Manuscript Hawk01-B@1 · Hawk01-B.tex · section · cfa976f1
Controlled island comparison without a reconstruction overclaim
Controlled island comparison without a reconstruction overclaim · lines 1583–1660
Hawk01-B@1#tab:islandaudit, Hawk01-B@1#eq:Petz
The preceding finite counterexample concerns scalar entropy data without additional structure. It is not a counterexample to the island prescription.
Replica-wormhole calculations supply additional gravitational saddles and derive island entropy formulas in controlled settings. Quantum-extremal-surface and entanglement-wedge analyses additionally address reconstruction and its code-space dependence AlmheiriEtAl2020,Penington2020,EngelhardtWall2015. The replica-wormhole work of Penington, Shenker, Stanford, and Yang also gives a direct gravitational reconstruction argument using a Petz-map formula PeningtonEtAl2022. These advances are more than a fit to one scalar entropy curve.
For a concrete comparison, select the evaporating JT-plus-bath setting of Almheiri, Engelhardt, Marolf, and Maxfield AlmheiriEngelhardtMarolfMaxfield2019. Table tab:islandaudit records what must be extracted or added before making the particular finite-code diamond-norm statement of this paper. “Not derived here” is a boundary of this comparison, not a claim that the cited literature lacks reconstruction results.
audit for the selected controlled-gravity comparison.
| Required operational datum | Treatment in this comparison |
|---|---|
| Logical code and allowed inputs | A finite code, conserved-sector restriction, and fixed ancillary preparation must be selected. They are not identified with the -dimensional code of the constructed model. |
| Radiation algebra and access | Specify the bath region, regulator, and observer algebra in the JT-plus-bath setting. Do not replace them by an arbitrary tensor factor without justification. |
| Reference correlations | Introduce a reference to that selected code and determine the corresponding channel or complementary leakage. No such calculation is supplied by the finite example here. |
| QES and entanglement wedge | Use the gravitational model's stated entropy and wedge results. Their geometric and holographic assumptions remain explicit. |
| Uniform reconstruction | Determine whether one recovery map works on every allowed code state, and what uniform error the reconstruction argument supplies. |
| Decoder and diamond error | A specific channel or reconstruction map must be supplied and its finite-code error bounded. Not derived here. |
| Four-dimensional transfer | No identification with neutral Schwarzschild evaporation, its endpoint, or its microscopic interaction law is made. |
For example, once a channel and a full-rank code state are supplied, a Petz candidate can be written as
where inverse powers are on . The support-completion term makes this a CPTP map on the whole output space: the trace of the first term is .
Displayed formulas in this passage
Hawk01-B@1#eq:Petz · line 1645