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How to Minimize the Decoherence Caused by Black Holes

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read An echo protocol recovers quantum coherence already lost to a black hole.

desk verdict New closed-form optimal continuation for horizon decoherence, but the global optimality claim needs proof; worth refereeing. read the letter →

arxiv 2501.04773 v2 pith:2USUSAK5 submitted 2025-01-08 hep-th gr-qcquant-ph

Daine L. Danielson , Jonah Kudler-Flam , Gautam Satishchandran , Robert M. Wald This is my paper · ORCID
classification hep-thgr-qcquant-ph
keywords blackholedecoherenceoptimalpurificationquantumfidelityKillinghorizonHartle-Hawkingvacuumsoftradiationcoherentstatesmemoryeffect
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

Alice's quantum superposition outside a black hole inevitably loses coherence because radiation encoding which-path information falls through the event horizon. This paper solves the reverse problem: given the radiation $f$ that has already crossed the horizon by a time $t_c$, what should Alice emit afterward to recover as much coherence as possible? The answer is an explicit integral formula: reflect $f$ about the horizon cross-section $\mathcal{C}$ and convolve with a sech kernel, Eq. (34). The paper shows that even after Alice has closed her superposition she can reopen it and emit this 'optimal purification' radiation to improve fidelity, with the recoverable fraction decaying as $e^{-\kappa(t-t_c)}$. If correct, the formula gives the exact boundary between decoherence that is irrevocable by time $t_c$ and decoherence that can still be undone.

What carries the argument

The central object is the Klein-Gordon norm of the positive affine-frequency part of $f+g$ on the horizon null surface $H$, treated as an initial-data surface. Minimizing this norm over continuations $g$ is equivalent to maximizing the coherent state's overlap with the vacuum, since the overlap of a coherent state with the vacuum is controlled by the norm of its one-particle component. The calculation uses an orthonormal basis of Rindler wave packets $\{ \phi^I_{jk}, \phi^{II}_{jk} \}$ and the Unruh-type combinations $F^1_{jk}, F^2_{jk}$ that are purely positive affine frequency; the optimal mode condition $g_{jk} = \operatorname{sech}(\pi\omega_j/\kappa)\, \tilde f_{jk}$ then assembles, via a Fourier convolution, into the integral formula Eq. (34). The step from a bifurcate Killing horizon to a black hole formed by collapse is carried by replacing the bifurcation surface $B$ with the arbitrary cross-section $\mathcal{C}$ and defining a pseudo-Killing time $v$ through $V - V_c = e^{\kappa v}$, so that the same frequency decomposition applies to the future of $\mathcal{C}$.

What would settle it

Take a simple $f$ supported on the past of $\mathcal{C}$ with finite Klein-Gordon norm and numerically search over general (non-coherent) pure states of the future-horizon radiation for higher vacuum fidelity than Eq. (34) attains; a positive result would refute the optimality claim. A second check is to test the completeness of $\{F^1_{jk},F^2_{jk}\}$ by trying to approximate, in Klein-Gordon norm, a compactly supported positive affine-frequency solution that does not vanish at $V=0$ with solutions that do, since the paper's extension to $f(0)\neq 0$ rests on that density.

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Extended reading notes

Core claim

The paper's central result is that, given the entangling radiation $f$ that has already crossed the horizon before the cross-section $\mathcal{C}$ (time $t_c$ in Alice's lab), the optimal continuation $g$ for $t>t_c$ is the coherent radiation obtained by reflecting $f$ about $\mathcal{C}$ and convolving with a sech kernel: $g(v,y) = \frac{\kappa}{2\pi}\int_{-\infty}^{\infty} \operatorname{sech}[\kappa(v-v')/2]\, \tilde f(v',y)\, dv'$, where $v$ is the pseudo-Killing time defined by $V - V_c = e^{\kappa v}$ and $\tilde f$ is the reflection of $f$ about $V=V_c$. This choice minimizes the Klein-Gordon norm of the positive affine-frequency part of $f+g$, which is equivalent to maximizing the overlap of the purified state with the Hartle-Hawking or Unruh vacuum. In the mode decomposition, the optimal filter multiplies each reflected Rindler-frequency mode by $\operatorname{sech}(\pi\omega/\kappa)$: low-frequency modes are almost exactly reflected ('CRT purification') while high-frequency modes leave the vacuum untouched ('Minkowski purification'). Alice realizes this purification by re-opening her superposition and emitting $g$; doing so strictly increases the fidelity of her particle's state even after the superposition has been closed, although the additional recovery is bounded by $e^{-\kappa(t-t_c)}$ at late times. The derivation is carried out for coherent-state purifications, with the authors arguing this is also optimal among all purifications.

Load-bearing premise

The load-bearing premise is that the optimal purification can be found among coherent states (and that the mode set $\{F^1_{jk},F^2_{jk}\}$ is complete), so if a non-coherent purification achieves higher vacuum fidelity for the same past-horizon radiation, Eq. (34) is not the true global optimum.

Editorial extensions

If this is right

  • Acting on the abort order at $t_c$, Alice can recover additional coherence by re-opening her superposition and emitting $g$ from Eq. (34); the gain over doing nothing is largest for low Rindler frequencies $\omega \ll \kappa$.
  • The recoverable fraction decays exponentially with delay: for $t - t_c \gg \kappa^{-1}$, $|g(t)| \le (1/\pi) e^{-\kappa(t-t_c)} \max|f|$, so acting promptly matters.
  • Even if Alice closed her superposition before $t_c$, emitting the optimal $g$ still reduces decoherence; in the paper's examples, immediate action lowers the particle-number expectation by about $1.3\%$, while acting one Killing time later lowers it by only $0.07\%$.
  • For superpositions held open for $T \gg \kappa^{-1}$, the optimal recovery is too small to change the exponential-in-$T$ decoherence estimates of earlier work, so the known black-hole decoherence rates stand.
  • The optimal recovery channel saturates the information-disturbance tradeoff for this setup, meaning the coherence Alice can salvage is exactly the information about her superposition that no behind-the-horizon observer could have extracted before $\mathcal{C}$.

Reading between the lines

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

  • If the coherent-state restriction ever fails, Eq. (34) still supplies a rigorous upper bound on recoverable coherence, and the $e^{-\kappa(t-t_c)}$ decay timescale for the irreversible part is likely to survive; a finite-dimensional or analogue-horizon experiment could test whether non-coherent recovery beats this bound.
  • The paper states that generalization to Maxwell and linearized gravitational fields is straightforward; writing the explicit sech-kernel recovery in terms of horizon free data for each spin would make the result directly usable for proposals that superpose massive bodies.
  • The pseudo-Killing construction assumes a stationary horizon and a fixed vacuum; for evaporating or time-dependent horizons, the recovery window and the meaning of 'vacuum overlap' would need modification, so the formula's quantitative reach beyond stationary black holes is an open question.
  • Because the same analysis applies to null infinity with arbitrary $\kappa$, the protocol suggests that any causal horizon, not just a black hole, imposes the same sech-filtered echo as the optimal coherence-recovery strategy; checking this in an accelerated-mirror analogue would be a concrete test.
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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 / 5 minor

Summary. The paper addresses the question of how much decoherence of a quantum superposition near a black hole has irrevocably occurred by a time tc. Given the entangling radiation f that has crossed the horizon before the cross-section C, the authors solve a variational problem for the continuation g on the future horizon that maximizes the overlap of the global coherent state with the Hartle-Hawking or Unruh vacuum. The solution is g(v,y) = (κ/2π) ∫ sech[κ(v−v')/2] \tilde f(v',y) dv' (Eq. 34), where \tilde f is the reflection of f about C. They show that Alice can recover a fraction of coherence even after closing her superposition, but this fraction decays as e^{−κ(t−tc)} (Eq. 37). The derivation is based on a Rindler-space calculation and its transfer to black holes via a pseudo-Killing parameter v defined by Eq. (33).

Significance. If the claimed optimality holds, the paper gives a parameter-free, closed-form answer to a concrete optimization problem in black hole decoherence, with explicit examples and a clear qualitative prediction of exponential decay of recoverability. The core calculation from Eq. (14) to Eq. (26) is internally consistent, and the paper is careful to state many of its assumptions. The result also connects to information-disturbance tradeoffs, providing a bridge between quantum information and semiclassical gravity. The explicit acknowledgment that the coherent-state restriction is a belief but not a proof is a notable strength in transparency, even though it is also the main weakness.

major comments (3)
  1. [Sec. II, first paragraph and Eq. (20)] The global optimality claim is not established. The optimization is restricted to coherent-state purifications, with the sentence 'We will restrict our analysis of the optimal purification to coherent states, but we believe that the optimal coherent state that we will find will be optimal among all possible purifications' being an assertion rather than a theorem. Since the abstract and Sec. IV claim to have determined 'the optimal purification' and the true Uhlmann fidelity, this gap is load-bearing. Please either provide a proof (e.g., using the Gaussian structure of the reduced states) or weaken the claim to optimality within the coherent-state family and adjust the interpretation of 'irrevocable decoherence' accordingly.
  2. [Sec. II, paragraph after Eq. (31)] The completeness of the mode set {F1_jk, F2_jk} is essential for Eq. (34) to be the true minimum over all continuations, but the argument given is a plausibility sketch that cites Lemma 5.1 of [37] without demonstrating that the lemma's hypotheses are satisfied in this setting. The step 'It follows that...' should be expanded into a rigorous proof, or the unproven completeness should be stated explicitly as a standing assumption.
  3. [Sec. III, paragraph before Eq. (34)] The transfer from the Rindler bifurcate horizon to a physically formed black hole relies on the pseudo-Killing parameter v and is stated without detailed justification. In particular, the horizon is not a complete bifurcate Killing horizon, and the Unruh state differs from the Hartle-Hawking state; the paper only cites an earlier low-frequency equivalence. Please specify the exact domain of validity of Eq. (34) for the Unruh state and justify that the mode decomposition in v captures all relevant degrees of freedom on the future horizon.
minor comments (5)
  1. [Abstract and Sec. III] The abstract claims results for 'the Hartle-Hawking or Unruh vacuum' without qualification, but Sec. III states that the Unruh validity holds only at low Killing frequencies. Please make this caveat consistent across the abstract and the main text.
  2. [Sec. II, Eq. (12) and Eq. (14)] The sign conventions in the Klein-Gordon inner product and the basis expansion are terse; a brief remark on the orientation of the transverse coordinates y and the meaning of 'c.c.' would improve readability.
  3. [Sec. IV, Eq. (41)] The channel fidelity identity is stated without proof or precise definitions of the channels N and Nc. Please give the definitions used or a clear reference so that the connection to the variational calculation is transparent.
  4. [Page 7, sentence after Eq. (19)] The text contains a duplicated word: 'Equation Equation (19)'. Fix the typo.
  5. [Sec. IV, first paragraph] The term 'CRT purification' is introduced without prior definition; consider defining it earlier or adding a parenthetical explanation at first use.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (34) is the solution of a self-contained variational minimization; self-citations are contextual and not load-bearing.

full rationale

The derivation of the central result, Eq. (34), is a direct variational calculation: given f on the past horizon, the authors minimize the expression in Eq. (20) over coefficients g_jk, obtaining g_jk = sech(pi*omega_j/kappa) f*_jk (Eq. 22), and then convert this to a convolution with a sech kernel via the inverse Fourier transform (Eq. 28). No parameter is fitted to the quantity being predicted; the only inputs are the Klein-Gordon inner product, standard Rindler mode functions, and the physical surface gravity kappa. The transfer to a collapsing black hole uses an explicitly defined pseudo-Killing time v (Eq. 33) and applies the same minimization; this is an analogy/construction, not a renaming of an assumed answer. The completeness of the mode set {F1_jk, F2_jk}, needed for the non-vanishing-at-C case, is supported by an external lemma (Lemma 5.1 of Brasco et al. [37]), not by an author-specific uniqueness theorem. The paper does cite the authors' prior work [15,16,19] for the physical setup, the decoherence rate Eq. (4), and the low-frequency Hartle-Hawking/Unruh equivalence, but those results are input premises, and the optimal-purification formula does not reduce to them. The explicit restriction to coherent states ('We will restrict our analysis of the optimal purification to coherent states...') is an admitted proof gap for global optimality, not a circular step, because the optimization is still solved within the stated class and no answer is smuggled in through that restriction. No fitted-input-called-prediction, self-definitional, or author-imported-uniqueness pattern is present.

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

No parameters are fitted to data. The surface gravity kappa is a physical input; the illustrative parameters c, t1, T, tc in Eqs. (38)-(39) are examples, not fitted values. No new physical entities are introduced; the 'pseudo-Killing time' v is a coordinate reparameterization, not a new force, particle, or conserved quantity. The main unproven premises are the coherent-state restriction, the completeness of the mode basis, and the transfer from Rindler to collapsing black hole geometries.

assumptions (6)
  • domain assumption Null surface H can serve as an initial data surface for solutions with suitable falloff, so f and g can be characterized by their restrictions to H.
    Invoked in Sec. I and II with citation [32] to set up the variational problem on the horizon; standard in the null-initial-data formulation but not proved here.
  • standard math Unruh-mode decomposition: the sets {phi_I} and {phi_II} are orthonormal bases and the combinations F1,F2 in Eq. (17) are positive affine-frequency and orthonormal.
    Used in Sec. II to transform the norm; standard construction from Wald's QFT in curved spacetime [35], but the limit epsilon to 0 and the exactness of positive frequency are only sketched.
  • standard math Completeness of {F1,F2} in the positive affine-frequency Hilbert space.
    At end of Sec. II, the paper invokes Lemma 5.1 of [37] to argue that functions vanishing at V=0 are dense in the KG norm; this is load-bearing for the claim that the minimization over the basis is the global minimization.
  • ad hoc to paper The optimal coherent purification is optimal among all purifications.
    Stated as a belief in Sec. II, first paragraph; not proven. The abstract relies on this for the word 'optimal'.
  • ad hoc to paper The Rindler optimization transfers to a stationary black hole horizon formed by collapse via the pseudo-Killing parameter v defined by V - Vc = e^{kappa v}.
    Sec. III, Eq. (33); the paper argues that only affine translation invariance of the horizon matters, but this is not a rigorous derivation.
  • domain assumption Neglect of radiation to infinity, evaporation, and assumption of perfect control over Alice's emitted radiation.
    Sec. III: 'We will... neglect any radiation to infinity' and 'if we imagine that she can perfectly control the entangling radiation'; these set the idealized scenario.

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Pith. "Pith review of How to Minimize the Decoherence Caused by Black Holes." pith.science (2026). https://pith.science/paper/2USUSAK5

@misc{pith2026250104773,
  author       = {Pith},
  title        = {Pith review of: How to Minimize the Decoherence Caused by Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2USUSAK5}},
  note         = {Machine review of arXiv:2501.04773}
}
abstract

We consider an experimentalist, Alice, who creates a quantum superposition of a charged or massive body outside of a black hole (or, more generally, in the presence of a Killing horizon). It was previously shown that emission of soft photons/gravitons into the black hole will result in the decoherence of the components of the superposition if it is held open for a sufficiently long span of time. However, at any finite time, $t_c$, during the process, it is not obvious how much decoherence has irrevocably occurred. Equivalently, it is not obvious how much information an observer inside the black hole can extract about Alice's superposition prior to time $t_c$. In this paper, we solve for the optimal experimental protocol to be followed by Alice for $t > t_c$ so as to minimize the decoherence of the components of her superposition. More precisely, given the entangling radiation that has passed through the horizon prior to the cross-section $\mathcal C$ corresponding to the time $t = t_c$ in Alice's lab, we determine the "optimal purification" of this radiation beyond $\mathcal C$ such that the global quantum state of the radiation through the horizon has maximal overlap (quantum fidelity) with the Hartle-Hawking or Unruh vacuum. Due to the intricate low frequency entanglement structure of the quantum field theory vacuum state, we find this optimal purification to be nontrivial. In particular, even if Alice has already "closed" her superposition by bringing the components back together, we find that she can decrease the net decoherence of the components of her superposition somewhat by re-opening it and performing further manipulations.

Figures

Figures reproduced from arXiv: 2501.04773 by the authors.

Figure 2
Figure 2. FIG. 2. A spacetime diagram showing the exterior region of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The optimal “ramp down” is shown in orange for [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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    Satishchandran reviews three equivalent mechanisms by which black holes and other Killing horizons decohere nearby quantum superpositions, via interior entanglement, soft radiation, and fluctuating multipoles.

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