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Entanglement asymmetry in the Hayden-Preskill protocol

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Pith's one-line read The paper derives the averaged Rényi-2 entanglement asymmetry of the emitted radiation in the Hayden-Preskill protocol and finds a sharp transition to an emergent U(1) symmetry at $N_B=(N+N_A+s)/2$.

desk verdict A clean, modest extension of the emergent-symmetry result to mixed-state Hayden-Preskill, with a correctly computed threshold and a soft spot in the log-average concentration step. read the letter →

arxiv 2411.17695 v2 pith:BTECGPSA submitted 2024-11-26 hep-th cond-mat.stat-mechquant-ph

Hui-Huang Chen , Zi-Jun Tang This is my paper · ORCID
classification hep-thcond-mat.stat-mechquant-ph
keywords entanglementasymmetryHayden-PreskillprotocolblackholeradiationemergentsymmetryRényi-2entropydecouplinginequalityrandomunitaryU(1)
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

This paper asks whether the radiation emitted by an old black hole can acquire an emergent symmetry as information escapes, in the Hayden-Preskill setup where Alice's diary, maximally entangled with a reference, is thrown into a black hole that is already entangled with early radiation. The authors compute the averaged Rényi-2 entanglement asymmetry of the radiation and find a sharp transition: a U(1) symmetry of the radiation emerges, with vanishing asymmetry, until the radiation size crosses $N_B=(N+N_A+s)/2$, after which the asymmetry saturates to $\frac{1}{2}\log(\pi N_B)$. The transition point depends on the diary size $N_A$ and the initial Rényi-2 entropy $s$ of the black hole, and when the black hole starts maximally mixed, the emergent symmetry persists throughout evaporation. The mechanism is traced to a decoupling inequality, showing that before the transition the radiation is almost maximally mixed and therefore carries no symmetry-breaking signature. If correct, this gives a simple diagnostic of when information about a thrown-in diary becomes accessible in the outgoing radiation.

What carries the argument

The load-bearing object is the Rényi-2 entanglement asymmetry $\Delta S^{(2)}(\rho_B) = \log\operatorname{tr}(\rho_{B,Q}^2) - \log\operatorname{tr}(\rho_B^2)$, where $\rho_B$ is the reduced state of the emitted radiation and $\rho_{B,Q}$ is obtained by deleting off-diagonal blocks between charge sectors of a U(1) generator $Q$ counting excitations. The calculation proceeds by Haar-averaging the two purities using a four-point unitary integral, summing over charge sectors with binomial dimensions $\binom{N_B}{q}$, and then using Stirling's formula; the transition location $(N+N_A+s)/2$ emerges from comparing the two purity terms. A decoupling inequality bounds the averaged trace distance between $\rho_B$ and the maximally mixed state by $2^{-(N+N_A+s-2N_B)/2}$, which is the paper's explanation for why the symmetry is exact before the transition.

What would settle it

Compute the exact Haar average of $\log\operatorname{tr}(\rho_B^2)$ and $\log\operatorname{tr}(\rho_{B,Q}^2)$ for modest qubit numbers (e.g., $N \le 12$) by direct numerical integration or Monte Carlo sampling of random unitaries; if the exact averages differ materially from the logarithms of the averaged purities near $N_B=(N+N_A+s)/2$, the sharp transition claimed in (5.5) is an artifact of the approximation rather than a finite-system property.

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Extended reading notes

Core claim

On the authors' own terms, the central result is the closed-form average in equation (5.5): $E[\Delta S^{(2)}(\rho_B)] = -\log\left(1 + (1/\sqrt{\pi N_B}-1)/(2^{N+N_A+s-2N_B}+1)\right)$ in the thermodynamic limit, with $s=-\log_2 \operatorname{tr}(\rho^2)$ the initial Rényi-2 entropy of the black hole. They read from this expression that for $N_B < (N+N_A+s)/2$ the averaged asymmetry is essentially zero, meaning the reduced state of the radiation obeys an emergent U(1) symmetry, exactly in the thermodynamic limit, while for larger $N_B$ it rises sharply to the value $\frac{1}{2}\log(\pi N_B)$. They also show the special case of a maximally mixed initial black hole, $s=N-N_A$, gives identically vanishing asymmetry at every radiation size, and they reproduce the transition condition from a decoupling inequality on the trace distance between the radiation state and the maximally mixed state.

Load-bearing premise

The sharp transition is obtained by approximating the average of a logarithm by the logarithm of an average; if the two purities fluctuate strongly around their means at finite system size, the step would be smeared out even though a crossover remains.

Editorial extensions

If this is right

  • Before the transition the radiation state is nearly maximally mixed, so any U(1) charge imbalance in the infalling diary is washed out; the emergent symmetry is exact in the thermodynamic limit.
  • After the transition the Rényi-2 entanglement asymmetry saturates at $\frac{1}{2}\log(\pi N_B)$, matching the known maximal value for product states at $\alpha=2$.
  • The transition time depends on both the diary size and the initial black hole entropy, so the appearance of the symmetry is shifted away from the Page time by $\frac{1}{2}(N_A+s)$.
  • When the initial black hole is maximally mixed with early radiation, the radiation has identically zero averaged asymmetry for all $N_B$; the U(1) symmetry survives the whole evaporation process.
  • The decoupling inequality supplies the same transition condition, so the emergent symmetry can be understood as a consequence of decoupling rather than of any symmetry in the Hamiltonian.

Reading between the lines

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

  • A direct finite-$N$ test of equation (5.5) would require the exact average of the logarithms rather than the logarithm of the averages; numerical Haar integration for small systems should show whether the sharp step survives or smears into a crossover.
  • The same pruned-purity method should extend to Rényi index $n>2$ and to non-Abelian symmetries, giving a family of transition curves with the same critical size but different saturation values.
  • The setup connects to quantum Mpemba studies: symmetry is restored from an asymmetric mixed state under random unitary evolution, so the transition could be observed in randomized circuits as a diagnostic of scrambling.
  • If emergent symmetry is read as a scrambling signature, the result suggests the diary information is recoverable only after the radiation size exceeds half of $N+N_A+s$, a bound that may be sharper than the original Hayden-Preskill recovery bound in mixed-state settings.
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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 / 4 minor

Summary. The paper studies the Rényi-2 entanglement asymmetry of the radiation in the Hayden-Preskill protocol, assuming that the initial black hole is in a general mixed state with Rényi-2 entropy s. The authors compute the Haar-averaged purity EU[tr ρ_B^2] and the averaged pruned purity EU[tr ρ_{B,Q}^2], obtaining the exact ratio (5.2). They then replace expectation values of logarithms by logarithms of expectation values, use Stirling's formula, and arrive at the main formula (5.5), which predicts that the averaged entanglement asymmetry is essentially zero for N_B < (N+N_A+s)/2 and saturates to (1/2) log(π N_B) for larger N_B. They also derive a decoupling inequality in Appendix A and use it to argue that the emergent U(1) symmetry below the transition is a genuine thermodynamic-limit effect. The maximally mixed case s = N−N_A is shown to yield exactly vanishing asymmetry for all N_B.

Significance. If the main formula (5.5) is established, the paper provides a new quantitative prediction for symmetry emergence in black-hole radiation, extending the pure-state analysis of Ref. [34] to the mixed-state Hayden-Preskill setup. The exact Haar integrations leading to Eqs. (3.7) and (4.4) are clean, parameter-free, and appear correct; the decoupling-inequality derivation in Appendix A is also a useful self-contained result. The qualitative claim that a maximally mixed initial black hole preserves the emergent symmetry exactly is solid and follows directly from the equality of the two averaged purities. The main uncertainty concerns the replacement E[log X] ≈ log E[X] for the pruned purity, which is load-bearing for the sharp transition and for the saturation value; this issue is fixable but requires either a concentration proof or a careful reframing of (5.5) as an annealed approximation.

major comments (3)
  1. [Section 5, Eqs. (5.1) and (5.5)] The replacement E[log X] ≈ log E[X] for X = tr ρ_B^2 and X = tr ρ_{B,Q}^2 is the only step that converts the exact ratio (5.2) into the sharp transition and saturation claimed in (5.5). The paper states in footnote 2 that this is proved in Refs. [37,38], but neither reference is shown to cover the specific observable tr ρ_{B,Q}^2, which is a sum over O(N_B) charge sectors in a mixed-state protocol rather than an entropy of a Haar-random pure state or a random tensor network. Since relative fluctuations of this pruned purity could in principle decay only polynomially, the validity of the log-average replacement is load-bearing for the central claim. Please either compute a variance bound for tr ρ_{B,Q}^2, or quote a theorem from Refs. [37,38] and verify its hypotheses, or present (5.5) explicitly as an annealed approximation and separate it from the rigorous statements.
  2. [Section 5, after Eq. (5.5)] The main formula (5.5) is not valid for s = N − N_A. The authors correctly observe immediately before (5.2) that for s = N − N_A one has EU[tr ρ_B^2] = EU[tr ρ_{B,Q}^2] exactly, hence E[ΔS^(2)] = 0 for every N_B. However, substituting s = N − N_A into (5.5) gives a nonzero value at N_B = N and a transition at N_B = N/2, in direct contradiction with the exact result. The approximation leading to (5.5) requires 2^{N−s−N_A} ≫ 1, which fails exactly in the maximally mixed case because the factor (2^{N−s−N_A} − 1) in the exact denominator vanishes. Please state the domain of validity of (5.5) explicitly and reconcile it with the exact maximally mixed statement.
  3. [Section 6, Eq. (6.4)] The inference from the decoupling bound to the conclusion that the entanglement asymmetry is almost vanishing is stated without a quantitative argument. A small average trace distance to the maximally mixed state does not by itself imply a small average of the difference of logarithms of purities unless one adds a Lipschitz or large-deviation estimate; the dimension-dependent Lipschitz constant of the logarithm of the purity near the maximally mixed state can be as large as O(d_B), so the rare-event contribution must be controlled explicitly. This can likely be repaired with a short Markov-inequality argument using the boundedness of the asymmetry, but as written the step is a heuristic. Please include that argument or state precisely which theorem justifies the conclusion.
minor comments (4)
  1. [Section 5, Eq. (5.5)] The equality sign in (5.5) should be replaced by an approximate sign, since the formula relies on Stirling's approximation and the log-average replacement. The domain of validity (large N, s not too close to N − N_A, and N_B not too close to N) should also be stated.
  2. [Section 5, Eq. (5.2)] The nested fraction in Eq. (5.2) is difficult to read as typeset; please rewrite it with explicit parentheses so that the denominator of the (C − 1) term is unambiguous.
  3. [Section 2] There is a typo in the sentence 'R´enyi-two entropy equals tos'; it should read 'Rényi-2 entropy equals s'.
  4. [Section 1] In the paragraph discussing Ref. [31], the name 'Hayden-Prskill' should be 'Hayden-Preskill'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central result follows from exact Haar averages with an externally cited concentration assumption, not from a fitted or self-referential input.

full rationale

The paper's derivation chain is self-contained and non-circular. Equations (3.7) and (4.4) compute the Haar-averaged purities E[tr(ρ_B^2)] and E[tr(ρ_{B,Q}^2)] directly from Haar moment identities, with no parameter fitted to any target result. The main result (5.5) is obtained by taking the ratio of these exact averages, applying Stirling's approximation, and using the approximation E[log X] ≈ log E[X] for the two purities. This concentration step is explicitly flagged in footnote 2 and justified by references [37,38], which are external works on typicality in random tensor networks, not self-citations of the present authors. Even if the applicability of that typicality result to the pruned purity could be questioned, that is a correctness or rigor concern, not circularity, because the approximation does not encode the claimed transition or the saturation value by construction. The transition threshold NB = (N + NA + s)/2 also follows directly from the decoupling inequality derived independently in Appendix A, and the parameter s is an input of the model rather than a quantity fitted to reproduce the final asymmetry. The only self-citation, [21], concerns Rényi entanglement asymmetry in CFT and is not load-bearing for any step in this paper. Thus there is no evidence that any 'prediction' reduces to an input by definition or by a self-citation chain.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No parameters are fitted to data; s (initial Rényi-2 entropy) and N_A (diary size) are model inputs. The derivation relies on Haar randomness and the log-average concentration approximation.

assumptions (3)
  • domain assumption The merged black-hole diary system evolves by a Haar-random unitary U (Section 2), which models maximal scrambling.
    Standard in the Hayden-Preskill literature; the averaging over U defines the protocol.
  • domain assumption E[log X] ≈ log E[X] for X = tr ρ_B^2 and tr ρ_{B,Q}^2 in the thermodynamic limit (footnote 2, Section 5).
    Used to go from averaged purities to averaged logarithm; cited to [37,38].
  • standard math The fourth moment Haar identity (eq. 3.3) and the decoupling inequality (Appendix A) are correct.
    Standard random matrix results; the decoupling inequality is derived in Appendix A.

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Cite this review

Pith. "Pith review of Entanglement asymmetry in the Hayden-Preskill protocol." pith.science (2026). https://pith.science/paper/BTECGPSA

@misc{pith2026241117695,
  author       = {Pith},
  title        = {Pith review of: Entanglement asymmetry in the Hayden-Preskill protocol},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BTECGPSA}},
  note         = {Machine review of arXiv:2411.17695}
}
abstract

In this paper, we consider the time evolution of entanglement asymmetry of the black hole radiation in the Hayden-Preskill thought experiment. We assume the black hole is initially in a mixed state since it is entangled with the early radiation. Alice throws a diary maximally entangled with a reference system into the black hole. After the black hole has absorbed the diary, Bob tries to recover the information that Alice thought should be destroyed by the black hole. In this protocol, we found that a $U(1)$ symmetry of the radiation emerges before a certain transition time (the time when the vanishing entanglement asymmetry begins to grow). This emergent symmetry is exact in the thermodynamic limit and can be characterized by the vanishing entanglement asymmetry of the radiation. The transition time depends on the initial entropy and the size of the diary. What's more, when the initial state of the black hole is maximally mixed, this emergent symmetry survives during the whole procedure of the black hole radiation. We successfully explained this novel phenomenon using the decoupling inequality.

Figures

Figures reproduced from arXiv: 2411.17695 by the authors.

Figure 1
Figure 1. The Hayden-Preskill protocol: Alice throws [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Diagram represent of identity 4.3. The averaged purity of ρB, can be obtained from EU [tr(ρ ⊗2 S )] as EU [trρ 2 B] = EU [tr(ρ ⊗2 S FB ⊗ ID)] = (dAd − 2 −s )dBd 2 D dAd(d 2 − 1) + (2−sd − dA)d 2 B dD dAd(d 2 − 1) = (2N − 2 −s−NA )2N−NB 2 2N − 1 + (2N−s−NA − 1)2NB 2 2N − 1 , (3.7) where we have used the fact dBdD = dAdC = d and denote the number of qubits in subsystem B as NB. 4 Averaged purity of “pruned” reduced st… view at source ↗
Figure 3
Figure 3. The averaged R´enyi-2 entanglement asymmetry as a function of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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