Approximate locality, black hole complementarity and overlapping qubits

Alexander Jahn; ChunJun Cao; Gong Cheng; Thomas Koutsikos

REVIEW 2 major objections 7 minor 56 references

Approximate locality, black hole complementarity and overlapping qubits

T0 review · 2 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Overlapping qubits make black hole interior and radiation one algebra

desk verdict A clean linear-algebraic toy model of complementarity without cloning, but the quantitative Page-curve claim is only established for Gaussian fermions, so the title's 'overlapping qubits' overreaches. read the letter →

arxiv 2608.10085 v1 pith:QMBEBQAA submitted 2026-08-10 hep-th quant-ph

ChunJun Cao , Gong Cheng , Alexander Jahn , Thomas Koutsikos This is my paper · ORCID

classification hep-thquant-ph

keywords blackholecomplementarityno-cloningtheoremoverlappingqubitsPagecurveinformationparadoxapproximatelocalitynon-isometriccodesMajoranafermions

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 argues that a toy model of an evaporating black hole built from overlapping, almost but not exactly independent, qubits makes black hole complementarity an automatic feature rather than an imposed principle. The key claim is that the interior and the radiation are two different representations of the same fundamental operator algebra in one small Hilbert space, so the same information is never cloned. The paper further claims that an observer who ignores the overlaps sees radiation entropy grow without bound, while an observer who accounts for the overlap sees a Page curve. This matters because it suggests a concrete information-theoretic mechanism by which unitary evaporation can coexist with approximate bulk locality, without a firewall.

What carries the argument

The central object is the overlapping qubit (or its fermionic cousin): a set of Pauli-like operators that obey the usual algebra within one qubit but almost commute with operators of other qubits, with commutator norm $O(\epsilon)$. The paper builds these operators by taking many approximately orthogonal unit vectors, guaranteed by a standard random-projection result, and mapping them through a Clifford algebra into a smaller fundamental Hilbert space. The overlap scale $\epsilon$ controls both the apparent nonlocality and the compression ratio: $O(e^{n\epsilon^2})$ overlapping qubits can fit into $n$ exact qubits. The argument is carried by the linear-algebra relationship between the approximate basis and the exact fundamental basis: entropy definitions are made through a pseudo-state reconstructed from measured expectation values, then a negativity-correction algorithm projects onto the nearest physical state. The quasi-local entropy is the von Neumann entropy of the corrected physical part; the effective entropy is what a naive observer computes when they mistake the leftover identity block for real degrees of freedom.

What would settle it

Run a direct numerical simulation of the original overlapping-qubit Pauli model with a non-Gaussian pure fundamental state, tracking the negativity-corrected quasi-local entropy of the radiation: if it fails to turn over near the expected Page time and return to zero at late times, the central claim that the overlapping-qubit model yields a Page curve is false.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that approximate locality is enough to dissolve the cloning paradox. In its roadmap section, interior operators and exterior radiation operators are shown to be equivalent up to a basis transformation once enough of the black hole has evaporated: past Page time, any interior operator can be written as a combination of exterior operators, because both sets live as overlapping degrees of freedom in the same finite fundamental Hilbert space. There is no cloning because there are not two independent copies of the information; there is one algebra seen through two nearly local lenses. In the later quantitative section, the paper claims that the entropy of the radiation computed with the overlaps ignored grows linearly with time, while the quasi-local entropy computed after correcting for the overlap follows a Page curve that eventually returns to zero. Those calculations are carried out explicitly for overlapping Majorana fermions, with the assertion that the gap to qubits is technical rather than physical. The paper also shows that distinguishing the overlapping description from a naively independent one requires resolving an identity block whose presence signals the compression of many apparent qubits into a much smaller Hilbert space.

Load-bearing premise

The whole quantitative story is computed for overlapping fermions, not for the overlapping qubits the physical narrative describes, so the argument depends on the transfer of the entropy behavior from one algebra to the other.

Editorial extensions

If this is right

Past Page time, any interior operator can be reconstructed from radiation operators, but the reconstruction is a basis change inside one Hilbert space; no-cloning violations do not arise.
A naive observer who ignores overlaps measures a linearly growing, Hawking-like entropy, while an observer who resolves the overlap sees a Page curve that returns to zero.
The fundamental Hilbert space can be exponentially smaller than the effective description suggests, providing a Hilbert-space mechanism for holographic compression.
The model places a complexity barrier between interior and exterior reconstruction: early reconstruction is impractical, and it becomes slowly easier long after Page time.
Approximate locality in this setting invalidates the tensor-factor and subalgebra assumptions behind standard firewall derivations, so the firewall tension may not apply.

Reading between the lines

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

Extension: if the fermionic Page-curve calculation transfers to the overlapping-qubit Pauli algebra, as the paper asserts is a technical gap, then small numerical tests on non-Gaussian states should be a clean next step; a failure there would separate the complementarity narrative from the actually computable model.
Extension: the identity block that separates effective from quasi-local entropy is structurally reminiscent of island contributions in gravitational replica computations; the paper only gestures at this connection, but it suggests a concrete dictionary between overlap maps and island prescriptions.
Extension: the fidelity decay of the compression map at high compression ratios implies that typical effective states become asymptotically orthogonal after compression, which, if accurate, predicts a sharp tradeoff between compression and recoverability in any physical realization.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First thing you should know: the no-cloning/complementarity mechanism in Sec. 3 is genuine and simple. If the radiation and interior are represented by two overcomplete, approximately orthogonal operator sets on one fundamental Hilbert space, then after Page time any interior operator is a linear combination of exterior operators, and there is no cloning because both are representations of the same algebra. That part is robust and well explained.

What is new: the application of CRSV overlapping qubits to black hole evaporation, the three entropy definitions (fundamental, effective, quasi-local), the negativity-correction analysis, and the numerical Page curve. The paper is honest in its limitations: Sec. 4 admits the fermionic model is used because the qubit mapping is analytically intractable; Sec. 5.1 admits the random single-particle map cannot reproduce low-point EFT correlators; Sec. 5.2 only recovers the identity-block structure, not purity or the Page curve for Pauli operators. These admissions are not buried.

Soft spots, in proportion. First, the title and abstract promise overlapping qubits, but the quantitative results are for overlapping Majorana fermions with fermionic Gaussian states. The authors call the gap technical, and they give intuition, but they do not prove that the transfer holds for Pauli algebras or for non-Gaussian states. That matters because black-hole microstates are not generically Gaussian. This is a load-bearing mismatch, not a cosmetic one. Second, the numerics: Fig. 4 has error bars, but Figs. 5 and 6 do not show error bars or a data/code deposit, so exact reproducibility is not yet possible. Third, the complexity discussion in Sec. 3.2 is heuristic, using Krylov entropy in a real-vector toy model; it is illustrative rather than rigorous, which is fine if read that way.

Where I disagree with a harsh reading: the Page curve is not fitted or reverse-engineered. epsilon, n, and the random Gaussian state are inputs; the entropy curves are computed forward. The circularity burden is low. The fundamental entropy peaking at half Page time is explained, and the three entropy definitions are physically motivated.

Who this is for: quantum information people working on black hole complementarity, non-isometric codes, and holographic compression. It deserves a serious referee. I would send it to review with a request that the authors either extend the proof to overlapping qubits/Pauli operators or sharpen the title and abstract to the fermionic setting. A conditional acceptance with that revision is appropriate.

Referee Report

2 major / 7 minor

Summary. The manuscript proposes a toy model of black hole evaporation built from CRSV overlapping qubits, in which the interior and radiation degrees of freedom are realized as two overcomplete, approximately commuting sets of operators acting on a single fundamental Hilbert space H_F of dimension 2^n. Section 3 argues via a rank argument that past Page time the radiation operators can reconstruct all interior operators, so that no cloning occurs; the two sets are different representations of the same operator algebra. Section 4 introduces three entropy notions: fundamental algebraic entropy, effective entropy (computed as if the overlapping fermions were independent), and quasi-local entropy (obtained after the negativity-correction algorithm removes the identity-block artifact). For a pure random fermionic Gaussian state in the fundamental space, the authors report numerically that the effective entropy grows linearly while the quasi-local entropy rises and then falls, i.e., a Page-like curve. The quantitative calculations are carried out for overlapping Majorana fermions because the authors state they are not technically equipped to handle the original overlapping-qubit mapping. Section 5.1 acknowledges that the single-particle overlap map fails to reproduce low-weight EFT correlation functions, and Sec. 5.2 sketches a general operator construction that recovers the identity-block factorization of the reconstructed state.

Significance. The algebraic complementarity mechanism of Sec. 3.1 is a clean and potentially useful observation: past the reconstruction time, interior and radiation operator algebras coincide, and the effective description's apparent cloning is an artifact of ignoring the overlaps. The fermionic calculations in Sec. 4 are coherent, with explicit proofs in the appendices and numerical support for two system sizes. If the fermionic results could be rigorously transferred to Pauli overlapping qubits and non-Gaussian states, the model would be a valuable toy realization of complementarity with finite Hilbert-space compression. However, the transfer is exactly the point where the paper makes an asserted rather than demonstrated jump, and the acknowledged failure of the map to reproduce EFT correlators in Sec. 5.1 limits the physical interpretation. The paper is transparent about many of its limitations, which is commendable, but the title and abstract overstate the scope of the quantitative Page-curve result.

major comments (2)

[Secs. 4.4, 5.2] The quantitative Page-curve claim is established only for overlapping Majorana fermions in fermionic Gaussian states. The paper explicitly states in Sec. 4 that "we are not technically equipped to perform direct analytical calculations using the original overlapping qubit mapping" and asserts that the gap is "a technical one and not a physical one"; this assertion is not a proof. Lemma 4.3 and Theorem 4.1 (Appendix C) rely on Williamson eigenvalues of Gaussian covariance matrices, and the general construction in Sec. 5.2 (Eq. 5.17) reproduces only the identity-block factorization of the reconstructed state. It does not establish positivity of the factor ρ'_d×d, purity after the negativity correction, or a Page curve for Pauli algebras or non-Gaussian states. Since the title and abstract claim a Page curve for "overlapping qubits", the central quantitative claim lacks direct support.
[Sec. 5.1 and Appendix D] The paper acknowledges that the single-particle overlap assumption (5.1) cannot reproduce low-weight EFT correlation functions: the minimized cost in Eq. (5.3) generically remains large (Appendix D). This is a load-bearing limitation because the model is motivated as a framework for approximate locality (Sec. 1) and the firewall discussion in Sec. 6 relies on the model's physical fidelity. As written, the toy model does not yet reproduce even low-point correlators of a local EFT, so the physical relevance of the complementarity mechanism is restricted to the algebraic level. The paper should either provide a more structured overlap map that passes this check or explicitly state that the locality/no-drama aspects are not yet addressed.

minor comments (7)

[Sec. 4.2, Eq. (4.11)] The sentence "the difference between the two digamma functions are limiting to ψ(2d_M+1)-ψ(d_M+1) → log(2)" is ungrammatical; it should read "the difference limits to log 2 as d_M → ∞".
[Sec. 4.3.2, Eq. (4.29)] The formula S_eff = S(ρ~') + N_R - n should explicitly state the entropy base; if the base is 2, this equals S(ρ~') + log_2(2^{N_R-n}), which should be said for clarity.
[Sec. 4.4, Fig. 5] The text states that the average overlap is chosen as |V_I · V_J| = 0.1, while the caption reports ϵ = 0.12; these numbers should be reconciled.
[Sec. 5.2, Eq. (5.5)] The vectors v_Ia are described as drawn from R^{d^2}, but the sum runs over a=1,...,d^2-1; they should be drawn from R^{d^2-1}.
[Appendix C, Eq. (C.3)] The transformation that block-diagonalizes \tilde{Γ} should use the orthogonal matrix O from the SVD of V (as in Eq. (4.14)), not S; as written, the notation is inconsistent, although the singular-value argument is unaffected.
[Secs. 4.4 and 6] The quasi-local entropy curve peaks after the reconstruction time t∼2n rather than at the conventional Page time; the phrase "recovery of a Page curve" in the abstract should be qualified to note this shift.
[Sec. 2.1, Eq. (2.1)] The commutator bound should specify I≠J to avoid applying O(ϵ) to the same-qubit commutator.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Page curve is a forward computation from random fermionic Gaussian states; the only by-construction element is the transparent linear growth of the effective entropy.

full rationale

The central quantitative claims are computed forward, not fitted. Sec. 4.4 fixes random fermionic Gaussian pure states, a random orthogonal-projection overlap matrix V(t), and an assumed linear emission rate 2N_R(t)=t, then constructs the pseudo-state covariance matrix \tilde{\Gamma}=V\Gamma_F V^T and applies the negativity-correction algorithm; the Page-like quasi-local entropy curves in Figs. 5-6 are outputs of this calculation, not targets used to tune parameters. The identity-block decomposition (Eq. 4.20) that yields S_eff \approx N_R-n is a mathematically displayed consequence of the SVD of V, and the paper openly states the resulting linear growth rather than presenting it as an independent prediction. Self-citations ([28], [29], [36]) are contextual pointers or related-work citations; no load-bearing theorem is imported from the authors' own prior work, and the explicit overlapping-qubit construction is credited to the external CRSV construction [11]. The paper also explicitly flags its main limitation: Sec. 4 says direct calculations with the overlapping qubit mapping are not performed and Sec. 6 states that spectral analyses are still needed to verify whether the fermionic purity result transfers to other systems. That is a scope or correctness gap for the title's qubit claim, not circularity, since the gap is admitted and no hidden identification is used to close it. No step in the derivation is equivalent to its own input by construction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

No new particles, forces, dimensions, or conserved quantities are introduced. The paper borrows overlapping qubits and fermions from CRSV and introduces new entropy functionals and the pseudo-state reconstruction, which are mathematical objects rather than physical entities.

free parameters (3)

n (fundamental Hilbert space dimension) = 2n=40 and 2n=1000 in numerics
Sets the black hole entropy e^{S_BH} and controls the Page time scale t~2n; chosen by hand for the numerical examples, not fitted to an external target.
epsilon (average overlap between randomly chosen overlapping vectors) = 0.12 (Fig. 5), 0.02 (Fig. 6)
Controls the strength of approximate locality; realized by choosing the number of columns M of the random projection matrix. It is a model parameter, not fitted to the Page curve, but the entropy curve shape depends on it.
M (number of columns in the global random matrix V') = not stated explicitly; chosen so that average overlap equals the reported epsilon
Fixes the overlap epsilon and is kept fixed while selecting subsets V(t); a free parameter of the numerical construction.

assumptions (6)

domain assumption The formation and evaporation of a black hole are governed by a unitary process on the fundamental Hilbert space H_F, dim H_F = 2^n.
Stated in Sec. 3.1 as the starting point of the toy model; needed for complementarity without cloning.
ad hoc to paper The interior and radiation degrees of freedom can be represented as two overcomplete sets of overlapping operators embedded in H_F via a time-dependent emergence map E(t).
This is the central modeling postulate of the paper (Sec. 3.1); no derivation from a UV theory is given.
standard math Johnson-Lindenstrauss lemma guarantees the existence of 2N approximately orthogonal unit vectors in R^{2n} with overlaps O(epsilon).
Used in Sec. 2.2 to construct the overlap matrix V with G_{IJ}=delta_{IJ}+epsilon R_{IJ}.
standard math A random orthogonal projection followed by column renormalization yields V whose singular values are >=1 (Appendix B).
Underlies Lemma 4.3 and Theorem 4.1; the proof assumes the projected column norms are strictly less than 1, which holds with probability 1 for random matrices.
domain assumption The fundamental state rho_F can be taken to be a random fermionic Gaussian pure state for numerical entropy calculations.
Introduced in Sec. 4.4 to make the negativity correction and entropy computations tractable; the paper argues the qualitative behavior persists but does not prove it for general states.
ad hoc to paper Physical conclusions from overlapping Majorana fermions carry over to overlapping qubits.
Stated as a 'technical' gap in Sec. 4; this is the weakest load-bearing assumption and is explicitly flagged by the authors as not yet proven.

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Cite this review

Pith. "Pith review of Approximate locality, black hole complementarity and overlapping qubits." pith.science (2026). https://pith.science/paper/QMBEBQAA

@misc{pith2026260810085,
  author       = {Pith},
  title        = {Pith review of: Approximate locality, black hole complementarity and overlapping qubits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QMBEBQAA}},
  note         = {Machine review of arXiv:2608.10085}
}

read the original abstract

We construct a toy model of an evaporating black hole using approximately local degrees of freedom acting on ``overlapping" qubits in which a version of black hole complementarity arises naturally. The operators corresponding to the radiation and the interior are identified as two distinct representations of the same fundamental algebra, thereby preventing the exact factorization of the Hilbert space into interior and exterior and avoiding the conventional no-cloning violations. We show how this toy model captures several qualitative and quantitative features of black hole evaporation and how the ability to account for this ``overlap" in the entropy calculation leads to the recovery of a Page curve.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Wootters and W.H

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