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A fresh review of “Complexity Equals Action” (arXiv v3) by Brown, Roberts, Susskind, Swingle, and Zhao, generated by our updated system. Read the findings, evidence, and review limitations below. This is a retrospective analysis of a published paper, not a Physical Review Letters editorial decision.
AI peer review · retrospective sample
Adam R. Brown; Daniel A. Roberts; Leonard Susskind; Brian Swingle; Ying Zhao
arXiv:1509.07876v3 · 10 May 2016 · 2026-09-18 (UTC)
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The manuscript contains an original and highly relevant conjecture supported by a substantial companion analysis. The requested revisions do not require expanding the letter into a full technical paper. They principally require sharpening the novelty claim, specifying the conventional status of the complexity normalization, limiting Lloyd-bound and dimensional statements to the regimes actually examined, and presenting charged hair as a proposed rather than demonstrated resolution.
With these scope and wording corrections, the letter would give a more accurate and compelling account of the evidence for CA duality.
This short letter proposes the complexity–action (CA) conjecture, C=A_WDW/(πℏ), identifying holographic state complexity with the on-shell action of the corresponding Wheeler–DeWitt patch. It presents late-time tests for neutral, rotating, and charged AdS black holes, together with perturbative tests involving static shells and shock waves. Its strongest result is the universal late-time neutral-black-hole rate dA/d(t_L+t_R)=2M, independent of horizon size and spacetime dimension.
The cited companion paper supplies the detailed calculations intentionally omitted from the letter, including the neutral, charged, rotating, shell, shock-wave, and tensor-network analyses [E1]. The central claims are important and suitable for a short conjecture letter, but several statements should be narrowed so that conventionally normalized, late-time evidence is not presented as an unrestricted theorem.
The historical novelty is substantial but should be stated precisely. The general association between black-hole interior growth and complexity, the maximal-volume prescription, and the shock-wave/switchback benchmarks predate this manuscript [E2]. The new contribution is the replacement of maximal ERB volume by the action of the full WDW patch, eliminating the configuration-dependent auxiliary length scale and producing a universal late-time coefficient for neutral black holes [E1][E2].
The claim that CA “subsumes” CV is stronger than demonstrated: the two are distinct bulk prescriptions that share scaling behavior and pass many of the same tests, rather than one being derived from the other. Later work on null-boundary actions, formation complexity, divergences, and complete time dependence provides retrospective qualification rather than contemporaneous prior art [E3][E4][E5][E6].
The manuscript appropriately presents a conjecture supported by nontrivial semiclassical checks rather than claiming a derivation from boundary CFT complexity. The detailed action calculations delegated to [9] are present in the companion paper: they derive the arbitrary-dimensional neutral result, the four-dimensional charged result, the rotating BTZ result, static-shell time dilation, and single, multiple, finite-energy, and localized shock-wave behavior [E1].
Thus, their omission from this short letter is not a methodological gap in the research program. The principal unresolved issue is the boundary definition of complexity: the numerical equality depends on the gate set, approximation tolerance, reference state, and continuum regulator, while the factor of π is a normalization convention [E1]. Accordingly, the results presently support a consistently normalized relative growth prescription, not a uniquely defined microscopic equality.
The neutral universality is established only at late times; the companion paper reports zero early-time action growth in the two-sided neutral case [E1], and later full-time analysis finds additional transient behavior [E6]. The rotating and charged calculations are dimension-specific—BTZ in 2+1 dimensions and charged black holes in 3+1 dimensions—so their extension to arbitrary dimension remains conjectural.
Finally, the large charged-black-hole discrepancy is not resolved quantitatively: hair is a plausible proposed explanation, but the companion paper’s superconducting analysis is qualitative rather than a complete hairy-WDW calculation [E1].
The manuscript addresses a foundational problem in holography: how continuing black-hole interior growth may be encoded after ordinary boundary observables have equilibrated. The proposed relation connects quantum information, semiclassical gravity, tensor networks, and black-hole dynamics. The universal neutral result and the simpler treatment of shock-wave configurations make the proposal especially relevant.
The “fastest computers” interpretation is potentially influential, provided it is consistently described as a conjectural, late-time conclusion for the semiclassical examples considered rather than as an all-time statement about all black holes.
The presentation is concise and generally effective for a short letter. Figures 1 and 2 clearly depict the relevant WDW patches for neutral/collapsing and charged geometries, respectively, and the progression from the conjecture to neutral, rotating, charged, shock-wave, and shell tests is coherent. The division of technical labor with the companion paper is legitimate, because that paper explicitly supplies the detailed derivations [E1].
A brief roadmap to the companion paper would nevertheless make this division clearer. The prose requires two small corrections: delete the extra “be” in “good reasons to be believe,” and replace the awkward shock-wave sentence with a direct formulation such as “Both the complexity-volume duality of [3,7] and the complexity-action duality proposed here reproduce the matching of these two growths.”
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