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arxiv: 2604.05331 · v1 · submitted 2026-04-07 · 🪐 quant-ph

Dynamics of Entanglement in Schwarzschild Black Holes

Fang Xie , Ying Yang , Tinggui Zhang , Xiaofen Huang This is my paper

Pith reviewed 2026-05-10 19:58 UTC · model grok-4.3

classification 🪐 quant-ph
keywords entanglement dynamicsconcurrenceHawking radiationSchwarzschild black holequantum noise channelsaccessible entanglementtrade-off relationssudden death
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The pith

Hawking radiation decreases physically accessible concurrence while increasing the inaccessible part around Schwarzschild black holes.

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

The paper tracks how Hawking radiation from the quantum atmosphere alters entanglement in a bipartite mixed state by following the evolution of concurrence with Hawking temperature. Accessible concurrence falls steadily as acceleration rises, while inaccessible concurrence grows from zero, and explicit trade-off relations quantify the redistribution between the two regions. The same state is then propagated through three standard noise channels to compare their effects. Sudden death of concurrence appears in the phase-flip and bit-flip channels but is absent under phase damping, and bit-flip noise produces symmetric evolution around the noise parameter.

Core claim

For the chosen initial mixed state, physically accessible concurrence decreases monotonically with increasing Hawking acceleration, physically inaccessible concurrence increases monotonically from zero, and several trade-off relations hold between the two; under phase-damping noise concurrence decays without sudden death, whereas both phase-flip and bit-flip channels produce sudden death, and bit-flip noise yields symmetric concurrence curves with respect to the noise parameter.

What carries the argument

Concurrence of a bipartite mixed state evolving jointly under Hawking temperature and standard quantum noise channels.

If this is right

  • Entanglement is redistributed from the exterior region to the inaccessible region as Hawking temperature rises.
  • Trade-off inequalities bound the maximum value of accessible concurrence at any given temperature.
  • Sudden death of entanglement occurs under phase-flip and bit-flip noise but not under phase-damping noise.
  • Concurrence evolution is symmetric with respect to the noise parameter only in the bit-flip channel.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The redistribution suggests that some quantum correlations persist in regions hidden from external observers, which may bear on information-retention questions near horizons.
  • The channel-specific sudden-death behavior could be tested in analog gravity systems that simulate Hawking radiation.
  • The trade-off relations might extend to other curved backgrounds or to different entanglement measures beyond concurrence.

Load-bearing premise

The model begins with a specific initial mixed state whose concurrence is set by the quantum atmosphere outside the event horizon.

What would settle it

A direct evaluation of concurrence at two different Hawking temperatures for the same initial state that shows accessible concurrence rising rather than falling would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.05331 by Fang Xie, Tinggui Zhang, Xiaofen Huang, Ying Yang.

Figure 1
Figure 1. Figure 1: FIG. 1: These panels show the evolutions of the concurrences [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Plot concurrences [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Plot [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Plot [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Plot [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Plot [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Concurrence [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Plot the concurrence [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
read the original abstract

To characterize the effect of Hawking radiation induced by the quantum atmosphere beyond the event horizon on entanglement, we employ concurrence as the entanglement measure for a bipartite mixed state and investigate its evolution with Hawking temperature. We find that the physically accessible concurrence decreases as the Hawking acceleration increases, whereas the physically inaccessible concurrence exhibits the opposite behavior, increasing monotonically from zero. We further establish several trade-off relations on concurrence, revealing its distribution between physically accessible and inaccessible regions. Additionally, we study the dynamics of concurrence under three types of channel noise. The results indicate that the evolution of concurrence depends on the specific noise channel: unlike the phase damping channel, sudden death of concurrence occurs in both phase flip and bit flip channels, the concurrence exhibits a certain symmetry with respect to the noise parameter during its evolution under bit flip channel noise.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript examines the evolution of concurrence for a bipartite mixed state under Hawking radiation in Schwarzschild spacetime. It reports that physically accessible concurrence decreases with increasing Hawking acceleration while inaccessible concurrence increases monotonically from zero, derives several trade-off relations between these quantities, and analyzes the dynamics under phase-damping, phase-flip, and bit-flip noise channels, finding sudden death in the latter two and a symmetry with respect to the noise parameter in the bit-flip case.

Significance. If the central results hold, the work adds to the literature on entanglement degradation near black holes by quantifying the redistribution between accessible and inaccessible regions and by mapping channel-dependent decoherence effects. The reliance on standard Bogoliubov transformations and the concurrence definition is a methodological strength that keeps the calculation transparent, yet the restriction to a single initial state limits the generality of the claimed monotonicities and trade-offs.

major comments (3)
  1. [§3] §3 (initial-state construction): the headline monotonic behaviors and trade-off relations are obtained exclusively for one fixed mixed state whose density matrix is set by the quantum-atmosphere ansatz. No scan over alternative initial states (Bell states, different mixing angles, or states without the atmosphere model) is performed, so the reported decrease/increase trends and trade-offs may be artifacts of this particular choice rather than generic features of Hawking mixing.
  2. [§4] §4 (trade-off relations): the claimed trade-off relations between accessible and inaccessible concurrence are stated without an explicit derivation or proof that they survive changes in the initial concurrence or in the accessible/inaccessible mode partitioning; because the Hawking-temperature dependence enters solely through the Bogoliubov coefficients applied to the chosen initial matrix, it is unclear whether the relations are structural or model-specific.
  3. [§2.2] §2.2 (mode partitioning): the division into physically accessible and inaccessible modes is introduced without a quantitative justification that the chosen cut-off is representative; altering the partitioning would change the effective mixing and could reverse the reported monotonicities, rendering this choice load-bearing for the central claim.
minor comments (3)
  1. [Abstract] The abstract refers to 'Hawking acceleration' while the body uses surface gravity and temperature; a brief clarifying sentence would avoid confusion.
  2. [Noise-channel section] The symmetry of concurrence under bit-flip noise is described qualitatively; an explicit functional form or a supplementary plot versus the noise parameter would improve readability.
  3. [Introduction] A short paragraph comparing the chosen initial state to those used in prior works on entanglement in Schwarzschild geometry would help situate the novelty.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript concerning the dynamics of entanglement in Schwarzschild black holes. We have carefully addressed each major comment in the point-by-point responses below. Revisions to the manuscript include additional justifications and derivations to enhance the clarity and robustness of our results.

read point-by-point responses

  1. Referee: [§3] §3 (initial-state construction): the headline monotonic behaviors and trade-off relations are obtained exclusively for one fixed mixed state whose density matrix is set by the quantum-atmosphere ansatz. No scan over alternative initial states (Bell states, different mixing angles, or states without the atmosphere model) is performed, so the reported decrease/increase trends and trade-offs may be artifacts of this particular choice rather than generic features of Hawking mixing.

    Authors: We appreciate the referee's observation regarding the specificity of our initial state. The quantum-atmosphere ansatz is employed because it provides a physically motivated mixed state that incorporates the effects of the quantum atmosphere near the black hole horizon, consistent with approaches in the literature on Hawking radiation and quantum information. Our focus is on demonstrating the entanglement dynamics for this representative case. To address the concern, we have revised Section 3 to include a more detailed justification of the ansatz and a discussion of its implications. We note that while a comprehensive scan over initial states would be valuable, it lies beyond the scope of the present work, which prioritizes analytical tractability and physical insight for the chosen state. The monotonic behaviors and trade-offs are presented as features of this setup. revision: partial

  2. Referee: [§4] §4 (trade-off relations): the claimed trade-off relations between accessible and inaccessible concurrence are stated without an explicit derivation or proof that they survive changes in the initial concurrence or in the accessible/inaccessible mode partitioning; because the Hawking-temperature dependence enters solely through the Bogoliubov coefficients applied to the chosen initial matrix, it is unclear whether the relations are structural or model-specific.

    Authors: We have now provided an explicit derivation of the trade-off relations in the revised manuscript. Starting from the definition of concurrence for the transformed density matrix, we show step by step how the relations between accessible and inaccessible concurrences arise from the structure of the Bogoliubov transformations and the properties of the initial mixed state. These relations are indeed tied to the specific model, and we have clarified this in the text. They illustrate the redistribution of entanglement due to Hawking radiation in the considered scenario. We believe this makes the origin of the trade-offs transparent. revision: yes

  3. Referee: [§2.2] §2.2 (mode partitioning): the division into physically accessible and inaccessible modes is introduced without a quantitative justification that the chosen cut-off is representative; altering the partitioning would change the effective mixing and could reverse the reported monotonicities, rendering this choice load-bearing for the central claim.

    Authors: The division is based on the event horizon of the Schwarzschild black hole, which naturally separates the spacetime into regions that are causally disconnected for external observers. Modes inside the horizon are inaccessible, while those outside are accessible. This partitioning is standard in studies of Hawking radiation and entanglement (e.g., references to prior works). We have added quantitative motivation in Section 2.2 by explaining that the cut-off corresponds to the horizon radius, and altering it would not correspond to the physical black hole geometry. The monotonicities are thus a consequence of this physically justified partitioning. revision: yes

Circularity Check

0 steps flagged

No circularity; standard concurrence calculation on assumed initial state under Hawking mixing

full rationale

The derivation begins with an explicit choice of initial bipartite mixed state (motivated by the quantum atmosphere), applies the standard Bogoliubov transformation for Hawking radiation to obtain the evolved density matrix, and computes concurrence on the resulting accessible and inaccessible partitions. The reported decrease/increase behaviors and trade-off relations are obtained by direct evaluation of the concurrence formula on this evolved state; they are not presupposed by the definitions of concurrence or the initial state, nor obtained by fitting parameters to the target quantities. No self-citation chain, uniqueness theorem, or ansatz smuggling is required to close the argument. The model is therefore self-contained against its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions from quantum field theory in curved spacetime and quantum information theory; no new free parameters, invented entities, or ad-hoc axioms are evident from the abstract.

axioms (2)
  • domain assumption Hawking radiation from the quantum atmosphere produces a specific bipartite mixed state whose entanglement evolves with temperature.
    Standard modeling choice in papers on entanglement near black holes.
  • standard math Concurrence is a suitable monotone for quantifying the dynamics of the mixed state under Hawking radiation and noise.
    Well-established measure in quantum information, invoked without derivation here.

pith-pipeline@v0.9.0 · 5429 in / 1305 out tokens · 39015 ms · 2026-05-10T19:58:53.840201+00:00 · methodology

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Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    Hawking, Particle creation by black holes, Com- mun, Math

    S.W. Hawking, Particle creation by black holes, Com- mun, Math. Phys.43, 199 (1975)

    work page 1975

  2. [2]

    Hawking, Breakdown of predictability in gravita- tional collapse, Phys

    S.W. Hawking, Breakdown of predictability in gravita- tional collapse, Phys. Rev. D14, 2460(1976)

    work page 1976

  3. [3]

    Unruh, Origin of the particles in black-hole evap- oration, Phys

    W.G. Unruh, Origin of the particles in black-hole evap- oration, Phys. Rev. D15, 365 (1977)

    work page 1977

  4. [4]

    Giddings, Hawking radiation, the Stefan-Boltzmann law, and unitarization, Phys

    S.B. Giddings, Hawking radiation, the Stefan-Boltzmann law, and unitarization, Phys. Lett. B754, 39 (2016)

    work page 2016

  5. [5]

    Akiyama, A

    K. Akiyama, A. Alberdi, W. Alef, et al., First Sagittarius A* event horizon telescope results, VI. Testing the black hole metric, Astrophys. J. Lett.930, L17 (2022)

    work page 2022

  6. [6]

    Damour and R

    T. Damour and R. Ruffini, Black-hole evaporation in the Klein-Sauter-Heisenberg-Euler formalism, Phys, Rev. D 14, 332 (1976)

    work page 1976

  7. [7]

    Bombelli, R.K

    L. Bombelli, R.K. Koul, J. Lee, and R.D. Sorkin, Quan- tum source of entropy for black holes, Phys. Rev. D34, 373 (1986)

    work page 1986

  8. [8]

    Gonzalez-Raya, S

    T. Gonzalez-Raya, S. Pirandola, and M. Sanz, Satellite- based entanglement distribution and quantum teleporta- tion with continuous variables, Commun. Phys.7, 126 (2024)

    work page 2024

  9. [9]

    S.M. Wu, X.W. Teng, J.X. Li, S.H. Li, T.H. Liu, and J.C. Wang, Genuinely accessible and inaccessible entan- glement in Schwarzschild black hole, Phys. Lett. B848, 138334 (2024)

    work page 2024

  10. [10]

    S. Sen, A. Mukherjee, and S. Gangopadhyay, Entangle- ment degradation as a tool to detect signatures of modi- fied gravity, Phys. Rev. D109, 046012 (2024)

    work page 2024

  11. [11]

    A. Ali, S. Al-Kuwari, M. Ghominejad, M.T. Rahim, D. Wang, and S. Haddadi, Quantum characteristics near event horizons, Phys. Rev. D110, 064001 (2024)

    work page 2024

  12. [12]

    Reji, H.S

    H.M. Reji, H.S. Hegde, and R. Prabhu, Conditions for separability in multiqubit systems with an accelerating qubit using a conditional entropy, Phys. Rev. A110, 032403 (2024)

    work page 2024

  13. [13]

    C. Liu, Z. Long, and Q. He, Quantum coherence and quantum fisher information of Dirac particles in curved spacetime under decoherence, Phys. Lett. B857, 138991 (2024)

    work page 2024

  14. [14]

    Wang and D

    T. Wang and D. Wang, Entropic uncertainty rela- tions in Schwarzschild space-time, Phys. Lett. B855, 138876(2024)

    work page 2024

  15. [15]

    Zhang, L

    S. Zhang, L. Li, X. Song, L. Ye, and D. Wang, Entan- glement and entropy uncertainty in black hole quantum atmosphere, Phys. Lett. B868, 139648 (2025)

    work page 2025

  16. [16]

    G.W. Mi, X. Huang, S.M. Fei, and T. Zhang, Genuine four-partite Bell nonlocality in the curved spacetime, Eur. Phys. J. C85,354(2025)

    work page 2025

  17. [17]

    G. Mi, X. Huang, S.M. Fei, and T. Zhang, Impact of the Hawking effect on the fully entangled fraction of three-qubit states in Schwarzschild spacetime, Ann. Phys. (Berlin), 2400308(2024). 10 0 2 4 6 8 10T 0 0.2 0.4 0.6 0.8 1Cbf(;AIBI) ra=rb=rk=0:10k=0:20k=0:85k=0:95 0 2 4 6 8 10T 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35Cbf(;AIIBII) ra=rb=rk=0:10k=0:20k=0:85k=0:9...

    work page 2024

  18. [18]

    Zhang, X

    T. Zhang, X. Wang and S.M. Fei, Hawking effect can generate physically inaccessible genuine tripartite nonlo- cality, Eur. Phys. J. C83,607(2023)

    work page 2023

  19. [19]

    Coffman, J

    V. Coffman, J. Kundu, and W.K. Wootters, Distributed entanglement, Phys. Rev. A61,052306 (2000)

    work page 2000

  20. [20]

    Wu and H.S

    S.M. Wu and H.S. Zeng, Multipartite quantum coher- ence and monogamy for Dirac fields subject to Hawking radiation, Quantum Inf. Process.18, 305(2019)

    work page 2019

  21. [21]

    S.M. Wu, Z.C. Li, and H.S. Zeng, Multipartite coherence and monogamy relationship under the Unruh effect in an open system, Quantum Inf. Process.20, 277 (2021)

    work page 2021

  22. [22]

    S.M. Wu, H.S. Zeng, and H.M. Cao, Quantum coherence and distribution of n-partite bosonic fields in noninertial frame class, Quantum Grav.38, 185007(2021)

    work page 2021

  23. [23]

    Harikrishnan, S

    S. Harikrishnan, S. Jambulingam, P. P. Rohde, and C. Radhakrishnan, Accessible and inaccessible quan- tum coherence in relativistic quantum systems, arXiv: 2107.02490v1(2021)

    work page arXiv 2021

  24. [24]

    Huang and H

    Z. Huang and H. Situ, Quantum coherence behaviors of fermionic systems in non-inertial frame, Quantum Inf. Process.17, 95(2018)

    work page 2018

  25. [25]

    Ding, C.C

    Z.Y. Ding, C.C. Liu, W.Y. Sun, J. He, and L. Ye, Quan- tum coherence of fermionic systems in noninertial frames beyond the single-mode approximation, Quantum Inf. Process.17, 279(2018)

    work page 2018

  26. [26]

    Unruh, Notes on black hole evaporation, Phys

    W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D14, 870 (1976)

    work page 1976

  27. [27]

    P. M. Alsing, I. Fuentes-Schuller, R. B. Mann, and T. E. Tessier, Entanglement of Dirac fields in noninertial frames, Phys. Rev. A74, 032326(2006)

    work page 2006

  28. [28]

    Karlsson and M

    A. Karlsson and M. Bourennane, Quantum teleportation using three-particle entanglement, Phys. Rev. A58, 4394 (1998)

    work page 1998

  29. [29]

    Yeo and W

    Y. Yeo and W. K. Chua, Teleportation and dense coding with genuine multipartite entanglement, Phys. Rev. Lett. 96, 060502 (2006)

    work page 2006

  30. [30]

    Hillery, V

    M. Hillery, V. Buˇzek, and A. Berthiaume, Quantum se- cret sharing, Phys. Rev. A59, 1829 (1999)

    work page 1999

  31. [31]

    Hill and W

    S. Hill and W. K. Wootters, Entanglement of a pair of quantum bits, Phys. Rev. Lett.78, 5022 (1997)

    work page 1997

  32. [32]

    Vidal and R.F

    G. Vidal and R.F. Werner, Computable measure of en- tanglement, Phys. Rev. A65, 032314 (2002)

    work page 2002

  33. [33]

    Wootters, Entanglement of formation of an ar- bitrary state of two qubits, Phys

    W.K. Wootters, Entanglement of formation of an ar- bitrary state of two qubits, Phys. Rev. Lett.80, 2245(1998)

    work page 1998

  34. [34]

    Carvalho, F

    A.R.R. Carvalho, F. Mintert and A. Buchleitner, Deco- herence and multipartite entanglement, Phys. Rev. Lett. 93, 230501 (2004)

    work page 2004

  35. [35]

    Qiang, G

    W. Qiang, G. Sun, Q. Dong, and S. Dong, Genuine mul- tipartite concurrence for entanglement of Dirac fields in noninertial frames, Phys. Rev. A98, 022320 (2018)

    work page 2018

  36. [36]

    S. Wu, X. Teng, J. Li, S. Li, T. Liu, and J. Wang, Genuinely accessible and inaccessible entangle- ment in Schwarzschild black hole, Phys. Lett. B848, 138334(2024)

    work page 2024

  37. [37]

    K. Kim, M. C. Pak, O. S. An, U. G. Ri, M. Ko and N. Kim, Quantum entanglement and coherence of tripartite W state for Dirac fields under noisy channels in non- inertial frames, Phys. Scr.97, 075101 (2022)

    work page 2022

  38. [38]

    Bennett, D.P

    C.H. Bennett, D.P. Di Vincenzo, J.A. Smolin, and W.K. Wootters, Mixed-state entanglement and quantum error correction, Phys. Rev. A54, 3824 (1996)

    work page 1996

  39. [39]

    Vedral and M.B

    V. Vedral and M.B. Plenio, Entanglement measures and purification procedures, Phys. Rev. A,57, 1619 (1998)

    work page 1998

  40. [40]

    de Vicente, Lower bounds on concurrence and sepa- rability conditions, Phys

    J.I. de Vicente, Lower bounds on concurrence and sepa- rability conditions, Phys. Rev. A75, 052320 (2007). 11

    work page 2007

  41. [41]

    Lee, D.P

    S. Lee, D.P. Chi, and J. Kim, Convex-roof extended naga- tivity as an extanglement measure for bipartite quantum systems, Phys. Rev. A68, 062304 (2003)

    work page 2003

  42. [42]

    Horodeki and P

    M. Horodeki and P. Horodeki, Reduction criterion of sep- arability and limits for a class of distillation protocols, Phys. Rev. A59, 4206 (1999)

    work page 1999

  43. [43]

    M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge: Cambridge Uni- versity Press)(2010). APPENDIX For the initial isotropic stateρ= 1−p 4 I⊗I+p|ψ +⟩⟨ψ+|, with two observers Alice and Bob located in the event horizon region and simultaneously affected to the channel noise, the four reduced density matrices obtained after...

    work page 2010