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Can spacetime fluctuations generate entanglement between co-moving accelerated detectors?

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Pith's one-line read A tiny spacetime shift entangles co-accelerated detectors

desk verdict A solid entanglement-harvesting calculation whose advertised Planck-scale probe rests on an unjustified state-preparation assumption: the nonzero concurrence requires the field to be in Bob's Rindler vacuum, and a coordinate shift alone does not prepare that state. read the letter →

arxiv 2504.12674 v2 pith:RXBFNDAR submitted 2025-04-17 gr-qc hep-thquant-ph

Dipankar Barman , Bibhas Ranjan Majhi This is my paper · ORCID
classification gr-qchep-thquant-ph PACS 03.67.Mn04.62.+v
keywords entanglementharvestingUnruh-DeWittdetectorsRindlervacuumnestedframesPlancklengthspacetimefluctuationsconcurrencequantumdiscord
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 proposes that two accelerated particle detectors following the same classical trajectory become entangled if a Planck-scale fluctuation shifts their Rindler frames apart, and that the amount of entanglement is the same for any nonzero shift. It calculates this for two eternally switched-on Unruh-DeWitt detectors in nested Rindler frames in (1+1) dimensions, both coupled to the vacuum of the earlier frame. The resulting concurrence is $\mathcal{C}=2\lambda^2|E|=\frac{\lambda^2}{2\Omega^2}\sqrt{\frac{2\pi\Omega/a_1}{\sinh(2\pi\Omega/a_1)}}$, which vanishes only at zero shift and does not depend on the shift length. A sympathetic reader would care because this gives a specific, in-principle observable signal of Planck-length physics that is not contaminated by ordinary Unruh noise. The paper also finds that mutual information and quantum discord vanish to second order, isolating entanglement as the only correlation produced.

What carries the argument

The central object is a pair of nested Rindler frames $R_0$ and $R_1$, whose underlying Minkowski frames are related by a constant spatial shift $\ell$, with each detector sitting at $\xi=0$ so that its proper time equals its Rindler time. The workhorse is the entangling term $E$ in the second-order perturbative density matrix, computed from the Feynman propagator $iG_F(\tau_0,\tau_1)$ between the two trajectories. Mode decomposition of the $R_0$-vacuum field expressed in $R_1$ coordinates converts $E$ into frequency integrals of Beta functions; the residue contributions from chains of poles cancel except for a single term, leaving the shift-independent concurrence formula. The absence of any dependence on $\ell$ traces to how the factors $(a_1\ell)^{\pm i 2\Omega/a_1}$ combine with gamma-function duplication identities.

What would settle it

A direct check is to evaluate the concurrence for two identical accelerated detectors coupled to the $R_0$ vacuum at two different nonzero shifts and at zero shift: the formula predicts identical nonzero values for all $\ell\neq 0$ and exactly zero at $\ell=0$. Finding any dependence of $\mathcal{C}$ on $\ell$ for $\ell\neq 0$, or any nonzero value in the $\ell\to 0$ limit, would refute the central claim. A qualitatively different test is to couple both detectors to the global Minkowski vacuum in the same geometry: the claim requires no entanglement, because the effect is tied specifically to the Rindler vacuum.

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Extended reading notes

Core claim

In the setup of two uniformly accelerated detectors, Alice in the shifted frame $R_1$ and Bob in the earlier frame $R_0$, both coupling to the $R_0$ vacuum, Bob's transition probability is exactly zero while Alice perceives the $R_0$ vacuum as a thermal bath. Since $\sqrt{P_A P_B}=0$, any nonzero entangling term $E$ yields entanglement. Evaluating the Feynman propagator between the two trajectories and performing the frequency integrals gives the concurrence of Eq. (24), which is independent of the shift $\ell$ for every $\ell\neq 0$, increases monotonically with acceleration, and vanishes at $\ell=0$. The paper interprets this as follows: if two detectors on the same accelerated trajectory become entangled, the only available source is a nonvanishing separation of their Rindler horizons due to spacetime fluctuations, because co-accelerated detectors in the global Minkowski vacuum never harvest entanglement. Entanglement harvesting therefore offers a cleaner probe of Planck-scale spacetime than particle-excitation measurements, which cannot distinguish Rindler-vacuum effects from the Unruh effect.

Load-bearing premise

The load-bearing assumption is that a Planck-scale spacetime fluctuation is faithfully represented by a constant shift $\ell$ between two Rindler frames and that both detectors still couple to the unshifted frame's $R_0$ vacuum; if the true fluctuation changes the field state or is not a static shift, the predicted entanglement need not appear.

Editorial extensions

If this is right

  • If the prediction is correct, observing entanglement between two detectors that were on the same accelerated trajectory would be an unambiguous signature that their Rindler horizons separated by a nonzero, possibly Planck-scale, amount.
  • The same amount of entanglement is harvested for any nonzero shift, so an experiment would not need to control the shift size: once the detectors are separated at all, the signal saturates at the predicted value.
  • Because mutual information and quantum discord vanish to second order, any measured correlation between the detectors would be purely entanglement, not classical correlation or discord-type nonclassical correlation.
  • At exactly zero shift the concurrence vanishes, matching the known result that co-moving accelerated detectors cannot harvest entanglement from the Minkowski vacuum; this gives a sharp on/off test.
  • The concurrence grows monotonically with acceleration, so sufficiently strong acceleration is the practical lever for making the effect observable.

Reading between the lines

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

  • If this mechanism is robust, the Rindler shift acts as a switch: any nonzero Planck-scale separation, however tiny, turns on a fixed amount of entanglement. That threshold behaviour could be tested in analogue systems with accelerated detectors, such as trapped-ion or circuit-QED simulators.
  • The derivation is explicitly in (1+1) dimensions; in (3+1) dimensions the angular modes and additional phase factors could modify the beta-function pole cancellations, so the exact shift independence and closed-form concurrence are the paper's own result rather than a demonstrated general property.
  • The paper assumes the field state remains the $R_0$ vacuum after the shift; if quantum-gravity fluctuations source a different state, the entanglement signal would be altered. Comparing detector response with entanglement measurements could therefore constrain candidate vacuum states of spacetime.
  • The shift independence suggests a converse diagnostic: measuring the same concurrence for two different nominal separations would itself be evidence that the separation is nonzero but irrelevant to the entanglement, consistent with the Planck-scale picture.
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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

1 major / 1 minor

Summary. The paper studies entanglement harvesting between two Unruh-DeWitt detectors in (1+1) Minkowski spacetime, each uniformly accelerated in one of two nested Rindler frames R0 and R1 separated by a constant shift ℓ. The field is taken to be in the R0 vacuum (the vacuum of Bob's Rindler modes), and both detectors eternally couple to this same field state. The authors compute the second-order density-matrix element E (the entangling term) via a contour integral in Appendix B and obtain the concurrence C = 2λ²|E| = (λ²/(2Ω²)) sqrt(2πΩ/a₁ / sinh(2πΩ/a₁)), which is independent of ℓ for any nonzero ℓ and vanishes exactly at ℓ=0. They further show that the mutual information and the quantum discord vanish to O(λ²). The paper interprets the ℓ-independence, with ℓ as small as the Planck length, as a robust signature of Planck-scale spacetime fluctuations: if two co-moving accelerated detectors become entangled, their trajectories must be separated by at least a Planck-scale shift.

Significance. If the result holds, it provides a concrete, quantitative prediction for a Planck-scale effect that could in principle be tested with accelerated detectors: the concurrence depends only on the detector gap Ω and the acceleration a₁, not on the shift ℓ. The derivation is carried out in detail, and the final formula is simple and falsifiable. The authors correctly credit previous work [56,57] for the thermal response of a shifted Rindler observer, and they explicitly contrast their setup with the known null result for Minkowski-vacuum harvesting [28]. However, the physical interpretation hinges on the assumption that the field state is the R0 vacuum, and this premise is not justified by the coordinate shift itself.

major comments (1)
  1. [Secs. III–IV and VI] The formal divergence in P_A also undermines the well-definedness of the perturbative density matrix. Equation (20) contains a term πδ(0), so P_A is infinite. While the concurrence formula (7) involves only √(P_A P_B) and P_B=0, the reduced density matrix (3) and the mutual-information/discord calculation in Section V use P_A directly in the eigenvalue expansion (C.25)–(C.26). A δ(0) in P_A makes the state non-normalizable and the O(λ²) perturbative expansion uncontrolled. The authors should discuss how to regularize this divergence (e.g., via a finite-time switching or an infrared cutoff) and confirm that the concurrence result (24) survives the regularization.
minor comments (1)
  1. [Appendix B, Eq. (B.20)] The phrase 'ℓ can be taken in the order of ℓ_p' (Sec. IV) is informal; the authors should specify units (e.g., ℓ = O(ℓ_p) in natural units) and clarify what exactly is implied about the Planck length.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the entanglement result is derived from an explicit mode-function and contour-integral calculation, with no fitted parameter or self-citation chain bearing the central claim.

full rationale

The paper's central result, Eq. (24), is obtained by explicit computation: the R0-vacuum mode functions are transformed to the shifted Rindler frame (Eq. (16)), the Feynman propagator is constructed (Eq. (21)), the entangling term E is evaluated by contour integration in Appendix B, and the concurrence follows from the standard formula Eq. (7). The shift-independence of the concurrence is a mathematical output, not an input: the shift length ℓ enters only through phase factors (a1ℓ)^{±i2Ω/a1}, whose modulus is 1, so the final expression is manifestly independent of ℓ for ℓ≠0. The vanishing for ℓ=0 is checked separately in Eq. (25)-(26). No parameter is fitted to the target concurrence, and no quantity defined in terms of the target result is used to derive it. The cited prior results [56,57] for shift-independent thermality motivate the setup but are not substitutes for the entanglement calculation; moreover, those citations are not self-citations of the present authors. The Minkowski-vacuum no-entanglement result [28] is used as a comparison and interpretive contrast, not as a step in deriving Eq. (24). The physical assumption that the field is in the R0 vacuum is a modeling premise rather than a circular reduction; whether spacetime fluctuations can prepare that state is a correctness or interpretation concern, not a circularity. The paper itself notes the theoretical and (1+1)-dimensional scope limits, which further supports treating the remaining issues as physical-modeling risks rather than circular derivations.

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

The central calculation introduces no new free parameters or entities; the shift ℓ, acceleration a1, energy gap Ω, and coupling λ are model inputs from prior work. The critical assumptions are the choice of R0 vacuum as the field state and the representation of Planck-scale fluctuations as a constant Rindler shift.

assumptions (4)
  • domain assumption The field is prepared in the R0 vacuum, the vacuum of Bob's Rindler frame, and Alice's detector in the shifted frame couples to this same vacuum.
    Used throughout Secs. III-V; it is the key state choice that makes PA nonzero and PB=0. This is an idealized non-Minkowski state and is not justified physically beyond the motivation from [56,57].
  • ad hoc to paper Spacetime fluctuations are modelled as a constant nonzero shift ℓ between two Rindler frames, with the shift as small as the Planck length.
    Introduced in the Abstract and Sec. VI; it is a modeling choice about the form of quantum gravity fluctuations and is not derived.
  • domain assumption The detectors interact with the field for infinite proper time (switching function κ=1), which is standard in [26,28,57] to avoid transients.
    Stated in Sec. II after Eq. (1); it leads to a formal δ(0) divergence in PA (Eq. 20) and needs regularization.
  • standard math Second-order perturbation theory in the coupling λ is valid for the density matrix and concurrence.
    Used in Secs. II and IV; standard for UDW entanglement harvesting, but the presence of δ(0) in PA raises questions about the perturbative regime.

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Pith. "Pith review of Can spacetime fluctuations generate entanglement between co-moving accelerated detectors?." pith.science (2026). https://pith.science/paper/RXBFNDAR

@misc{pith2026250412674,
  author       = {Pith},
  title        = {Pith review of: Can spacetime fluctuations generate entanglement between co-moving accelerated detectors?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RXBFNDAR}},
  note         = {Machine review of arXiv:2504.12674}
}
read the original abstract

Recent studies [Class. Quant. Grav. 42, 03LT01 (2025); Phys. Rev. D 111, 045023 (2025)] indicate that in a nested sequence of Rindler wedges, vacuum of former Rindler frame appears to be thermally populated for an observer in shifted Rindler frame. Interestingly, this thermality is independent of shift parameter as long as it is non-zero and therefore arises even if the shift parameter is as small as Planck length. Building on this insight, we propose a set-up involving two atoms accelerating with identical acceleration. We find that if their Rindler frames (consequently their trajectories) get infinitesimally separated, the atoms become entangled. Remarkably again, this entanglement, like the perceived thermality, is independent of the shift parameter, provided it is non-vanishing. Further we observe the vanishing of mutual information and discord. It implies the absence of both classical and non-classical correlations which are not related to entanglement. We investigate the dependence of entanglement on acceleration of the detectors. The present study indicates that the entanglement between two detectors, moving on the same Rindler wedge, is possible. Moreover, small spacetime fluctuations can lead to entanglement between detectors, moving along same classical trajectory. Hence we feel that such theoretical prediction has potential to probe the Planck length nature of spacetime.

Figures

Figures reproduced from arXiv: 2504.12674 by the authors.

Figure 1
Figure 1. FIG. 1. Nested Rindler frames with observers Alice and Bob. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. We plotted dimensionless concurrence [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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Works this paper leans on

83 extracted references · 31 canonical work pages

  1. [28]

    Reznik, Foundations of Physics33, 167 (2003), URL https://doi.org/10.1023/A:1022875910744

    B. Reznik, Foundations of Physics33, 167 (2003), URL https://doi.org/10.1023/A:1022875910744

    doi:10.1023/a:1022875910744 2003

  2. [1]

    = 0R0 ˆφ(η 0) ˆφ(η′ 0) 0R0 = Z dω0[uω0u′⋆ ω0 +u−ω0u′⋆ −ω0]. (18) To calculate the Wightman propagators consisting coordinates of both observers (particularly when the attention is on the cross correlator), we assign the primed modes to Alice and therefore express this primed modes in terms ofR 1 coordi- nates. We assign the non-primed modes to Bob and kee...

    2021

  3. [2]

    (23) In this Appendix, we calculate the entangling termEin Eq

    Evaluation of Eq. (23) In this Appendix, we calculate the entangling termEin Eq. (22). One can use the Wightman propagator in Eq. (19) to evaluate the entangling term. We obtain E=− Z dω0 4πω0 Z ∞ −∞ dτ1 Z ∞ τ1 dτ0 eiΩ(τ0+τ1)e−iω0τ0 a0 a1 − iω0 a0 e iω0 a0 a1τ1(1+a 1ℓea1τ1)− iω0 a0 − Z dω0 4πω0 Z ∞ −∞ dτ1 Z ∞ τ1 dτ0 eiΩ(τ0+τ1)e−iω0τ0 a0 a1 iω0 a0 e iω0 a0...

  4. [3]

    (23) Let us denoteI AC andI BD as the integrands of the integrals in Eq

    Evaluation of contour integrals in Eq. (23) Let us denoteI AC andI BD as the integrands of the integrals in Eq. (23), respectively: IAC = i 4πω0 (a1ℓ)−i( 2Ω a1 )B[i( 2Ω a1 ),ν−i( 2Ω−ω0 a1 )] a1(Ω−ω 0 +iε) ; IBD = i 4πω0 (a1ℓ)+i( 2Ω a1 )B[−i( 2Ω a1 ),ν+i( 2Ω−ω0 a1 )] a1(Ω−ω 0 +iε) . (B.19) We can express the beta functions in terms of gamma functions (asB[...

  5. [4]

    I+a 3σz +b 3I(±z3) +c3σz(±z3) +Re(c1)σx(±z1)−Im(c 1)σy(±z1) +Re(c2)σy(±z2) +Im(c2)σx(±z2) # ⊗Π k .(C.17) Here±signs are fork=0,1, respectively. Thus Eq. (C.9) becomes pkρk = 1 4

    =k+l=1. (C.6) In the expression of the unitary matrixV,Iis the identity operator andσ’s are Pauli matrices. Here for later convenience, we defined k= (t 2 +y 2 3);l= (y 2 1 +y 2 2). (C.7) After the measurement{Bk}, the stateρ AB will change to the ensemble{pk,ρ k} with ρk := 1 pk (I⊗B k)ρ AB (I⊗B k),(C.8) andp k =Tr(I⊗B k)ρ AB (I⊗B k). We need to evaluate...

    work page

  6. [5]

    Hotta, Phys

    M. Hotta, Phys. Rev. D78, 045006 (2008), arXiv:0803.2272

    arXiv 2008

  7. [6]

    Hotta, Journal of the Physical Society of Japan78, 034001 (2009), https://doi.org/10.1143/JPSJ.78.034001, URLhttps: //doi.org/10.1143/JPSJ.78.034001

    M. Hotta, Journal of the Physical Society of Japan78, 034001 (2009), https://doi.org/10.1143/JPSJ.78.034001, URLhttps: //doi.org/10.1143/JPSJ.78.034001

    work page doi:10.1143/jpsj.78.034001 2009

  8. [7]

    Matson, Nature (2012), URLhttps://doi.org/10.1038/ nature.2012.11163

    J. Matson, Nature (2012), URLhttps://doi.org/10.1038/ nature.2012.11163

    arXiv 2012

Show all 83 references
  1. [8]

    M. Frey, K. Funo, and M. Hotta, Phys. Rev. E90, 012127 (2014), URLhttps://link.aps.org/doi/10. 1103/PhysRevE.90.012127

    2014

  2. [9]

    A. K. Ekert, Phys. Rev. Lett.67, 661 (1991), URLhttps:// link.aps.org/doi/10.1103/PhysRevLett.67.661

    1991 doi

  3. [10]

    Tittel, J

    W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, Phys. Rev. Lett. 81, 3563 (1998), arXiv:quant-ph/9806043

    1998 arXiv

  4. [11]

    Salart, A

    D. Salart, A. Baas, C. Branciard, N. Gisin, and H. Zbinden, 15 Nature454, 861 (2008), URLhttps://doi.org/10.1038/ nature07121

    2008

  5. [12]

    Yin, Y .-H

    J. Yin, Y .-H. Li, S.-K. Liao, M. Yang, Y . Cao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y . Liu, S.-L. Li, et al., Na- ture582, 501 (2020), URLhttps://doi.org/10.1038/ s41586-020-2401-y

    2020

  6. [13]

    G. J. Mooney, C. D. Hill, and L. C. L. Hollenberg, Scientific Reports9, 13465 (2019), URLhttps://doi.org/10.1038/ s41598-019-49805-7

    2019

  7. [14]

    S. J. Summers and R. Werner, Physics Letters A110, 257 (1985), ISSN 0375-9601

    1985

  8. [15]

    S. J. Summers and R. Werner, Commun. Math. Phys.110, 247 (1987)

    1987

  9. [16]

    S. J. Summers and R. Werner, Journal of Mathematical Physics28, 2448 (1987), ISSN 0022-2488, https://pubs.aip.org/aip/jmp/article- pdf/28/10/2448/11150126/2448 1 online.pdf, URL https://doi.org/10.1063/1.527734

    1987 doi

  10. [17]

    S. J. Summers and R. Werner, Journal of Mathematical Physics 28, 2440 (1987)

    1987

  11. [18]

    Peres and D

    A. Peres and D. R. Terno, Rev. Mod. Phys.76, 93 (2004), URL https://link.aps.org/doi/10.1103/RevModPhys.76. 93

    2004 doi

  12. [19]

    S. W. HAWKING, Nature248, 30 (1974), URLhttps://doi. org/10.1038/248030a0

    1974 doi

  13. [20]

    S. W. Hawking, Communications in Mathematical Physics43, 199 (1975)

    1975

  14. [21]

    W. G. Unruh, Phys. Rev. D14, 870 (1976)

    1976

  15. [22]

    N. D. Birrell and P. C. W. Davies,Quantum fields in curved space, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 1982)

    1982

  16. [23]

    W. G. Unruh and R. M. Wald, Phys. Rev.D29, 1047 (1984)

    1984

  17. [24]

    L. C. B. Crispino, A. Higuchi, and G. E. A. Matsas, Rev. Mod. Phys.80, 787 (2008), 0710.5373

    2008 arXiv

  18. [25]

    Takagi, Progress of Theoreti- cal Physics Supplement88, 1 (1986), http://ptps.oxfordjournals.org/content/88/1.full.pdf+html

    S. Takagi, Progress of Theoreti- cal Physics Supplement88, 1 (1986), http://ptps.oxfordjournals.org/content/88/1.full.pdf+html

    1986

  19. [26]

    Higuchi, S

    A. Higuchi, S. Iso, K. Ueda, and K. Yamamoto, Phys. Rev. D 96, 083531 (2017), arXiv:1709.05757

    2017 arXiv

  20. [27]

    Valentini, Physics Letters A153, 321 (1991), ISSN 0375- 9601

    A. Valentini, Physics Letters A153, 321 (1991), ISSN 0375- 9601

    1991

  21. [29]

    Reznik, A

    B. Reznik, A. Retzker, and J. Silman, Phys. Rev. A71, 042104 (2005), arXiv:quant-ph/0310058

    2005 arXiv

  22. [30]

    J.-I. Koga, G. Kimura, and K. Maeda, Phys. Rev. A97, 062338 (2018), arXiv:1804.01183

    2018 arXiv

  23. [31]

    K. K. Ng, R. B. Mann, and E. Mart ´ın-Mart´ınez, Phys. Rev. D 97, 125011 (2018), arXiv:1805.01096

    2018 arXiv

  24. [32]

    J.-i. Koga, K. Maeda, and G. Kimura, Phys. Rev. D100, 065013 (2019), 1906.02843

    2019 arXiv

  25. [33]

    Barman, S

    D. Barman, S. Barman, and B. R. Majhi, JHEP07, 124 (2021), arXiv:2104.11269

    2021 arXiv

  26. [34]

    Barman, S

    D. Barman, S. Barman, and B. R. Majhi, Phys. Rev. D106, 045005 (2022), 2205.08505

    2022 arXiv

  27. [35]

    Wu, R.-D

    S.-M. Wu, R.-D. Wang, X.-L. Huang, and Z. Wang (2024), 2408.11277

    2024 arXiv

  28. [36]

    Pozas-Kerstjens and E

    A. Pozas-Kerstjens and E. Martin-Martinez, Phys. Rev. D94, 064074 (2016), 1605.07180

    2016 arXiv

  29. [37]

    T. R. Perche, C. Lima, and E. Mart ´ın-Mart´ınez, Phys. Rev. D 105, 065016 (2022), 2111.12779

    2022 arXiv

  30. [38]

    Z. Liu, J. Zhang, and H. Yu, JHEP08, 020 (2021), 2101.00114

    2021 arXiv

  31. [40]

    Menezes, Phys

    G. Menezes, Phys. Rev.D97, 085021 (2018), arXiv:1712.07151

    2018 arXiv

  32. [41]

    Tjoa and R

    E. Tjoa and R. B. Mann, JHEP08, 155 (2020), arXiv:2007.02955

    2020 arXiv

  33. [42]

    Gallock-Yoshimura, E

    K. Gallock-Yoshimura, E. Tjoa, and R. B. Mann, Phys. Rev. D 104, 025001 (2021), URLhttps://link.aps.org/doi/10. 1103/PhysRevD.104.025001

    2021

  34. [43]

    Barman, D

    S. Barman, D. Barman, and B. R. Majhi, JHEP09, 106 (2022), 2112.01308

    2022 arXiv

  35. [44]

    Barman and B

    S. Barman and B. R. Majhi, Gen. Rel. Grav.56, 70 (2024), 2301.06764

    2024 arXiv

  36. [45]

    Kukita and Y

    S. Kukita and Y . Nambu, Entropy19, 449 (2017), arXiv:1708.01359

    2017 arXiv

  37. [46]

    Martin-Martinez, A

    E. Martin-Martinez, A. R. H. Smith, and D. R. Terno, Phys. Rev. D93, 044001 (2016), arXiv:1507.02688

    2016 arXiv

  38. [47]

    H. K, S. Barman, and D. Kothawala, Phys. Rev. D109, 065017 (2024), 2311.15019

    2024 arXiv

  39. [48]

    Y . Ji, J. Zhang, and H. Yu, JHEP06, 161 (2024), 2401.13406

    2024 arXiv

  40. [49]

    Lin and S

    F.-L. Lin and S. Mondal, JHEP08, 159 (2024), 2406.19125

    2024 arXiv

  41. [50]

    Mayank, K. Hari, S. Barman, and D. Kothawala (2025), 2502.20874

    2025 arXiv

  42. [51]

    Barman, A

    D. Barman, A. Choudhury, B. Kad, and B. R. Majhi, Phys. Rev. D107, 045001 (2023), 2211.00383

    2023 arXiv

  43. [52]

    Chowdhury and B

    P. Chowdhury and B. R. Majhi, JHEP05, 025 (2022), 2110.11260

    2022 arXiv

  44. [53]

    Nandi, B

    P. Nandi, B. R. Majhi, N. Debnath, and S. Kala, Phys. Lett. B 853, 138706 (2024), 2401.02778

    2024 arXiv

  45. [54]

    Nandi and B

    P. Nandi and B. R. Majhi, Phys. Lett. B857, 138988 (2024), 2403.11253

    2024 arXiv

  46. [55]

    Dutta, P

    M. Dutta, P. Nandi, and B. R. Majhi (2025), 2503.19688

    2025 arXiv

  47. [56]

    Pozas-Kerstjens, J

    A. Pozas-Kerstjens, J. Louko, and E. Martin-Martinez, Phys. Rev. D95, 105009 (2017), 1703.02982

    2017 arXiv

  48. [57]

    L. J. Henderson and N. C. Menicucci, Phys. Rev. D102, 125026 (2020), arXiv:2005.05330

    2020 arXiv

  49. [58]

    C. J. Shallue and S. M. Carroll (2025), 2501.06609

    2025

  50. [59]

    D. J. Stargen (2025), 2501.09676

    2025 arXiv

  51. [60]

    Lochan and T

    K. Lochan and T. Padmanabhan, Class. Quant. Grav.42, 03LT01 (2025)

    2025

  52. [61]

    H. S. Sahota and K. Lochan, Phys. Rev. D111, 045023 (2025), URLhttps://link.aps.org/doi/10.1103/PhysRevD. 111.045023

    2025 doi

  53. [62]

    Gutti, A

    S. Gutti, A. U. Nair, and P. Samantray, Phys. Rev. D108, 025010 (2023), 2210.08925

    2023 arXiv

  54. [63]

    A. U. Nair, R. K. Jha, P. Samantray, and S. Gutti (2024), 2412.02560

    2024 arXiv

  55. [64]

    I. L. C. Michael A. Nielsen,Quantum computation and quantum information, Cambridge Series on Information and the Natural Sciences (Cambridge University Press, 2004), 1st ed., ISBN 0521635039,9780521635035,0521632358, URLhttp://gen.lib.rus.ec/book/index.php?md5= 2badbdeda09e90d...

    2004

  56. [65]

    Simidzija and E

    P. Simidzija and E. Mart ´ın-Mart´ınez, Phys. Rev. D98, 085007 (2018), arXiv:1809.05547

    2018 arXiv

  57. [66]

    Ollivier and W

    H. Ollivier and W. H. Zurek, Phys. Rev. Lett.88, 017901 (2001), URLhttps://link.aps.org/doi/10. 1103/PhysRevLett.88.017901

    2001

  58. [67]

    Henderson and V

    L. Henderson and V . Vedral, Journal of Physics A: Mathe- matical and General34, 6899 (2001), ISSN 1361-6447, URL http://dx.doi.org/10.1088/0305-4470/34/35/315

    2001 doi

  59. [68]

    M. S. Sarandy, Phys. Rev. A80, 022108 (2009), URLhttps: //link.aps.org/doi/10.1103/PhysRevA.80.022108

    2009 doi

  60. [69]

    Luo, Phys

    S. Luo, Phys. Rev. A77, 042303 (2008), URLhttps://link. aps.org/doi/10.1103/PhysRevA.77.042303. 16

    2008 doi

  61. [70]

    M. Ali, A. R. P. Rau, and G. Alber, Phys. Rev. A 81, 042105 (2010), URLhttps://link.aps.org/doi/10. 1103/PhysRevA.81.042105

    2010

  62. [71]

    Q. Chen, C. Zhang, S. Yu, X. X. Yi, and C. H. Oh, Phys. Rev. A 84, 042313 (2011), URLhttps://link.aps.org/doi/10. 1103/PhysRevA.84.042313

    2011

  63. [72]

    A. K. Pal and I. Bose, Journal of Physics B: Atomic, Molecular and Optical Physics44, 045101 (2011), ISSN 1361-6455, URL http://dx.doi.org/10.1088/0953-4075/44/4/045101

    2011 doi

  64. [73]

    Huang, Phys

    Y . Huang, Phys. Rev. A88, 014302 (2013), URLhttps:// link.aps.org/doi/10.1103/PhysRevA.88.014302

    2013 doi

  65. [74]

    M. A. Yurischev, Quantum Information Processing14, 3399 (2015), ISSN 1573-1332, URLhttp://dx.doi.org/10. 1007/s11128-015-1046-5

    2015

  66. [75]

    Guo, Scientific Reports6, 25241 (2016), URLhttps:// doi.org/10.1038/srep25241

    Y . Guo, Scientific Reports6, 25241 (2016), URLhttps:// doi.org/10.1038/srep25241

    2016 doi

  67. [76]

    Riedel G ˚arding, N

    E. Riedel G ˚arding, N. Schwaller, C. L. Chan, S. Y . Chang, S. Bosch, F. Gessler, W. R. Laborde, J. N. Hernandez, X. Si, M.-A. Dupertuis, et al., Entropy23(2021), ISSN 1099-4300, URLhttps://www.mdpi.com/1099-4300/23/7/797

    2021

  68. [77]

    Mart ´ın-Mart´ınez and P

    E. Mart ´ın-Mart´ınez and P. Rodriguez-Lopez, Phys. Rev. D 97, 105026 (2018), URLhttps://link.aps.org/doi/10. 1103/PhysRevD.97.105026

    2018

  69. [78]

    Peres, Phys

    A. Peres, Phys. Rev. Lett.77, 1413 (1996), arXiv:quant- ph/9604005

    1996

  70. [79]

    Horodecki, P

    M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 223, 1 (1996), arXiv:quant-ph/9605038

    1996 arXiv

  71. [80]

    C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Woot- ters, Phys. Rev. A54, 3824 (1996), arXiv:quant-ph/9604024

    1996 arXiv

  72. [81]

    Hill and W

    S. Hill and W. K. Wootters, Phys. Rev. Lett.78, 5022 (1997), arXiv:quant-ph/9703041

    1997 arXiv

  73. [82]

    W. K. Wootters, Phys. Rev. Lett.80, 2245 (1998), arXiv:quant- ph/9709029

    1998

  74. [83]

    Oberhettinger,Tables of Fourier Transforms and Fourier Transforms of Distributions, vol

    F. Oberhettinger,Tables of Fourier Transforms and Fourier Transforms of Distributions, vol. pp. 204, Chap. 3, 10.1007/978-3-642-74349-8 (Springer, Berlin, Heidelberg, 1990)

    1990 doi

  75. [84]

    DLMF,NIST Digital Library of Mathematical Functions, f. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schnei- der, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V . Saunders, eds

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