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The Hayden–Preskill recovery threshold jumps from a constant to Θ(n^{2/3}) when the decoder loses the absolute positions of collected qubits.

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2026-08-01 23:01 UTC pith:NF43JXAB

load-bearing objection A genuine new scaling law for Hayden–Preskill recovery when subsystem labels are erased; the Θ(n^{2/3}) threshold is well supported and the paper deserves a serious referee despite a few lemma-level typos. the 2 major comments →

arxiv 2607.15537 v1 pith:NF43JXAB submitted 2026-07-17 quant-ph

Microscopic Side Information Controls Ordered Hayden--Preskill Recovery

Joao V. R. Alencar , Allan R. P. Moreira , Joao B. R. Silva This is my paper
classification quant-ph MSC 81P4581P4060C0594A17 PACS 03.67.-a03.67.Mn04.70.Dy
keywords Hayden–Preskill protocolordered deletion channelquantum recoverydecouplingplanted subsequencemicroscopic side informationorder statisticsChoi purity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper shows that erasing the absolute positions of received qubits changes the Hayden–Preskill recovery problem from a constant-size task into one that needs Θ(n^{2/3}) output qubits. The channel it studies keeps a uniformly random set of ℓ survivors from an n-qubit scrambled register but presents them only in their original relative order; the decoder never learns their microscopic locations. The authors prove a sharp threshold: when ℓ = o(n^{2/3}), no recovery map can beat the trivial no-output fidelity 4^{-k} for any scrambling unitary, and when ℓ = ω(n^{2/3}) (still o(n)), a Haar-random scrambler allows near-perfect recovery with probability tending to one. The exponent 2/3 is traced to the order statistics of random subsets: rank-aligned coincidences between two independent subsequences have total strength ℓ^{3/2}/n, which becomes of order one exactly at ℓ ≈ n^{2/3}. A block-resolved variant, in which the decoder is told which of B consecutive blocks each survivor came from, interpolates between the no-label law and the labeled law, with recovery scale n^{2/3}B^{-1/3} for coarse blocks and n/B for fine blocks.

Core claim

The paper's central claim is a recovery phase transition for the ordered deletion channel acting on an n-qubit scrambled register. When the decoder receives ℓ surviving qubits in their original relative order without their absolute positions, the optimal entanglement fidelity for a fixed k-qubit diary converges to the no-output value 4^{-k} for all scrambling unitaries once ℓ = o(n^{2/3}); if ℓ = ω(n^{2/3}) and ℓ = o(n), the same fidelity converges to 1 for Haar-random scrambling unitaries, implying the operational recovery scale ℓ_rec = Θ(n^{2/3}). The paper also derives the block-resolved phase diagram: revealing only which of B consecutive blocks each survivor came from changes the recove

What carries the argument

The central object is the ordered deletion channel D_{n→ℓ}: a random ℓ-subset of positions survives, the qubits are reordered canonically, and the set S is hidden. The proof is carried by an exact identity for the channel's normalized Choi-state purity, G_{n,ℓ} = E 4^{M_{n,ℓ}}, where M_{n,ℓ} counts rank-aligned coincidences between two independent random ℓ-subsets; because both subsets are increasing, only same-rank overlaps form closed contraction loops. The controlling scale is the alignment strength V_{n,ℓ} = ∑_{a,i} p_{a i}^2 ≍ ℓ^{3/2}/n, the total collision probability of the order-statistic kernel. In the subcritical regime this quantity forces the Choi state to be maximally mixed (uni

Load-bearing premise

The supercritical recovery theorem holds in probability over Haar-random scrambling unitaries; if the scrambler is fixed adversarially (e.g., the identity), ordered deletion may reveal nothing, so no sublinear recovery length is guaranteed.

What would settle it

Run ordered-deletion recovery at ℓ = n^{2/3} log n with Haar-random U: Theorem 1 predicts entanglement fidelity →1, so seeing fidelity stuck near 4^{-k} would falsify the supercritical claim. At ℓ = n^{2/3}/log n, any scrambling unitary must stay at baseline 4^{-k}; a single unitary beating it would falsify the universal converse.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For the ordered deletion channel, the fixed-error recovery length is Θ(n^{2/3}) for any fixed diary size k — a mesoscopic scale that grows with n but remains far smaller than n.
  • Below the n^{2/3} scale, every recovery map, even optimized per unitary, is asymptotically equivalent to discarding the output: the optimal fidelity is pinned to 4^{-k} uniformly over scramblers.
  • Above the scale, a Haar-random scrambler gives near-perfect recovery with probability tending to one; the guarantee is probabilistic in the scrambler, not worst-case.
  • Revealing only coarse block labels (B blocks) changes the law to n^{2/3}B^{-1/3} for B ≤ √n and to n/B for B ≥ √n, so each extra bit of coarse position label reduces the required output by a fixed factor.
  • The permutation-twirled chronological channel has a different symmetry class with an exact Pauli-degree spectrum and purity limit (1-α)^{-3}, showing that the notion of order itself changes the channel.

Where Pith is reading between the lines

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

  • Inference: The same alignment-strength statistic should control recovery under any scrambling ensemble that is close enough to Haar for second moments; random circuits of sufficient depth would likely exhibit the same n^{2/3} threshold, making the prediction testable on near-term devices.
  • Inference: Because V_{n,ℓ} ≍ ℓ^{3/2}/n assumes a uniformly random deletion set, non-uniform deletion processes (e.g., clustered or adversarial losses) should shift the exponent; measuring the recovery curve could therefore serve as a diagnostic of deletion statistics.
  • Inference: The block-resolved phase diagram suggests a whole family of side-information resolutions; noisy or probabilistic labels should interpolate between the n^{2/3} and labeled regimes, and the optimal trade-off may be governed by the same collision quantity.
  • Inference: The planted-subsequence separation used here is a classical hypothesis-testing problem; it may transfer to classical synchronization channels, where ordered but unlabeled outputs could exhibit an analogous ℓ^{3/2}/n transition in detection performance.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies a Hayden–Preskill variant in which the decoder receives ℓ output qubits in their original relative order but without their microscopic position labels. For a fixed diary size k, it proves that for the resulting ordered deletion channel the optimal entanglement fidelity tends to the no-output baseline 4^{-k} whenever ℓ=o(n^{2/3}), uniformly over the scrambling unitary, and tends to 1 for ℓ=ω(n^{2/3}), ℓ=o(n), in probability over a Haar-random scrambling unitary. Monotonicity then gives the fixed-error recovery length ℓ_rec=Θ(n^{2/3}). A block-resolved version with B consecutive block labels gives ℓ_rec(n,B)≍ n^{2/3}B^{-1/3} for B≤√n and n/B for B≥√n. The proofs are based on an exact Choi-purity identity expressed as E4^{M_{n,ℓ}}, factorial-moment estimates for rank-aligned coincidences, a planted-subsequence separation argument, and the joint state–channel decoupling theorem of Cheng–Dupuis–Gao.

Significance. If correct, the paper provides a clean information-theoretic demonstration that microscopic subsystem identity is an operational resource in Hayden–Preskill recovery: removing the position labels changes the recovery scale from O(1) to Θ(n^{2/3}). The order-statistic mechanism (rank-aligned collisions of strength ℓ^{3/2}/n) is well identified and appears on both the converse and achievability sides. The appendices give detailed, self-contained proofs, and the derivations are parameter-free in the sense that no quantities are fitted to data. The block-resolved phase diagram is a useful additional result. The main caveat is that the supercritical recovery statement is a typicality result over Haar-random unitaries, not a uniform-in-U guarantee; the paper is explicit about this in its definitions, but the presentation would benefit from stating this limitation more prominently.

major comments (2)
  1. [Appendix A.2, Lemma 7 (Eqs. A6–A7)] The stated bounds are incorrect as written: the proof yields sup_z b_{d,p}(z) ≤ C p/√(dq) and b_{D,p}(N) ≥ c p/√(Dq), not the displayed C p√(dq) and c p√(Dq). The displayed versions are not only inconsistent with the proof but in some parameter regimes false (e.g., the stated lower bound fails at D=100, p=1/2, N=D−1). Moreover, Lemma 9's derivation, especially Eq. (A16), explicitly uses the p/√(dq) form. The statement must be corrected and then used consistently.
  2. [Section III and IV, Theorem 1(ii)/Corollary 2] The supercritical achievability holds only in probability over a Haar-random scrambling unitary, and this is not a removable technicality: a fixed permutation that moves the k diary qubits to the last k positions leaves the output independent of the diary with probability 1−o(1) whenever ℓ=o(n), so no uniform-in-U supercritical law can hold. The paper's definitions (14)–(15) already encode this by defining ℓ_rec probabilistically, but the abstract and the phrase 'the fixed-error recovery law ℓ_rec=Θ(n^{2/3})' should be qualified explicitly as a typicality statement over U. I recommend adding a short remark in Section VI giving the permutation counterexample and explaining why the n^{2/3} law is not a guarantee for a fixed non-scrambling unitary.
minor comments (5)
  1. [Appendix A.3, Eq. (A44)] The third condition in (A44) is typeset ambiguously as 'a_n p ℓ/n'; from the proof it should be a_n√(n/ℓ) → ∞. Please correct the notation to avoid confusion.
  2. [Section II, 'scrambling unitary'] The term 'scrambling unitary' is used informally. In Theorem 1(i), 'uniformly over the scrambling unitary' appears to mean uniformly over all unitaries U, not only a subset of 'scrambling' unitaries; this should be stated explicitly.
  3. [Figure 1] The axes labels are somewhat hard to read and the exponent notation in the caption appears garbled ('x 1/3' and 'x 1'). Please ensure the scaling branches are displayed as x^{-1/3} and x^{-1}, matching the text.
  4. [Section I, Eq. (2)] The alignment strength V_{n,ℓ} ≍ ℓ^{3/2}/n is introduced informally before its formal definition in Lemma 11. A forward reference would help the reader.
  5. [Appendix A.4, Proposition 16] The step 'Monotonicity in the Rényi order gives I_{2/3}^S ≥ I_{1/2}^S' would benefit from a citation or a one-line justification, since monotonicity of the optimized Sibson information is not as immediate as monotonicity of the unoptimized divergence.

Circularity Check

0 steps flagged

No significant circularity: the n^{2/3} law is derived from order-statistic collision bounds and external decoupling results, not from fitted inputs or self-citations.

full rationale

The paper's derivation chain is self-contained at the level of its claimed theorems. The subcritical converse rests on the exact Choi-purity identity (A5), G_{n,ell}=E 4^{M_{n,ell}}, and on the uniform factorial-moment bound (A11), both proved in Appendix A from elementary combinatorics of ordered subsets; no parameter is fitted to the target fidelity. The supercritical achievability uses a planted-subsequence model, a weighted alignment score (A33), Freedman/Hanson-Wright concentration (Lemmas 12-15), and the external joint state-channel decoupling theorem [18]; the Haar-random quantification is explicit in the statement of Theorem 1(ii) and is not silently presented as a uniform guarantee. The block-resolved results are derived from the same single-block collision bounds (B5) and a deterministic lower bound (B8) on X(L), so the phase diagram s(n,B) is obtained by solving Lambda_{n,ell,B} ~ 1 rather than by assuming it. There are no author self-citations at all: all cited theorems (decoupling, Renyi data processing, Hanson-Wright, Freedman) are due to other authors and are standard external tools. The only caveat raised in the reader's take—that supercritical recovery is typical over the scrambling unitary rather than uniform—is a limitation of the theorem's quantification, not a circular step, and the paper states this probability quantification explicitly. No load-bearing step reduces to its own input.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper introduces no free parameters or invented physical entities. It uses standard axioms of quantum information theory, the HP state model, Haar-random scrambling, and known decoupling/Rényi tools. The only non-standard input is the ordered-deletion channel definition itself, which is the object under study rather than a way to force the conclusion.

axioms (5)
  • domain assumption The Hayden–Preskill input state ρ_CR = Φ_MR ⊗ π_H (Eq. 4) with E purifying H.
    The old-black-hole model is the standard HP setup; the recovery analysis depends on this specific state.
  • domain assumption Haar-random scrambling unitary for the supercritical achievability.
    Theorem 1(ii) and Theorem 3(ii) are statements in probability over Haar-random U; this randomness is essential for the decoupling argument.
  • domain assumption Joint state–channel decoupling theorem of Cheng, Dupuis, and Gao (Ref. [18]).
    The final step lifting classical separation to quantum recovery uses this one-shot decoupling result as a black box.
  • standard math Sandwiched Rényi divergence data processing inequalities (Refs. [20,22,23]).
    Used in Proposition 16 and the block-resolved achievability to transfer classical information separation to the Choi state.
  • standard math Hanson–Wright and Freedman concentration inequalities (Refs. [24,25]).
    Control the fluctuations of Rademacher quadratic forms and martingales in the planted-subsequence argument.

pith-pipeline@v1.3.0-alltime-deepseek · 25668 in / 24525 out tokens · 215274 ms · 2026-08-01T23:01:55.489323+00:00 · methodology

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read the original abstract

In the Hayden--Preskill protocol, the decoder is usually assumed to know the microscopic identity of the collected output qubits. We study what happens when these labels are unavailable and only the relative order of the received qubits is preserved. The resulting order-preserving deletion channel maps an $n$-qubit scrambled register to a subsequence of length $\ell$. For a diary of fixed size $k$, we prove that the optimal entanglement fidelity converges to the no-output value $4^{-k}$ when $\ell=o(n^{2/3})$, uniformly over the scrambling unitary. For a Haar-random scrambling unitary, it converges to one when $\ell=\omega(n^{2/3})$ and $\ell=o(n)$. Monotonicity then gives the fixed-error recovery scale $\ell_{\mathrm{rec}}=\Theta(n^{2/3})$. We also consider partial position information obtained by dividing the register into $B$ consecutive blocks and revealing the block of origin of each received qubit. The recovery scale becomes $\ell_{\mathrm{rec}}(n,B)\asymp n^{2/3}B^{-1/3}$ for $B\leq\sqrt n$ and $\ell_{\mathrm{rec}}(n,B)\asymp n/B$ for $B\geq\sqrt n$. The exponent $2/3$ is traced to rank-aligned coincidences between random subsequences, which control both the converse and the recovery argument. Thus, even purely classical information about the origin of the received subsystems can change the amount of quantum output required for Hayden--Preskill recovery.

Figures

Figures reproduced from arXiv: 2607.15537 by Allan R. P. Moreira, Joao B. R. Silva, Joao V. R. Alencar.

Figure 1
Figure 1. Figure 1: FIG. 1. Scaling form of the fixed-error recovery phase diagram. With [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

51 extracted references · 28 linked inside Pith

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    Model and main statement The ordered deletion channel is Dn→ℓ(ρ) = n ℓ −1 X S⊂[n] |S|=ℓ JS TrSc (ρ)J † S.(A1) Thus the surviving qubits are presented in their original relative order, while the set S is not revealed. Logarithms and entropies are measured in bits unless stated otherwise. Theorem 5(Ordered-deletion Hayden–Preskill threshold).Fix the diary s...

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    Operational fixed-error threshold The degradation identity Dn→ℓ =D ℓ+1→ℓ ◦ Dn→ℓ+1 shows thatF opt e (U;ℓ) is nondecreasing inℓ. Fix ε and δ as in (14). Suppose first that no positive constant lower bound in (19) existed. There would then be a sequencen j such that ℓrec(nj;ε, δ) n2/3 j − →0. The universal subcritical theorem implies that, along this sequen...

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