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REVIEW 3 major objections 4 minor 28 references

Projections and teleportation of operator quenches in CFT

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A boundary-state projection teleports quench information, but not fully

desk verdict 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. read the letter →

arxiv 2412.17059 v1 pith:YMIM57Z4 submitted 2024-12-22 hep-th

Zsolt Gyongyosi , Timothy J. Hollowood , S. Prem Kumar This is my paper · ORCID
classification hep-th
keywords operatorquenchRenyientropyteleportationCardystateboundaryconformalfieldtheorylargecentralchargeblockblackholefinal
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

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.

What carries the argument

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 $h_n = \frac{c}{24}(n-1/n)$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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).
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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. 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.

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 (3)
  1. [Sec. 4.1, Eqs. (4.12)-(4.17)] 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.
  2. [Sec. 4.3, Eqs. (4.25)-(4.29) and Sec. 6] 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.
  3. [Sec. 3.4, Eqs. (3.39)-(3.43)] 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.
minor comments (4)
  1. [Sec. 3.4, Eq. (3.42)] 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.
  2. [Sec. 5, after Eq. (5.8)] 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.
  3. [Reference list] Reference [2] contains a typo in its title: 'projection operatorls' should be 'projection operators'.
  4. [Sec. 4.2, Eq. (4.19)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; free-scalar Renyi limits are derived from replica correlators and permutation counting, and the large-c comparison is explicitly qualified and rests on external conformal-block results.

full rationale

The derivation chain is self-contained and does not reduce to its inputs. In the free scalar theory, the Renyi entropy excess is computed directly from the explicit correlator formula (2.9), the uniformizing maps (3.24), and OPE-channel dominance leading to (3.26)-(3.28); the universal limits (3.33)-(3.37) follow from the permutation/Cayley-distance counting in (2.17) and contain no fitted parameters. The even/odd n dependence is a nontrivial combinatorial consequence of the cyclic permutations eta and eta^{-1}, not an input. The analytic continuation in Sec. 3.4 is presented as ambiguous and is not used to force a von Neumann entropy value. For the large-c treatment, the vacuum-block result (4.13) is quoted from [26], and the OPE-channel assumptions are supported by citations to [27,28] within the same computation; these are external works rather than self-citations by the present authors. Moreover, the paper explicitly flags the limitation: 'This is not universal' (Sec. 4.1) and later states that the large-c analysis 'precludes a clear conclusion on the efficacy of teleportation' (Sec. 6). Therefore the central free-scalar claim is independently derived, and the large-c claim is appropriately qualified rather than circularly forced.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard CFT techniques and a few domain assumptions. No free parameters are fitted. The most fragile input is the identity-block dominance assumed for the large-c six-point correlator.

assumptions (5)
  • domain assumption Projection onto a Cardy state over region P is equivalent to cutting the spacetime along P and imposing BCFT boundary conditions (Sec. 3, eqs. (3.2)-(3.3)).
    The BCFT mapping underpins all entropy computations in the projected state; it is adopted from Refs. [4-6] rather than proved here.
  • domain assumption In the epsilon to 0 OPE limits, the replica correlators are dominated by the pairing channels e and eta (and sigma = eta^{-1} deep in P), so the Cayley-distance formula (2.17) applies (Secs. 2.2-2.3, 3.3).
    The universal even/odd n result (3.37) depends on this channel dominance; the paper argues it but does not present a control over subleading corrections.
  • ad hoc to paper Identity conformal block dominance in the HHLLLL six-point correlator at large c (Sec. 4.1, eq. (4.13)).
    The paper assumes the vacuum block dominates: 'This is not universal, but we can analyse it in a large-c semiclassical limit assuming identity conformal block dominance.' No proof or check of block subdominance is given.
  • domain assumption The entropy difference (1.9) is a valid CFT analogue of the finite-dimensional teleportation measure Delta S = S(V union A) - S(A) (Sec. 1, eqs. (1.2)-(1.9)).
    The paper motivates this by analogy with a finite-dimensional toy model; the correspondence is not rigorously established for the projected state.
  • domain assumption The von Neumann limit can be extracted from the Renyi entropies by the generating function (3.39), ignoring the branch-cut imaginary part (Sec. 3.4, eqs. (3.39)-(3.43)).
    The paper shows this continuation is ambiguous, then computes I(V,A)=log dO by dropping the imaginary part; whether this is the correct prescription is left to future work.

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Pith. "Pith review of Projections and teleportation of operator quenches in CFT." pith.science (2026). https://pith.science/paper/YMIM57Z4

@misc{pith2026241217059,
  author       = {Pith},
  title        = {Pith review of: Projections and teleportation of operator quenches in CFT},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YMIM57Z4}},
  note         = {Machine review of arXiv:2412.17059}
}
read the original abstract

Motivated by recent proposals for information recovery from black holes via non-isometric maps and post-selection in an effective description, we set up and investigate a teleportation scenario in a 2d CFT involving a local operator quench and projection on a portion of space onto a Cardy state with the theory in the vacuum state. Using conformal invariance the system can be mapped to CFT with boundary (BCFT). Renyi entropies for spatial intervals in the projected state can then be computed as a function of the location of the quench, either using the replica method, or using twist fields, the latter employing universal results for correlators at large c. We find qualitatively distinct behaviours in the two systems. Our replica computations reveal a surprising universal n dependence of Renyi entropies which implies that teleportation does occur but is not optimal as would be expected because the projector is not especially tuned. We also find that the curious n dependence of the Renyi entropies means that the limit to the von Neumann entropy is not straightforward.

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