influence of dark matter on quantum entanglement and coherence in curved spacetime

Shu-Min Wu; Si-Han Shang; Wentao Liu; Yu-Xuan Wang

REVIEW 2 major objections 5 minor 100 references

Influence of dark matter on quantum entanglement and coherence in curved spacetime

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A Schwarzschild black hole embedded in perfect fluid dark matter can either degrade or enhance the entanglement and coherence of quantum fields, depending on the dark-matter density, with bosonic entanglement and fermionic coherence as…

desk verdict A standard PFDM extension whose headline claim is tied to a WEC-violating sign branch; the calculations are otherwise solid and fixable in revision. read the letter →

arxiv 2507.16142 v1 pith:CLVNCILT submitted 2025-07-22 gr-qc quant-ph

Shu-Min Wu , Yu-Xuan Wang , Si-Han Shang , Wentao Liu This is my paper · ORCID

classification gr-qcquant-ph PACS 04.70.Dy03.65.Ud04.62.+v

keywords perfectfluiddarkmatterquantumentanglementcoherenceSchwarzschildblackholeHawkingtemperaturebosonicfieldfermionicrelativisticinformation

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

This paper argues that perfect fluid dark matter around a Schwarzschild black hole does not simply degrade quantum resources: it can suppress them or strengthen them depending on the dark-matter density. For an observer pair that initially shares a maximally entangled state, one far away and one hovering at the horizon, the authors derive closed-form expressions for entanglement and coherence of bosonic and fermionic fields. They find that both quantities fall and then rise as the dark-matter parameter $\alpha$ grows, because the dark matter first raises and then lowers the horizon temperature. The result matters because it identifies which field type and which resource respond most strongly to dark matter: bosonic entanglement for entanglement-based probes and fermionic coherence for coherence-based probes.

What carries the argument

The load-bearing object is the PFDM-modified Schwarzschild metric with line element $-F(r)\,dt^2+F(r)^{-1}dr^2+r^2d\Omega^2$, where $F(r)=1-\frac{2M}{r}-\frac{\alpha}{r}\ln(|\alpha|/r)$ and $r_h$ is the horizon radius solving $F(r_h)=0$. The surface gravity at the horizon is $\kappa=(\alpha+r_h)/(2r_h^2)$, and the horizon temperature is $\kappa/2\pi$. Quantizing scalar and Dirac fields in this background, the paper adopts the single-mode approximation: each field mode of frequency $\omega$ is entangled only with its partner mode across the horizon through a two-mode squeezed state whose thermal weight is set by $\kappa$. The interior modes are traced out to obtain the reduced states shared by the two observers, and logarithmic negativity (an entanglement monotone based on the trace norm of the partial transpose), the $l_1$-norm of coherence, and the relative entropy of coherence are evaluated in closed form. Every $\alpha$-dependence enters through the combination $4\pi\omega r_h^2/(r_h+\alpha)$ in the thermal factors, which produces the dip-then-recover curves.

What would settle it

A decisive check is to compute logarithmic negativity and the $l_1$-norm of coherence for wave packets with a small frequency spread around $\omega$ in the same PFDM metric, replacing the two-mode states of Eqs. (44)-(47) with a multi-mode decomposition. If the dip-then-rise in $\alpha$ and the ordering with fermionic entanglement above bosonic and fermionic coherence below bosonic survive for a reasonably narrow spread, the single-mode result is stable; if they vanish or flip, the reported sensitivity ordering is an artifact of the approximation.

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Extended reading notes

Core claim

The paper claims that in the spacetime of a Schwarzschild black hole surrounded by perfect fluid dark matter, the quantum entanglement and coherence of both scalar (bosonic) and Dirac (fermionic) fields are non-monotonic functions of the dark-matter density $\alpha$: for fixed black hole mass and mode frequency, they initially decrease and then increase as $\alpha$ grows. The mechanism is that $\alpha$ enters the surface gravity $\kappa=(\alpha+r_h)/(2r_h^2)$, so the effective horizon temperature first rises and then falls, and the thermal noise that degrades quantum resources follows the same rise and fall. The paper further claims that fermionic entanglement always exceeds bosonic entanglement for the same parameters, meaning the Hawking-type degradation is relatively stronger for bosons and bosonic entanglement is the more sensitive entanglement probe; conversely, fermionic coherence always lies below bosonic coherence, making fermionic coherence the more sensitive coherence probe.

Load-bearing premise

The calculation assumes each field mode is monochromatic and is entangled only with its single partner mode across the horizon; if the modes spread over a range of frequencies, the derived resource values and the boson-versus-fermion ordering could change.

Editorial extensions

If this is right

For fixed black hole mass and mode frequency, both entanglement and coherence of bosonic and fermionic fields first decrease and then increase as the PFDM density $\alpha$ grows from zero.
Bosonic entanglement remains below fermionic entanglement across the parameter range, so bosons are more strongly degraded by the horizon thermal effect and serve as the more sensitive entanglement-based probe of PFDM.
Fermionic coherence remains below bosonic coherence, so fermionic coherence responds more strongly to PFDM and is the better coherence-based probe.
All four resources degrade more severely for smaller black hole mass (smaller $\omega M$), because the horizon temperature is higher for lighter black holes.
The non-monotonic behavior is driven by the PFDM-induced modulation of the horizon temperature; there is no separate decoherence channel beyond this temperature shift.

Reading between the lines

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

The same $\alpha$-dependent thermal factors appear in every quantity built from the reduced density matrices, so other resources, such as quantum Fisher information, teleportation fidelity, or Bell-nonlocality, should also show the dip-then-recover pattern; measuring one of those would provide an independent check.
Because $r_h$ itself shifts with $\alpha$, comparing observers at fixed horizon radius rather than fixed black hole mass would separate the density-induced temperature change from the horizon-radius shift and give a cleaner test of the mechanism.
If the single-mode approximation is relaxed so that a Hawking mode couples to a spread of partner frequencies, the exact resource values and possibly the boson-fermion ordering change; the paper's probe-selection advice is therefore conditional on that approximation holding.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. The paper is the first PFDM-specific calculation of entanglement and coherence for scalar and Dirac fields in a black-hole background. That part is real. But the central claim — that PFDM density non-monotonically modulates quantum resources — is demonstrated only for the α>0 branch, which by the paper's own Eq. (2) has negative energy density. The text says α>0 arises from the weak energy condition and is positive; that is backwards. This is not a typo: with metric (6), G^t_t = α/r³, so T^t_t = α/(8πr³) and ρ = −α/(8πr³). Positive energy density requires α<0, and no resource curves are shown there.

What is actually new: the α-dependent formulas for l1 coherence, relative entropy of coherence, and logarithmic negativity, for both bosonic and fermionic fields, with the PFDM horizon radius in the Boltzmann weights. The derivations are standard Bogoliubov-and-trace business, executed carefully. The boson/fermion sensitivity ordering (bosonic entanglement, fermionic coherence) is a clear, plausible inference from the formulas, and the mechanism — nonmonotonic Hawking temperature in α — is sensible.

Where it gets soft. Besides the sign error, the single-mode approximation is invoked without a word of justification. For a monochromatic mode near a black hole it's the usual idealization, but ref [52] shows multimode corrections can change entanglement behavior in the closely related Unruh setting; a sentence or footnote acknowledging that would be fair. The reader's report also claimed the fermionic partial-transpose eigenvalues are listed incorrectly. I checked the 2×2 block: eigenvalues (1/2, 1/2, 1/[2(e^{-x}+1)], −1/[2(e^{-x}+1)]) — exactly what the paper displays. The negativity formula also follows. So that one complaint is off the mark.

Bottom line: this is the right community's incremental but legitimate extension. It deserves a referee, but the referee should demand a fix of the sign/WEC discussion and either an explicit limitation paragraph on the single-mode approximation or a multimode check. If the α<0 branch (positive energy density) reproduces the nonmonotonicity and the boson/fermion ordering, the paper will be a solid contribution. As is, the headline 'PFDM can enhance or degrade quantum resources' only holds for an exotic negative-energy fluid under the paper's own conventions.

Referee Report

2 major / 5 minor

Summary. The paper studies quantum entanglement and coherence of bosonic and fermionic fields in a Schwarzschild black hole surrounded by perfect fluid dark matter (PFDM). Starting from the PFDM-modified metric F(r)=1-2M/r - (α/r)ln(|α|/r), it constructs Kruskal vacuum states via Bogoliubov transformations for a single monochromatic mode, forms Alice-Bob maximally entangled states with Bob near the horizon, and derives analytic expressions for the l1-norm and relative entropy of coherence and for logarithmic negativity as functions of the dimensionless PFDM density α/M and the mode frequency ωM. The main reported results are that entanglement and coherence vary non-monotonically with α, that fermionic entanglement exceeds bosonic entanglement, and that bosonic entanglement and fermionic coherence are respectively the most PFDM-sensitive resources (Secs. II-IV, Figs. 3-4). I checked the fermionic partial-transpose eigenvalues listed in Sec. IV B and found them consistent with the displayed matrix.

Significance. If the results hold, the paper provides an analytic demonstration that dark-matter density can either degrade or restore quantum resources in a black-hole spacetime, and that the optimal field type depends on which quantum resource is used. This gives a concrete criterion for designing relativistic-quantum-information probes of PFDM. The strengths of the manuscript include explicit analytic formulas for both field types, the use of the Chandrasekhar transformation to reduce the Dirac equation, and the systematic comparison of bosonic and fermionic behavior across a wide α/M range. The analysis is parameter-free in the sense that no constants are fitted to data; the only inputs are α/M and ωM. The main caveats are the sign consistency of the energy-momentum tensor and the unstated reliance on the single-mode approximation; both affect the physical interpretation of the central claims.

major comments (2)

[Sec. II, Eq. (2) and text following Eq. (6)] Equation (2) defines ρ = -α/(8πr³), but the text states that α>0 arises from the weak energy condition and ensures a positive energy density. These statements are mutually inconsistent. Substituting the metric (6) into the standard Einstein equations with the convention T^t_t = -ρ gives the consistent relation ρ = +α/(8πr³); equivalently, Eq. (5) as written is inconsistent with Eq. (2). Since Figs. 3 and 4 plot only α/M>0, the manuscript must correct the sign convention and confirm that the plotted branch is the positive-energy-density WEC branch. As written, all numerical results presented for α/M>0 correspond to negative energy density under Eq. (2), so the central physical claim about positive-energy-density PFDM is not established for the plotted values.
[Sec. II C, Eqs. (44)-(47), and Sec. IV] The quantitative results assume that Bob's detector is sensitive to a single monochromatic mode and that the inside/outside horizon modes form a two-mode squeezed state with a single frequency parameter. This single-mode approximation is neither stated nor justified, even though Ref. [52] shows that in the closely related Unruh effect going beyond the single-mode approximation can change entanglement behavior. Because the non-monotonic α-dependence and the boson/fermion sensitivity ordering in Figs. 3-4 are computed within this approximation, the authors should either justify its validity for the PFDM-Schwarzschild background or demonstrate robustness with a multi-mode or wave-packet treatment, at least for representative parameters.

minor comments (5)

[Sec. II, Eq. (2)] The stress tensor is described as a perfect fluid, but Eq. (2) has p_r ≠ p_θ = p_φ; this should be reworded (e.g., as an anisotropic PFDM fluid).
[Throughout] The manuscript contains numerous typographical errors, including 'coheren ce' in the title, 'CONCLUTIONS' in Sec. V, 'obtin' near Eq. (4), and 'V 1/κ e^{κv}' missing an equals sign in Eq. (37); a full proofread is required.
[Sec. II and Figs. 1, 3, 4] The dimensionless parameter is introduced as α~ = α/M after Eq. (6), but the figures and later text use α/M; please standardize the notation and axis labels.
[Sec. III, Eq. (48)] The basis with respect to which coherence is defined (the Fock basis {|n⟩_A |m⟩_I}) should be stated explicitly before Eq. (48), since both the l1-norm and the relative entropy of coherence are basis-dependent.
[Sec. IV] The term 'sensitivity' is used qualitatively when comparing bosonic and fermionic resources; defining a quantitative measure (e.g., the derivative of the resource with respect to α/M or its relative change from α=0) would make the central comparison more robust.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the entanglement and coherence results are closed-form consequences of an external PFDM metric and standard relativistic quantum information methods.

full rationale

The derivation chain starts from the Kiselev/Li-Yang PFDM metric (external, cited) and standard surface-gravity/Hawking-temperature relations. The Bogoliubov transformations for bosonic and fermionic fields are re-derived from those metric functions in Eqs. (42)-(43), and the entanglement and coherence measures are evaluated in closed form in Eqs. (54), (57), (59), and (62). No parameter is fitted to data; alpha is a free parameter scanned over. No 'prediction' is selected by the results. The paper's self-citations (e.g. Refs. [35,47,49,51,54]) are contextual or methodological, and none is load-bearing; the metric and quantization procedures are independently sourced. The single-mode approximation is a stated modeling assumption (Ref. [52] is cited exactly for the basis states), and relaxing it would be a robustness or correctness concern, not circularity. The flagged sign-convention issue, where rho = -alpha/(8 pi r^3) implies alpha > 0 gives negative energy density despite the weak-energy-condition claim, is an internal consistency problem rather than a circular reduction. Therefore no circular step is identified.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central calculation adds no new entities; PFDM, the metric, and the field quantization are taken from prior literature. The only inputs are the free parameters alpha and omega, and a set of physical assumptions about the vacuum and the single-mode approximation. The paper's own internal sign inconsistency for alpha is treated as an unflagged assumption.

free parameters (2)

PFDM density alpha (dimensionless alpha/M) = scanned from 0 to 20; no fit
Control parameter of the PFDM metric; the paper scans it to demonstrate non-monotonic behavior. It is not fitted to data, but the paper also does not constrain it observationally.
Mode frequency omega M = 0.05, 0.1, 0.15, 0.2
Frequency of the entangled field mode as seen by the observers; the paper scans it to show higher-frequency modes are more robust. It is an input of the observer model, not fitted.

assumptions (4)

domain assumption The PFDM metric Eq. (6) solves the effective gravitational field equations (3)-(5).
The paper adopts the Kiselev/Li-Yang perfect fluid dark matter solution without deriving it; the central calculation is done in this background.
standard math The Kruskal-like coordinates and Damour-Ruffini analytic continuation (Eqs. (37)-(39)) correctly define the vacuum structure for the PFDM black hole.
This is the standard black-hole quantization method; the paper applies it directly to the PFDM horizon.
ad hoc to paper The single-mode approximation: Bob's detector couples to a single monochromatic mode.
The paper restricts to monochromatic modes and uses the resulting two-mode squeezed state without discussing the beyond-single-mode approximation; all quantitative results depend on this.
ad hoc to paper The sign and energy conditions for alpha: the paper asserts alpha > 0 corresponds to positive energy density, but Eq. (2) implies rho = -alpha/(8 pi r^3), so the physical range of alpha is ambiguous.
This internal inconsistency is an unflagged assumption about which sign of alpha is physical.

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Cite this review

Pith. "Pith review of Influence of dark matter on quantum entanglement and coherence in curved spacetime." pith.science (2026). https://pith.science/paper/CLVNCILT

@misc{pith2026250716142,
  author       = {Pith},
  title        = {Pith review of: Influence of dark matter on quantum entanglement and coherence in curved spacetime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CLVNCILT}},
  note         = {Machine review of arXiv:2507.16142}
}

read the original abstract

Dark matter (DM) remains undetected, and developing theoretical models such as the promising perfect fluid dark matter (PFDM) is a key challenge in modern cosmology. In this work, we investigate the quantum characteristics of PFDM by analyzing the behavior of quantum entanglement and coherence for both fermionic and bosonic fields near a Schwarzschild black hole embedded in a PFDM halo. Our results reveal that PFDM can either enhance or degrade quantum entanglement and coherence, depending sensitively on its density. Notably, bosonic entanglement shows greater susceptibility to PFDM effects compared to fermionic entanglement, while fermionic coherence exhibits a stronger dependence on PFDM than its bosonic counterpart. These findings highlight the necessity of selecting appropriate quantum probes for DM detection based on the type of quantum resources, as different quantum fields exhibit significantly different responses to PFDM in curved spacetime.

Figures

Figures reproduced from arXiv: 2507.16142 by the authors.

**Figure 2.** FIG. 2: The impact of PFDM density [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3: Bosonic entanglement and coherence as a function of t [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4: Fermionic entanglement and coherence as a function o [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

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