{"id":"75038226-9948-4904-9c7b-ed2438808b1c","arxiv_id":"2411.17695","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The averaged Rényi-2 entanglement asymmetry of radiation in the Hayden-Preskill protocol vanishes below a transition set by the initial Rényi-2 entropy and diary size, signaling an emergent U(1) symmetry.","lead":"Black holes that rapidly scramble information can still emit radiation that looks symmetric. This paper computes when the radiation in the Hayden-Preskill protocol develops a hidden U(1) symmetry, in terms of the black hole's initial entropy and the diary's size.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central formula (5.5) hinges on replacing disorder averages of logarithms by logarithms of disorder-averaged purities; without a concentration proof for the pruned purity, the sharp transition is read from the ratio of averages rather than from the true average asymmetry.","rationale":"I read the paper as a clean analytic computation of averaged purities in the Hayden-Preskill protocol, with the main physical conclusion being an emergent U(1) symmetry of the radiation before a transition that depends on the initial Rényi-2 entropy s. The Haar-averaged purity calculations (Sections 3 and 4) are internally consistent: the identities (3.3), (4.3), and (4.5) are applied correctly, and the exact ratio (5.2) follows. The central approximation is the log-average step in Section 5, exactly the weakest assumption identified by the reader. The paper cites [37,38] for thermodynamic-limit validity, but those references do not transparently cover the pruned purity observable in this mixed-state, symmetry-nonconserving setting, so the concentration claim is an assumption rather than a demonstrated theorem. A secondary, less severe issue is that the simplification (5.4) is not uniform as s approaches N-N_A: for s=N-N_A-δ with δ=O(1), the exact factor in (5.2) diverges as 1/(δ ln2) times the asymptotic power, shifting the transition by O(log δ) qubits. This boundary-layer effect is subleading in the thermodynamic limit for generic fixed δ/N and does not change the qualitative conclusion. Overall, the central claim is well supported in the intended regime, and the main caveat is a standard concentration assumption that is likely valid but not proved here. The reader's ACCEPT verdict remains appropriate.","tokens_in":10250,"tokens_out":35471,"duration_ms":303685,"concrete_test":"Perform exact Haar sampling for N=8,10,12,14 with N_A=2 and s=(N-N_A)/2, evaluating for many U the two logarithms log trρ_B^2 and log trρ_{B,Q}^2 at N_B values straddling (N+N_A+s)/2. Compare the sample means E[log trρ_B^2] and E[log trρ_{B,Q}^2] with log of the analytic averages in Eqs. (3.7) and (4.4). If the differences do not shrink toward zero as N grows (e.g., do not decay faster than 1/N), then the log-average approximation behind Eq. (5.5) fails and the sharp-transition claim needs to be restated in terms of the ratio of averages only.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's main result, Eq. (5.5), is obtained from Eq. (5.1) by the replacement E[log trρ_B^2] ≈ log E[trρ_B^2] and likewise for trρ_{B,Q}^2 (footnote 2, Section 5). The exact Haar averages in Eqs. (3.7) and (4.4) are correct, so the ratio in Eq. (5.2) is exact as a ratio of averaged purities. But the quantity whose average defines the Rényi-2 entanglement asymmetry is the difference of logarithms, not the logarithm of the ratio of averages. The sharp transition at N_B=(N+N_A+s)/2 and the saturation value (1/2)log(πN_B) are properties of log(E trρ_{B,Q}^2 / E trρ_B^2). Matching these to E[ΔS^(2)] requires both purities to be self-averaging, with fluctuations that vanish relative to their means in the thermodynamic limit. The cited references [37,38] establish typicality for entropies in random tensor networks, but not explicitly for the pruned purity trρ_{B,Q}^2 in this mixed-state protocol, where the observable is a sum over O(N_B) charge sectors. If the relative variance of the pruned purity does not decay exponentially, the crossover could be broadened and Eq. (5.5) would not describe the actual average asymmetry, although the decoupling argument still supports the qualitative emergent-symmetry claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":116,"tokens_out":41232,"duration_ms":453826,"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":[{"comment":"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.","section":"Section 5, Eqs. (5.1) and (5.5)"},{"comment":"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.","section":"Section 5, after Eq. (5.5)"},{"comment":"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.","section":"Section 6, Eq. (6.4)"}],"minor_comments":[{"comment":"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.","section":"Section 5, Eq. (5.5)"},{"comment":"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.","section":"Section 5, Eq. (5.2)"},{"comment":"There is a typo in the sentence 'R´enyi-two entropy equals tos'; it should read 'Rényi-2 entropy equals s'.","section":"Section 2"},{"comment":"In the paragraph discussing Ref. [31], the name 'Hayden-Prskill' should be 'Hayden-Preskill'.","section":"Section 1"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Chen and Tang extend the emergent-symmetry result of Ares et al. to the Hayden-Preskill protocol with a general mixed initial black hole. The work is direct: they compute Haar averages of the purity and the pruned purity of the radiation subsystem B, then take the log-ratio to get the Rényi-2 entanglement asymmetry. I checked the derivations of (3.7) and (4.4), and they are explicit and correct as far as I can see. The decoupling inequality in Appendix A gives a clean independent bound that yields the same threshold N_B = (N + N_A + s)/2, so the central qualitative result does not rest on any dubious step.\n\nCredit where due: the paper is honest, flags the key approximation in a footnote, and the main claim about emergent symmetry below threshold is supported by two independent arguments. The dependence of the transition on the initial Rényi-2 entropy s and the diary size is a real new result. No parameters are fitted, no circularity is hiding anywhere.\n\nSoft spots: the main formula (5.5) replaces E[log tr ρ_B²] with log E[tr ρ_B²], and likewise for the pruned purity. The footnote cites [37,38] for validity in the thermodynamic limit, but those references do not obviously cover the pruned purity in this mixed-state setting. The pruned purity is a sum over O(N_B) charge sectors; concentration is plausible but not proven here. So (5.5) is a conjecture about the true averaged asymmetry. The exact ratio of averages in (5.2) still gives a crossover, but the sharpness of the transition and the saturation value (1/2)log(π N_B) depend on the concentration assumption. This is a genuine caveat, but it is not a load-bearing flaw for the paper's central qualitative conclusion, because the decoupling argument already establishes vanishing asymmetry in the thermodynamic limit below threshold. A related but minor point: the paper works with Rényi-2 only and does not discuss the n→1 limit to von Neumann entanglement asymmetry; the emergent-symmetry statement is therefore specifically about the Rényi-2 version.\n\nWho is this for? People working on symmetry in scrambling dynamics, black hole information, or random circuits. It is a modest but solid extension of [34], with correct math and a clearly stated regime of validity. The concentration caveat should be pushed by the referee, but the core result is very likely correct.\n\nRecommendation: send this to a serious referee. It deserves referee time, and it will likely come back with minor revisions asking for a more careful statement about the concentration assumption in Eq. (5.5). I would not desk-reject it.","headline":"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.","tokens_in":11095,"tokens_out":1877,"would_cite":true,"duration_ms":18717,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $N_B=(N+N_A+s)/2$.","keywords":["entanglement asymmetry","Hayden-Preskill protocol","black hole radiation","emergent symmetry","Rényi-2 entropy","decoupling inequality","random unitary","U(1) symmetry"],"falsifier":"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.","tokens_in":10082,"feed_emoji":"🕳️","tokens_out":6610,"duration_ms":57585,"temperature":0.7,"pith_summary":"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.","feed_headline":"Black hole radiation gains a U(1) symmetry before a sharp transition","feed_subtitle":"The effect is exact in the thermodynamic limit and vanishes when the black hole starts maximally mixed.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the Hayden-Preskill protocol and the information-recovery setting that this paper generalizes to mixed initial black hole states.","marker":"[7]"},{"why":"The random-pure-state entanglement asymmetry study that this paper extends to the mixed-state Hayden-Preskill setup.","marker":"[34]"},{"why":"Introduces entanglement asymmetry as a probe of symmetry breaking and supplies the maximal product-state value used for the saturation comparison.","marker":"[14]"},{"why":"Provides the pruned-state method used to compute $\\operatorname{tr}(\\rho_{B,Q}^2)$ by summing over charge-sector projectors.","marker":"[26]"},{"why":"Justifies replacing the expectation value of logarithms with logarithms of expectation values in the thermodynamic limit.","marker":"[37]"},{"why":"Supplies concentration results for random tensor networks that support the same logarithm-average approximation.","marker":"[38]"},{"why":"Gives the general decoupling inequality whose simplified version the appendix proves and uses to explain the transition.","marker":"[41]"}],"fun_headline_variants":["Radiation gets U(1) symmetry before a sharp transition","Emergent U(1) symmetry in radiation: sharp transition predicted","Black hole radiation: U(1) symmetry appears at a critical time","Hayden-Preskill: radiation symmetry emerges, exact in thermodynamic limit","U(1) symmetry in radiation: exact emergence before a transition"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Radiation gets U(1) symmetry before a sharp transition","Emergent U(1) symmetry in radiation: sharp transition predicted","Black hole radiation: U(1) symmetry appears at a critical time","Hayden-Preskill: radiation symmetry emerges, exact in thermodynamic limit","U(1) symmetry in radiation: exact emergence before a transition"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000319,"raw_usage":{"total_tokens":1800,"prompt_tokens":943,"completion_tokens":857,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":779}},"tokens_in":559,"tokens_out":857,"duration_ms":7861,"temperature":1.0,"reasoning_tokens":779,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:48:56.985220+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Black holes as mirrors: Quantum information in random subsystems,","cited_arxiv_id":null,"evidence_quote":"Defines the Hayden-Preskill protocol and the information-recovery setting that this paper generalizes to mixed initial black hole states."},{"cited_title":"Entanglement asymmetry study of black hole radiation,","cited_arxiv_id":null,"evidence_quote":"The random-pure-state entanglement asymmetry study that this paper extends to the mixed-state Hayden-Preskill setup."},{"cited_title":"Entanglement asymmetry as a probe of symmetry breaking,","cited_arxiv_id":null,"evidence_quote":"Introduces entanglement asymmetry as a probe of symmetry breaking and supplies the maximal product-state value used for the saturation comparison."},{"cited_title":"Symmetry Restoration and Quantum Mpemba Effect in Symmetric Random Circuits,","cited_arxiv_id":null,"evidence_quote":"Provides the pruned-state method used to compute $\\operatorname{tr}(\\rho_{B,Q}^2)$ by summing over charge-sector projectors."},{"cited_title":"Relative Entropy of Random States and Black Holes,","cited_arxiv_id":null,"evidence_quote":"Justifies replacing the expectation value of logarithms with logarithms of expectation values in the thermodynamic limit."},{"cited_title":"Holographic duality from random tensor networks,","cited_arxiv_id":null,"evidence_quote":"Supplies concentration results for random tensor networks that support the same logarithm-average approximation."},{"cited_title":"Lecture Notes on Quantum Computation,","cited_arxiv_id":null,"evidence_quote":"Gives the general decoupling inequality whose simplified version the appendix proves and uses to explain the transition."}],"review_version":1}