{"id":"ee617933-8bc2-4dc8-a289-36fc13d67a26","arxiv_id":"2412.17059","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a 2D CFT, projecting an operator quench onto a Cardy state produces Renyi entropies with a surprising even/odd n dependence, indicating imperfect teleportation and obstructing the n to 1 von Neumann limit.","lead":"This paper computes how much quantum information can be teleported when a local disturbance in a 2D quantum field theory is projected onto a special boundary state, a setup inspired by black hole final-state proposals. It finds teleportation happens but is not perfectly efficient, with a strange even-odd dependence in the Renyi entropies that makes the standard von Neumann limit ambiguous.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central free-scalar Renyi limits are robust, but the large-c teleportation comparison rests on an unchecked vacuum-block dominance in the six-point correlator; one numerical/analytic check would settle it.","rationale":"The reader's weakest assumption identifies exactly the condition on which the large-c result depends: in eqs. (4.12)-(4.17) the six-point correlator is reduced to the vacuum block in the HHLLLL limit, with the explicit caveat that this is 'not universal'. I agree that this is the most load-bearing concern, because the abstract and Sec. 4.3 use the large-c result to claim qualitatively distinct and more efficient teleportation compared with the free scalar. A direct check of the conformal block decomposition would settle whether the assumption lands. I did not find a comparable flaw in the free-scalar computation: the even/odd n limits in eq. (3.37) are obtained by straightforward permutation counting, and the paper's caution about the n -> 1 ambiguity is presented as a finding rather than hidden. I also considered whether the unspecified choice of the Cardy state in Sec. 3 could affect the free-scalar result, but the doubling-trick calculation implicitly fixes a gluing condition, and the paper's projector is defined relative to that construction; without evidence that different Cardy states give different limits, this is not a decisive objection. The reader's CONDITIONAL verdict is therefore appropriate, and my read does not change it.","tokens_in":21338,"tokens_out":28763,"duration_ms":280059,"concrete_test":"Fix an integer n, say n = 3, and take h_n = c/24 (n - 1/n) with Delta/c small. Compute the semiclassical six-point conformal block decomposition of the correlator in eq. (4.12) at the cross-ratios (4.16) with |u| = |v| = 1 and theta_v near -2*pi (the anti-chiral insertion deep in P), using Zamolodchikov recursion or an independent numerical bootstrap. Compare the contribution of the vacuum channel of fig. 7 with the first non-vacuum heavy-light block. If the non-vacuum block is not suppressed by powers of Delta/c and epsilon/a relative to eq. (4.13), then identity-block dominance is violated and the large-c conclusion of Sec. 4.3 should be withdrawn or substantially qualified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The free-scalar derivation of Sec. 3 is internally consistent: the universal limits (3.33)-(3.37) follow from the permutation counting in eq. (2.17) and from the explicit OPE-channel limits as gamma -> pi/2, so I do not find a defect there. The load-bearing weak point is the large-c comparison in Secs. 4.1-4.3. In eq. (4.12), the six-point chiral correlator is evaluated in the HHLLLL limit by assuming the OPE channel of fig. 7, i.e. each twist field fuses only with its BCFT image, so that the vacuum block (4.13) gives C_n. The paper itself flags 'This is not universal' (Sec. 4.1) and later says the large-c analysis is 'incomplete' and 'precludes a clear conclusion on the efficacy of teleportation' (Sec. 6). The unproven input is that all other semiclassical conformal blocks are subleading by powers of Delta/c or epsilon/a at the unit-modulus cross-ratios (4.16). If a non-vacuum heavy-light block contributes at order one, then the Renyi entropies (4.17), the asymptotics (4.29), and the qualitative contrast with the free scalar (fig. 11) do not follow. This does not undermine the free-scalar central claim, but it is the condition on which the large-c teleportation conclusion rests.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper sets up a two-dimensional CFT analogue of the black-hole final-state projection: a local operator quench creates a chiral/antichiral entangled state, and projection onto a Cardy state on P=(-infinity,0] is implemented by cutting the geometry and mapping to a BCFT. The authors compute the Renyi entropy of the interval A=[a,infinity) and the UV-safe difference Delta S_n^{|PO>}(bar w) between placing the chiral operator in A and outside A union P. For the free scalar CFT, explicit n=2,3,4 results are obtained, leading to the universal limits (3.37): Delta S = log d_O at bar w=0, and for bar w << -a, Delta S = log d_O/(1-n) for even n and 0 for odd n. The authors argue that the von Neumann limit is ambiguous via the generating-function method of Ref. [21]. For large-c sparse CFTs they evaluate a six-point chiral correlator under identity-block dominance, obtain Renyi excesses (4.17), find qualitatively different behavior from the free scalar, and treat finite intervals for n=2 using elliptic functions.","tokens_in":21652,"tokens_out":12845,"duration_ms":112671,"significance":"The free-scalar half of the paper is strong and genuinely useful: it is a parameter-free derivation in which the universal limits follow from explicit permutation counting and the Cayley distance, with reproducible formulas for n=3,4 given in Eqs. (3.29)-(3.32). The even/odd structure of (3.37) is a sharp, checkable prediction and is the paper's most valuable result. The large-c half is a promising proposal rather than an established calculation: the decisive vacuum-block-dominance assumption is explicitly non-universal, and the asymptotics never reach a saturating value that could be compared with log d_O. The authors' own caveats in Sec. 6 are accurate, but those caveats are load-bearing and prevent the large-c comparison from being used as evidence for the central qualitative contrast.","major_comments":[{"comment":"The entire large-c Renyi-entropy formula rests on the assumption that the six-point chiral correlator (4.12) is dominated by the single OPE channel of Fig. 7, in which each twist field fuses with its BCFT image. This is the load-bearing step: if any non-vacuum heavy-light block contributes at order one at the unit-modulus cross-ratios (4.16), then Eqs. (4.17), (4.25)-(4.29), and the curves in Figs. 10-11 do not follow. The text states 'This is not universal' but does not provide a bound, estimate, or numerical check of the next blocks. A concrete test, for example numerical conformal blocks for the HHLLLL six-point correlator or an independent semiclassical computation, is needed before the large-c results can support the qualitative conclusions.","section":"Sec. 4.1, Eqs. (4.12)-(4.17)"},{"comment":"Even granting the block-dominance assumption, the large-c calculation never produces the saturation value of Delta S_n^{|PO>}: in the relevant limit bar w -> -infinity the excess entropy in (4.29) grows linearly or logarithmically, and the quantum dimension is formally infinite/nonperturbative in c, as the paper notes below (4.12). The paper's own Sec. 6 states that this 'precludes a clear conclusion on the efficacy of teleportation.' I agree, but then the conclusion in Sec. 4.3 that large-c sparse CFTs teleport more efficiently than free scalars is not supported by the presented computation; it should be clearly labeled as a conjecture contingent on the uncomputed block corrections and on a finite-c treatment of d_O.","section":"Sec. 4.3, Eqs. (4.25)-(4.29) and Sec. 6"},{"comment":"The argument that the von Neumann limit is ambiguous relies on applying the generating-function reconstruction of Ref. [21] to the infinite sequence (3.41). A branch cut in the auxiliary function G(z) may indicate a limitation of this particular reconstruction method rather than a genuine ambiguity of the CFT von Neumann entropy. The paper should either supply a direct argument that no analytic continuation of the Renyi sequence exists (for example via Carlson's theorem or an explicit demonstration of non-uniqueness), or state clearly that the conclusion is conditional on the validity of the generating-function method in this infinite-dimensional setting.","section":"Sec. 3.4, Eqs. (3.39)-(3.43)"}],"minor_comments":[{"comment":"The closed form for G(z) does not appear to be the generating function of the sequence (3.41): expanding the right-hand side gives G(0) = (log d_O)/(2 d_O), which is nonzero, and the coefficients do not match the series generated by (3.39). The branch cuts are also at z = +- d_O^2 rather than +- d_O under the natural computation. Please recheck the algebra; the qualitative point about a real-axis obstruction may survive, but the formula as written should be corrected.","section":"Sec. 3.4, Eq. (3.42)"},{"comment":"The phrase that the cross ratio chi rises to a maximum equal to 'dO(1+k)^{-2}' is unclear; for the numerical example in Fig. 13 the maximum is (1+k)^{-2}, so the displayed expression appears to contain a typo and should be corrected.","section":"Sec. 5, after Eq. (5.8)"},{"comment":"Reference [2] contains a typo in its title: 'projection operatorls' should be 'projection operators'.","section":"Reference list"},{"comment":"The factorization of the eight-point correlator as a product of two six-point correlators is described only as 'schematically' valid; it would be helpful to state explicitly that this also relies on the same vacuum-block-dominance assumption and to indicate the precise conditions under which cross terms are negligible.","section":"Sec. 4.2, Eq. (4.19)"}],"recommendation":"major_revision","confidential_remarks":"The free-scalar core (Secs. 2-3) is self-contained and publishable, and the authors are appropriately candid about the gaps in the large-c section. My recommendation of major revision is driven by the fact that the large-c comparison is presented as a main result but depends on an unchecked conformal-block assumption and does not reach saturation; a focused revision that either supplies the missing block check or reclassifies the large-c conclusions as conjectures would make the paper acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Thanks for sending this. The paper is worth a serious look. The main thing you should know: the free scalar computation in Sec. 3 is solid and yields a genuinely new result — the UV-safe entropy difference goes from log d_O at the edge of the projection region to log d_O/(1-n) for even n and 0 for odd n deep inside, with the von Neumann limit ambiguous. That even/odd n behavior is a clean, universal combinatorial statement following from Cayley distance counting, and the explicit n=3,4 formulas check the replica calculation. No free parameters are fitted; the derivation is self-contained. Credit where due.\n\nThe weak spot is the large-c analysis in Sec. 4. The six-point chiral correlator is evaluated in the HHLLLL limit assuming that each twist field fuses only with its BCFT image, i.e. vacuum block dominance. The authors flag this as 'not universal' and later admit the analysis is incomplete and precludes a clear conclusion. The stress-test is right: if a non-vacuum heavy-light block contributes at order one at unit-modulus cross-ratios, then the Renyi entropies (4.17), the asymptotics (4.29), and the comparison with the free scalar (fig. 11) do not necessarily follow. That is the condition on which the large-c teleportation conclusion rests. This is exactly what a referee should probe: a numerical block computation or a consistency check would settle it.\n\nOne minor point: the 'teleportation' language is appropriately hedged, but the free-scalar claim rests on a single tractable example (uncompactified scalar with a specific O). That is fine for a proof-of-principle, but the universality of the even/odd n behavior beyond this theory is not established.\n\nOverall: the central free-scalar result is robust and worth publishing; the large-c part is honest about its own incompleteness. This paper deserves peer review, with a referee focused on Sec. 4 and the vacuum-block assumption. I would cite it for the even/odd n result, and it is a good reading-group candidate for discussing what counts as evidence of teleportation.","headline":"The free-scalar computation gives a genuinely new even/odd n signature in projected-state Renyi entropies, and it is robust; the large-c teleportation comparison is explicitly conditional on an unproven vacuum-block dominance.","tokens_in":22174,"tokens_out":2318,"would_cite":true,"duration_ms":21272,"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":"A boundary-state projection teleports quench information, but not fully","keywords":["operator quench","Renyi entropy","teleportation","Cardy state","boundary conformal field theory","large central charge","conformal block","black hole final state"],"falsifier":"Compute the full six-point chiral correlator with all conformal blocks summed numerically at large c in the HHLLLL limit; if the non-identity blocks contribute at order one, eq. (4.17) is wrong. On the free-scalar side, evaluate the n=3 and n=5 Renyi entropy difference in a lattice discretization of the scalar with a projection onto the Cardy state; the universal limits (3.37) require the odd-n plateau to be exactly 0 and the even-n plateau to be $\\log 2/(1-n)$, so any deviation would falsify the claimed limits.","tokens_in":21144,"feed_emoji":"🌀","tokens_out":11250,"duration_ms":94044,"temperature":0.7,"pith_summary":"The paper sets up a two-dimensional CFT analogue of a black-hole final-state projection: a local operator quench creates a chiral/anti-chiral entangled state, and a projection of a region onto a Cardy state is meant to teleport the anti-chiral half to another region. It establishes that this teleportation generically works but is not optimally efficient. For a free scalar CFT, the UV-safe Renyi entropy difference that measures teleportation has universal limits: $\\log d_O$ at the edge of the projection region, and $\\log d_O/(1-n)$ for even $n$ and $0$ for odd $n$ deep inside it (eq. (3.37)). The even/odd dependence is the paper's central result because it implies the $n\\to 1$ von Neumann limit is ambiguous, leaving the mutual information of the teleported state undefined from these Renyi data. For sparse large-$c$ CFTs the paper reaches a qualitatively different conclusion, with more efficient teleportation whose Renyi excess depends on the quench width and temperature, although the saturated values are not accessible in the large-$c$ limit.","feed_headline":"A boundary-state projection teleports quench information, but not fully","feed_subtitle":"Even-n and odd-n Renyi entropies plateau at different values, so the von Neumann limit is ambiguous.","key_machinery":"The load-bearing object is the conformal map $w=\\xi^2$, which opens the projection slit $P=[-\\infty,0]$ into the boundary of a half-plane and turns the projected state into a boundary CFT (BCFT) with Cardy boundary conditions; the doubling trick reflects anti-chiral insertions across this boundary so that all correlators become chiral. The Renyi entropy difference is then a ratio of $2n$-point correlators in which the OPE channel of the chiral points is fixed by whether the chiral insertion lies in $A$, and the channel of the anti-chiral points is fixed by where the anti-chiral insertion sits relative to $P$; these channels are recorded by permutations of the $n$ replicas, and the entropy is the Cayley distance $D(\\pi,\\sigma)$ times $\\log d_O/(n-1)$. For the large-$c$ computation, the central object is the six-point chiral correlator of two heavy twist fields and four light quench insertions (the HHLLLL limit), evaluated on the identity/vacuum conformal block, with twist dimension $h_n = \\frac{c}{24}(n-1/n)$.","core_discovery":"On its own terms, the paper's central claim is that the projected quench state has Renyi entanglement whose universal limits are governed by the Cayley distance (the least number of swaps between permutations) between replica permutation channels, not by any tuning of the projector. Concretely, in the free scalar CFT the entropy difference $\\Delta S_{|POy,n}(\\bar w)$ starts at $\\log d_O$ when the anti-chiral insertion is at the boundary of the projection slit and, deep inside the slit, instead of falling to $-\\log d_O$ it saturates at $\\log d_O/(1-n)$ for even $n$ and at $0$ for odd $n$ (eq. (3.37)). The mechanism is combinatorial: the anti-chiral OPE channel deep in the slit is the inverse cyclic permutation $\\eta^{-1}$, whose Cayley distance from the identity determines the even/odd split. Consequently teleportation occurs, but only the $n=2$ Renyi order reaches the value expected of a tuned projector, and the even/odd pattern makes analytic continuation to $n=1$ ambiguous. In large-$c$ CFTs the six-point twist-field correlator gives qualitatively different, width- and temperature-dependent Renyi excesses, indicating more efficient teleportation, with the caveat that the true saturation values lie beyond the vacuum-block approximation.","pith_inferences":["The even/odd ambiguity is likely a generic feature of post-selected replica calculations whenever the OPE channel deep in the projector is an inverse cyclic permutation; testing the thermal generalization of the free-scalar computation would show whether the plateau values persist.","For the black-hole motivation, this means untuned projections are not information-preserving at the level of Renyi entropies; a tuned or non-isometric projection, rather than a generic Cardy-state projector, is needed to recover perfect teleportation.","One could turn the even/odd pattern into a diagnostic of how well a given BCFT boundary condition acts as a teleportation channel: compute $\\Delta S_n$ for the Ising CFT or for a scalar of different $\\alpha$ and compare with eq. (3.37)."],"forward_implications":["For a free scalar CFT, no Cardy-state projector of this kind is 100 percent efficient: only the second Renyi entropy reaches $-\\log d_O$ deep in the projection region, while all other even $n$ plateau at $\\log d_O/(1-n)$ and odd $n$ plateau at $0$.","The even/odd pattern rules out a simple $n\\to 1$ continuation, so the von Neumann mutual information of the teleported state is not fixed by these Renyi entropies alone.","In large-$c$ sparse CFTs, the teleportation excess depends on the quench pulse width and temperature rather than on $\\log d_O$, and the entropy difference becomes negative when the anti-chiral mode enters $P$, signalling successful but non-universal teleportation.","When both $P$ and $A$ are finite intervals, the second Renyi entropy difference dips inside $P$ but returns to $\\log d_O$ at both ends, so teleportation of finite regions is only partial."],"supporting_citations":[{"why":"Supplies the operator-quench Renyi entropy method and the bootstrap-channel pairing rules that Sections 2 and 3 use to compute entropy differences.","marker":"[14]"},{"why":"Establishes the free scalar example and identifies the quench state's entanglement entropy with log dO, the baseline the projected computation must reproduce.","marker":"[16]"},{"why":"Gives the map from a partial projective measurement onto a Cardy state to a boundary CFT calculation, the core technology for the projection.","marker":"[4, 5]"},{"why":"Extends the projection-to-BCFT map to teleportation and final-state scenarios, defining the setup the paper adapts.","marker":"[6–8]"},{"why":"Defines the Cardy states onto which the projection is made; their boundary-condition data fix the BCFT.","marker":"[10]"},{"why":"Provides the generating-function method for taking the von Neumann limit of Renyi entropies, which the paper uses to expose the branch-cut ambiguity in Section 3.4.","marker":"[21]"},{"why":"Introduces twist operators for interval Renyi entropies in the replicated CFT, used in the large-c calculation.","marker":"[23, 24]"},{"why":"Gives the vacuum conformal block for the six-point HHLLLL correlator, the basis of the large-c Renyi entropy formula (4.17).","marker":"[26]"}],"fun_headline_variants":["Cayley distance explains suboptimal quench teleportation","Even-odd Renyi split: teleportation works, but not optimally","Boundary projection teleports quench but leaves entropy ambiguous","Quench teleportation via projections: only n=2 hits the mark","Odd-order Renyi entropy stalls: teleportation is imperfect"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The large-c results assume that in the limit where twist fields are heavy and quench operators light, the vacuum conformal block dominates the six-point correlator, with each twist field fusing only to its BCFT mirror image; if other conformal blocks contribute at the same order, the large-c entropies and the efficiency comparison would change.","fun_headline_variants_meta":{"raw":{"variants":["Cayley distance explains suboptimal quench teleportation","Even-odd Renyi split: teleportation works, but not optimally","Boundary projection teleports quench but leaves entropy ambiguous","Quench teleportation via projections: only n=2 hits the mark","Odd-order Renyi entropy stalls: teleportation is imperfect"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000395,"raw_usage":{"total_tokens":2106,"prompt_tokens":1014,"completion_tokens":1092,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":630,"completion_tokens_details":{"reasoning_tokens":1015}},"tokens_in":630,"tokens_out":1092,"duration_ms":10380,"temperature":1.0,"reasoning_tokens":1015,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:49:29.830344+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the full six-point chiral correlator with all conformal blocks summed numerically at large c in the HHLLLL limit; if the non-identity blocks contribute at order one, eq. (4.17) is wrong. On the free-scalar side, evaluate the n=3 and n=5 Renyi entropy difference in a lattice discretization of the scalar with a projection onto the Cardy state; the universal limits (3.37) require the odd-n plateau to be exactly 0 and the even-n plateau to be $\\log 2/(1-n)$, so any deviation would falsify the claimed limits.","supporting_citations":[{"cited_title":"Boundary Conditions, Fusion Rules and the Verlinde Formula,","cited_arxiv_id":null,"evidence_quote":"Defines the Cardy states onto which the projection is made; their boundary-condition data fix the BCFT."},{"cited_title":"An Alternative Method for Extracting the von Neumann Entropy from Renyi Entropies","cited_arxiv_id":"2008.10076","evidence_quote":"Provides the generating-function method for taking the von Neumann limit of Renyi entropies, which the paper uses to expose the branch-cut ambiguity in Section 3.4."}],"review_version":1}