Reconstruction Error Norms and Nonperturbative Precision

A reconstruction error is not a number until three choices are fixed: the target observable or channel, the states and ancillary systems on which it may be tested, and the norm. Small error in a few vacuum correlators need not imply small operator norm, channel distance, or error on an enlarged code sector. Conversely, a uniform full-Hilbert-space norm is often infinite or maximally pessimistic for unbounded bulk fields. The useful statement is therefore an energy- or code-constrained bound at a stated perturbative and nonperturbative precision. We work with Lorentzian asymptotically AdS fields, standard quantization, fixed gravitational dressing, and vacuum $i0$ unless another state is named.

Required background. Finite-$N$ reconstruction limits identifies the missing nonperturbative sectors, while energy-constrained channel distances supplies norms suitable for infinite-dimensional systems.

Helpful background. Approximate recovery relates reconstruction error to lost information, and nonperturbative finite-$N$ effects explains why a power-series remainder does not control $e^{-N^p}$ physics.

Four inequivalent error statements

Let $\Phi$ be a target dressed bulk observable and $\widehat\Phi$ its boundary reconstruction. Put $\Delta=\widehat\Phi-\Phi$. Common claims include:

1. a selected correlator bound, $|\langle A\Delta B\rangle|\leq\varepsilon_{A,B}$;
2. a state-vector bound, $\lVert\Delta|\psi\rangle\rVert\leq\varepsilon$ for $|\psi\rangle\in\mathcal S$;
3. a code-compressed operator bound, $$\lVert P_{\rm code}\Delta P_{\rm code}\rVert\leq\varepsilon_{\rm code};$$
4. a channel bound for the encoding and recovery maps, possibly allowing an arbitrary reference system, $$\frac12\lVert\mathcal R\circ\mathcal N-\operatorname{id}\rVert_{\diamond,E} \leq\varepsilon_{\diamond,E}.$$

The energy-constrained diamond norm takes the supremum only over input-reference states satisfying $\operatorname{Tr}(H\rho)\leq E$. Ancillas matter because a channel can look accurate on unentangled inputs while damaging correlations. For bounded operators, a code operator bound controls all code matrix elements. For unbounded fields, one instead specifies a common invariant domain, uses graph norms or energy bounds, or passes to bounded Weyl operators $e^{i\Phi(f)}$.

The inequalities run only under additional hypotheses. A correlator sample does not determine the operator norm. A small compressed norm $P\Delta P$ does not control leakage $(1-P)\Delta P$. A small error for one operator does not establish recovery of an entire noncommutative algebra.

A free-field cutoff calculation

Expand a normalizable smeared free field in orthonormal global-AdS modes,

$$\Phi(f)=\sum_j\left(f_j a_j+f_j^*a_j^\dagger\right), \qquad \widehat\Phi_\Lambda(f)= \sum_{\omega_j\leq\Lambda} \left(f_j a_j+f_j^*a_j^\dagger\right).$$

The omitted tail is $\Delta_\Lambda=-\sum_{\omega_j>\Lambda}(f_ja_j+f_j^*a_j^\dagger)$. On the vacuum,

$$\varepsilon_\Lambda^2 \equiv\lVert\Delta_\Lambda|0\rangle\rVert^2 =\sum_{\omega_j>\Lambda}|f_j|^2.$$

For a family of number-diagonal states with mean energy at most $E$,

$$\operatorname{Tr}(\rho\,\Delta_\Lambda^2) =\sum_{\omega_j>\Lambda}|f_j|^2(2n_j+1) \leq\sum_{\omega_j>\Lambda}|f_j|^2 \left(1+\frac{2E}{\omega_j}\right).$$

This explicit bound separates the ultraviolet smoothness of $f$ from the state-energy constraint. If $f$ is sufficiently smooth, its mode coefficients decay and the tail goes to zero as $\Lambda\to\infty$. A point field has no such finite bound without renormalization and smearing.

Now compare a target two-point function with the reconstructed one on a normalized state $|\psi\rangle$. If

$$\lVert\Delta_\Lambda|\psi\rangle\rVert\leq\varepsilon_\Lambda, \qquad \lVert\Delta_\Lambda^\dagger|\psi\rangle\rVert\leq\varepsilon_\Lambda,$$

then Cauchy–Schwarz gives

$$\begin{aligned} \big|\langle\Phi^\dagger\Phi\rangle_ \psi-\langle\widehat\Phi_\Lambda^\dagger \widehat\Phi_\Lambda\rangle_\psi\big| \leq{}&\varepsilon_\Lambda \left(\lVert\Phi|\psi\rangle\rVert +\lVert\widehat\Phi_\Lambda|\psi\rangle\rVert\right). \end{aligned}$$

This is the first application: a low-energy correlator error is auditable from the omitted-mode tail. It is not an unconstrained operator-norm result. Indeed $\Delta_\Lambda$ is unbounded on the full Fock space, so $\lVert\Delta_\Lambda\rVert=\infty$ whenever any omitted coefficient is nonzero.

Even compression can hide leakage. If the code contains only low-frequency excitations, then

$$P_{\rm code}\Delta_\Lambda P_{\rm code}=0, \qquad (1-P_{\rm code})\Delta_\Lambda P_{\rm code}\neq0,$$

because the creation part makes a high-frequency quantum. Which expression matters depends on whether the task asks only for code matrix elements or for stable implementation without leaving the code.

From operators to recovery channels

For a boundary erasure channel $\mathcal N$ and recovery $\mathcal R$, the operational quantity is a channel distance, not a list of reconstructed correlators. Approximate quantum error correction relates a small complementary-channel distinguishability to the existence of a recovery map. In holography, Almheiri, Dong, and Harlow used this perspective to formulate bulk reconstruction on a code subspace (Almheiri, Dong, and Harlow 2015, §§3–4). The continuity of Stinespring dilations gives a general bridge between channel distance and environment leakage, with square-root changes between some error conventions (Kretschmann, Schlingemann, and Werner 2008, Theorem 1).

Those results do not eliminate the need to specify the algebra. A gravitational observable’s dressing may cross the erased region or end at a boundary anchor, changing which channel is being corrected. The reference system and energy bound must also be included when the Hilbert space is infinite dimensional.

Enlarging the state set: an adversarial control

The cutoff example fails uniformly as soon as the allowed set includes a one-particle state in an omitted mode $k$. For $|1_k\rangle=a_k^\dagger|0\rangle$,

$$\lVert\Delta_\Lambda|1_k\rangle\rVert^2 \geq 3|f_k|^2, \qquad \omega_k>\Lambda,$$

whereas the reconstruction sets that mode to zero. Allow coherent states with arbitrarily large occupation and the error becomes unbounded. Thus a small vacuum tail does not control the full state space.

A second failure arises from requested precision. Suppose a perturbative construction has verified error $O(N^{-K})$, while the target is a finite-entropy signal $e^{-cN^p}$. For fixed $K$,

$$\frac{N^{-K}}{e^{-cN^p}}=N^{-K}e^{cN^p}\longrightarrow\infty.$$

The perturbative bound is parametrically larger than the signal. It licenses no conclusion about that nonperturbative feature even if every calculated coefficient is correct. An asymptotic estimate must not be silently read as an exponentially precise error bar.

A reconstruction statement that can be checked

A complete statement has the form:

For boundary region $A$, boundary algebra $\mathcal A(A)$, dressed bulk algebra $\mathcal M_V$, code projector $P_{E,k}$, and reconstruction through order $N^{-K}$, the map $\mathcal R_A$ has error at most $\varepsilon(E,k,K,N)$ in the stated code-operator or energy-constrained channel norm for times $|t|<t_{\max}$.

Each symbol is scientifically consequential. Changing $A$, the dressing $V$, energy $E$, excitation count $k$, perturbative order $K$, time window, or norm creates a different claim.

The controlled handoff to entanglement-wedge and error-correction arguments therefore carries this full specification. Exact finite-$N$ locality is not recovered by choosing a weaker diagnostic; it is replaced by an approximate algebraic statement whose error and domain can be tested.

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(4), 163. DOI.
• Kretschmann, D., Schlingemann, D., and Werner, R. F. (2008). “A continuity theorem for Stinespring’s dilation.” IEEE Transactions on Information Theory 54, 1708–1717. DOI.