Interior Reconstruction, State Dependence, Recovery, and Scrambling

An interior operator can have an exterior or radiation representative when a declared semiclassical code subspace is approximately correctable from that boundary algebra. The statement is relative to the encoding, state family, dressing, time interval, perturbative order, and error norm; it does not define one globally linear interior operator on the full microscopic Hilbert space.

Required background. Leading Semiclassical JLMS and Code-Subspace Claims supplies the relative-entropy relation. The Information Problem: Assumptions and Observables supplies the physical question.

Helpful background. Finite N, Horizons, State Dependence, and Reconstruction Limits fixes the reconstruction boundary. QEC Evidence, Current Disputes, and Status, Information Scrambling and Recovery Diagnostics, and Operator Growth versus Recoverability supply evidence and diagnostics.

Code-subspace representation

Let

$$V:\mathcal H_{\mathrm{code}}\longrightarrow \mathcal H_R\otimes\mathcal H_{\bar R}$$

be an isometric encoding for a finite-energy semiclassical sector. A logical interior operator $O_I$ is reconstructible on $R$ if there exists $O_R$ such that

$$\left\lVert (O_RV-VO_I)\lvert\psi\rangle \right\rVert \le\epsilon\,\lVert O_I\rVert$$

for every normalized $\lvert\psi\rangle$ in the declared code subspace. Equivalent channel statements bound the diamond norm of recovery on states possibly entangled with a reference.

JLMS equality of boundary and bulk relative entropies in an entanglement wedge motivates such recovery to leading semiclassical order Jafferis et al. 2016. The representative is nonunique: operators differing by something that annihilates $V\mathcal H_{\mathrm{code}}$ act identically on the code.

Application: commutators and reference changes

Choose a two-dimensional code spanned by semiclassical states $\lvert0\rangle_{\mathrm{code}}$ and $\lvert1\rangle_{\mathrm{code}}$, and let $O_I$ flip them. Suppose $O_R$ satisfies the error bound above. For any boundary operator $A_{\bar R}$ of unit norm,

$$\left| \langle\psi|V^\dagger[O_R,A_{\bar R}]V|\phi\rangle \right| \le 2\epsilon\,\lVert O_I\rVert$$

if $O_R$ and $A_{\bar R}$ are intended to represent commuting logical algebras. This code-matrix-element test is stronger than checking one expectation value.

Now change the reference state or enlarge the span. A Petz or modular-flow reconstruction built from a reference density matrix can change its explicit form, while representing the same logical operator on the overlap. Consistency requires the representatives to agree within error on that overlap. Disagreement there is not harmless “state dependence.”

After an entanglement-wedge transition, a logical region that was reconstructible from the remaining black hole may become reconstructible from radiation. Scrambling helps distribute the information, but small OTOCs alone do not prove the recovery inequality or construct the decoder.

Macroscopic enlargement test

Enlarge $\mathcal H_{\mathrm{code}}$ to include order-$N^2$ changes of geometry, both sides of a QES transition, or too many independent interior excitations. The entanglement wedge can become state-dependent, the encoding may cease to be approximately isometric, and no single $O_R$ need work. Demand a globally linear representative on the entire microscopic Hilbert space: known code-subspace arguments do not establish it.

Also change the gravitational dressing anchor. Commutators with asymptotic charges change, so the reconstructed object must be relabeled rather than treated as the same local field.

Controlled conclusion

Interior reconstruction is an approximate operator-algebra statement within a semiclassical sector. It can reconcile multiple boundary representatives without cloning because they agree only as logical operators on the code. It does not determine the black-hole endpoint or microscopic evaporation unitary. Proposed late-time maps that are not even isometric are analyzed on Non-Isometric Encoding in Evaporation Models.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Jafferis, D. L., A. Lewkowycz, J. Maldacena, and S. J. Suh. “Relative Entropy Equals Bulk Relative Entropy.” Journal of High Energy Physics 2016, 6 (2016): 004. DOI.
• Petz, D. “Sufficiency of Channels over von Neumann Algebras.” Quarterly Journal of Mathematics 39 (1988): 97–108. DOI.