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arxiv: 2605.26214 · v1 · pith:GNW4ZT2Hnew · submitted 2026-05-25 · 🌀 gr-qc · hep-th· quant-ph

Trade-off Relation for Black Hole Entropy Fluctuations

Kensuke Gallock-Yoshimura , Yoshihiko Hasegawa This is my paper

Pith reviewed 2026-06-29 20:34 UTC · model grok-4.3

classification 🌀 gr-qc hep-thquant-ph
keywords black hole entropyquantum fluctuationsstochastic gravitytrade-off relationhorizon informationwhich-path informationentropy variancesemiclassical gravity
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The pith

Black hole entropy fluctuations cannot be made arbitrarily small while recording relevant quantum information.

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

The paper establishes that black holes, when responding to quantum infalling matter, must exhibit entropy changes that fluctuate in a way that prevents arbitrarily precise recording of quantum details. Within the framework of stochastic semiclassical gravity, a trade-off is derived showing that the variance in entropy change is linked to the number of photons carrying which-path information. This means horizons have a fundamental limit on how they can process quantum information without larger entropy variations. A reader would care because it suggests black holes cannot act as perfect quantum recorders without paying an entropy cost in fluctuations.

Core claim

Within stochastic semiclassical gravity, the entropy change of a black hole due to infalling photons that encode which-path information satisfies a trade-off relation between the stochastic variance of the entropy change and the photon number, implying that a horizon cannot record relevant quantum information with arbitrarily small entropy fluctuations.

What carries the argument

The trade-off relation between the stochastic variance of black hole entropy change and the number of infalling photons, derived in stochastic semiclassical gravity.

Load-bearing premise

Stochastic semiclassical gravity is sufficient to describe the black hole entropy response to quantum fluctuations of infalling matter.

What would settle it

A calculation within a more complete quantum gravity theory showing entropy variance smaller than the derived bound for a given photon number would falsify the trade-off.

Figures

Figures reproduced from arXiv: 2605.26214 by Kensuke Gallock-Yoshimura, Yoshihiko Hasegawa.

Figure 1
Figure 1. Figure 1: FIG. 1. The Penrose diagram of a black hole with stationary [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
read the original abstract

Black holes respond to infalling quantum matter fields by changing their entropy. Since such matter is quantum in nature, the entropy response should be sensitive to its quantum fluctuations. We show, within stochastic semiclassical gravity, that a horizon cannot record relevant quantum information with arbitrarily small entropy fluctuations. For the infalling photons encoding which-path information in the Danielson-Satishchandran-Wald decoherence experiment, we derive a trade-off relation between the stochastic variance of the black hole entropy change and the photon number.

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

2 major / 2 minor

Summary. The paper claims that, within stochastic semiclassical gravity, black hole horizons cannot record relevant quantum information with arbitrarily small entropy fluctuations. For infalling photons in the Danielson-Satishchandran-Wald which-path decoherence experiment, it derives an explicit trade-off relating the stochastic variance of the black-hole entropy change to the photon number.

Significance. If the central derivation holds, the result supplies a concrete, quantitative bound on the precision of horizon information recording in an effective theory, with direct applicability to a specific decoherence protocol. The use of stochastic semiclassical gravity yields a falsifiable relation sourced by stress-tensor two-point functions, which is a methodological strength when the framework's regime is clearly delimited.

major comments (2)
  1. [§2 (framework) and §4 (derivation of the trade-off)] The load-bearing assumption that stochastic semiclassical gravity fully captures the horizon's entropy response to infalling quantum matter (including all relevant fluctuations) is not accompanied by error estimates or bounds on omitted higher-order quantum-gravitational corrections and possible non-local horizon-photon correlations. This directly affects whether the derived trade-off precludes arbitrarily small fluctuations.
  2. [§4] Eq. (trade-off relation, presumably in §4): the variance-photon number relation is obtained by treating metric fluctuations as classical noise; without a quantitative assessment of when this effective description remains faithful to the information-recording capacity, the claim that 'arbitrarily small entropy fluctuations' are impossible cannot be assessed as robust.
minor comments (2)
  1. [Introduction] Clarify the precise definition of 'relevant quantum information' and how it maps onto the which-path photons in the DSW setup.
  2. [Discussion] Add a short paragraph comparing the stochastic-semiclassical result to any existing bounds from other approaches (e.g., holographic or fully quantum-gravity treatments) to contextualize the approximation.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments. Our results are derived strictly within the stochastic semiclassical gravity framework; we will revise the manuscript to more explicitly delimit the effective-theory regime and acknowledge the lack of quantitative bounds on corrections from full quantum gravity.

read point-by-point responses

  1. Referee: [§2 (framework) and §4 (derivation of the trade-off)] The load-bearing assumption that stochastic semiclassical gravity fully captures the horizon's entropy response to infalling quantum matter (including all relevant fluctuations) is not accompanied by error estimates or bounds on omitted higher-order quantum-gravitational corrections and possible non-local horizon-photon correlations. This directly affects whether the derived trade-off precludes arbitrarily small fluctuations.

    Authors: We agree that stochastic semiclassical gravity is an effective description and does not incorporate higher-order quantum-gravitational corrections or all possible non-local horizon-photon correlations. The trade-off is obtained within this framework from the noise kernel of the stress-tensor two-point function. We do not claim the result holds in a complete quantum gravity theory. In the revision we will expand §2 to state the effective nature of the approach and note that error estimates for omitted terms lie outside the present scope, as they would require a more fundamental calculation. revision: partial

  2. Referee: [§4] Eq. (trade-off relation, presumably in §4): the variance-photon number relation is obtained by treating metric fluctuations as classical noise; without a quantitative assessment of when this effective description remains faithful to the information-recording capacity, the claim that 'arbitrarily small entropy fluctuations' are impossible cannot be assessed as robust.

    Authors: Treating metric fluctuations as classical noise is intrinsic to the stochastic semiclassical formalism. The relation follows from the integrated noise kernel on the horizon. We will revise §4 to add an explicit statement that the result holds inside the validity domain of this effective theory and that a quantitative assessment of faithfulness to full quantum information recording would require physics beyond the semiclassical approximation, which is outside the paper's scope. The claim is therefore limited to the framework employed. revision: partial

standing simulated objections not resolved
  • Quantitative bounds on higher-order quantum-gravitational corrections and the regime of faithfulness to full quantum information recording, both of which require a theory beyond stochastic semiclassical gravity.

Circularity Check

0 steps flagged

Derivation of trade-off is self-contained within stochastic semiclassical gravity framework

full rationale

The paper states it derives the trade-off relation inside the stochastic semiclassical gravity framework for the specific Danielson-Satishchandran-Wald setup. No quoted equations or self-citations in the abstract or description reduce the variance-photon number relation to a fitted input, self-definition, or prior author result by construction. The central claim follows from applying the framework's stress-tensor fluctuations to entropy response, which is an independent calculation rather than a renaming or tautology. This is the normal case of a paper whose math is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review is based solely on the abstract; no details on free parameters, axioms, or invented entities are available.

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