Code Subspaces, Logical Algebras, and Encoding Maps

A holographic recovery claim is defined only after fixing three objects: a code domain of low-energy states, a logical algebra of gauge-invariant bulk observables, and an encoding into the boundary physical Hilbert space. These choices determine whether “reconstruction” means an exact intertwining relation, an approximation in a named norm, or merely agreement of selected correlators. A global tensor factorization of the bulk Hilbert space is neither required nor generally available.

Required background. Operator-Algebra Quantum Error Correction supplies correctable-algebra conditions. Choosing a Continuum Subsystem: Algebra, Split, or Regulator supplies the continuum subsystem choices.

Helpful background. Error Models, Codes, and Recovery Conditions fixes the recovery task. Entanglement-Wedge Reconstruction gives the holographic target. Channel–State Correspondence in Infinite Dimensions explains why finite-dimensional Choi manipulations need qualifications.

The code-domain triple

Let $P$ project onto a code subspace $\mathcal H_{\mathrm{code}}$ and let

$$V:\mathcal H_{\mathrm{code}}\longrightarrow\mathcal H_{\mathrm{CFT}}, \qquad V^\dagger V=I_{\mathrm{code}},$$

be an isometric encoding. In gravity, $\mathcal H_{\mathrm{code}}$ is normally a perturbative state space around a chosen semiclassical background, not the complete bulk Hilbert space. Its definition must state the energy cutoff $E_*$, allowed particle number, asymptotic charges, topology and black-hole sector, and accuracy in $G_N$, $\alpha'$, and Kaluza–Klein truncation.

The logical object is a von Neumann algebra $\mathcal M$, not automatically the full matrix algebra $B(\mathcal H_{\mathrm{code}})$. Its center can record superselection or area-sector data. For a boundary region $A$, restriction is a channel or algebra inclusion $\mathcal E_A$; in a regulated factorization one may write

$$\mathcal E_A(\rho)=\operatorname{Tr}_{\bar A}(V\rho V^\dagger).$$

An operator $O\in\mathcal M$ is reconstructible on $A$ if there exists $O_A$ in the physical region algebra such that $O_AVP=VOP$. Approximate reconstruction must replace this equality by a state-uniform norm or an energy-constrained channel distance.

This separation was already essential in the original holographic QEC formulation: the same perturbative bulk operator can have different boundary representatives only on a restricted low-energy sector Almheiri, Dong, and Harlow 2015. Operator-algebra QEC then makes the reconstructible algebra, rather than a fictitious factor, primary Harlow 2017.

A finite low-energy model

Choose a background $|\Omega\rangle$ and two orthogonal, gravitationally dressed one-particle wavepackets $a_1^\dagger|\Omega\rangle$ and $a_2^\dagger|\Omega\rangle$, all with energies below $E_*$. Define

$$\mathcal H_{\mathrm{code}} =\operatorname{span}\left\{ |\Omega\rangle, a_1^\dagger|\Omega\rangle, a_2^\dagger|\Omega\rangle \right\}.$$

For a wedge containing packet 1 but not packet 2, a sensible logical algebra is generated by the number and bounded transition operators of mode 1 together with any declared central charge sector. It is not $M_3(\mathbb C)$: an operator that interchanges the two wavepackets is not localized in that wedge. The boundary restriction map can be tested on the nine matrix units $|i\rangle\langle j|$; this finite basis turns an abstract claim into a reproducible channel comparison.

If $V$ is extracted perturbatively from boundary correlators, normalization requires

$$\langle i|V^\dagger V|j\rangle =\delta_{ij}+O(N^{-2},\alpha'/L^2,E_*/E_{\mathrm{br}}),$$

where $E_{\mathrm{br}}$ is the backreaction scale. This is a controlled isometry statement only on the declared span.

First application

Take the three-state code above and a boundary region $A$. Compute the matrices

$$\rho_A^{ij}=\operatorname{Tr}_{\bar A}\!\left(V|i\rangle\langle j|V^\dagger\right).$$

To test recovery of the mode-1 algebra, solve for a completely positive unital adjoint map $\mathcal R_A^*$ whose action satisfies

$$\max_{O\in\mathcal M,\,\lVert O\rVert\leq1} \left\lVert P\!\left(\mathcal R_A^*(O_A)-O\right)P\right\rVert \leq\varepsilon.$$

The report includes the dressing anchor, center sector, $E_*$, state basis, norm, and $N$. It thereby distinguishes a recovered algebra from agreement of one two-point function.

Adversarial control

Enlarge the code by adding states with $O(N^2)$ energy, distinct black-hole microcanonical bands, or a different asymptotic charge. The Gram matrix can cease to be perturbatively close to the identity, the wedge can move, and an operator that preserved the old code can leak out of the new one. A recovery channel optimized on the three-state span need not have any small error on the enlarged domain.

This control falsifies the common shortcut of treating $\mathcal H_{\mathrm{code}}$ as “all semiclassical states.” The strongest surviving claim is tied to a named sector and energy window.

Regime, evidence ceiling, and handoff

At large $N$, $G_N/L^{d-1}\sim N^{-2}$ controls perturbative gravity; $\alpha'/L^2$ controls string corrections; the compactification scale controls omitted KK modes. Code dimension and energy must remain low enough that accumulated errors and backreaction do not compete with the leading wedge data. At finite $N$, continuum algebras and gravitational constraints make an exact factorization especially suspect.

The evidence ceiling is an encoding and recovery statement on a specified domain. It is not a construction of the full quantum-gravity Hilbert space. Continue to Exact Toy-Code Reconstruction Theorems for an exact finite-dimensional example and to Continuum Factorization and Type-III Obstacles for the continuum obstruction.

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

• Almheiri, A., Dong, X., and Harlow, D. (2015), “Bulk Locality and Quantum Error Correction in AdS/CFT,” Journal of High Energy Physics 2015(04), 163. DOI; arXiv:1411.7041.
• Harlow, D. (2017), “The Ryu–Takayanagi Formula from Quantum Error Correction,” Communications in Mathematical Physics 354, 865–912. DOI; arXiv:1607.03901.
• Kribs, D. W., Laflamme, R., Poulin, D., and Lesosky, M. (2006), “Operator Quantum Error Correction,” Quantum Information & Computation 6, 382–399. arXiv:quant-ph/0504189.