Non-Isometric Encoding Proposals

A non-isometric bulk-to-boundary map can model an effective interior state space larger than the fundamental boundary space, but it cannot preserve every inner product or every logical observable. A viable proposal must expose its kernel, the subset on which it is approximately isometric, the decoding rule, and any postselection or state dependence. Normalizing $W|\psi\rangle$ when $W^\dagger W\neq I$ generally makes the state map nonlinear and is not by itself a quantum channel.

Required background. Approximate Finite-N Recovery, Alpha-Bits, and Error Bounds supplies the norm and domain requirements. Continuum Factorization and Type-III Obstacles prevents a naive dimension count from replacing the continuum algebra.

Helpful background. Completely Positive Maps and Causal Quantum Channels supplies physical-channel conditions. Symmetry, Covariance, and QEC Constraints and Renormalization and Coarse Graining as Encoding give consistency checks.

Linear maps, kernels, and approximate isometry

Let

$$W:\mathcal H_{\mathrm{EFT}}\longrightarrow\mathcal H_{\mathrm{fund}}$$

be linear. Its Gram operator $G=W^\dagger W$ measures inner-product distortion:

$$\langle W\phi|W\psi\rangle-\langle\phi|\psi\rangle =\langle\phi|(G-I)|\psi\rangle.$$

If $\dim\mathcal H_{\mathrm{EFT}}>\dim\mathcal H_{\mathrm{fund}}$ in a regulated model, $W$ necessarily has a nonempty kernel. Vectors that differ by a kernel vector represent the same fundamental state, so only observables preserving equivalence classes can descend. For $O$ to have a well-defined encoded action, a necessary condition is

$$O\,\ker W\subseteq\ker W.$$

On a restricted subspace $S$ with projector $P_S$, approximate isometry is the quantitative statement

$$\left\lVert P_S(G-I)P_S\right\rVert\leq\delta.$$

If $\delta<1$, the polar part of $W|_S$ defines a nearby isometry. This is very different from a map with a large operationally accessible kernel.

Non-isometric code proposals use computational restrictions to argue that simple observers cannot detect certain null states or inner-product failures Akers et al. 2022. Later work relates quantitative non-isometry to state dependence: a trivial-kernel map can be approximately isometric on the relevant set, whereas a nonempty kernel forces some state-dependent reconstruction Antonini et al. 2025.

Decoding and physicality

A formal pseudoinverse $W^+$ gives $O_{\mathrm{fund}}=WOW^+$ on $\operatorname{im}W$, but this is not automatically bounded, local, causal, or efficiently implementable. Nor is it a state-independent representation if $O$ fails to preserve $\ker W$. If the proposal uses postselection, the success probability and the unnormalized completely positive map must be stated; conditioning on success is nonlinear at the state level.

Computational protection adds another parameter: a class $\mathcal C_{\mathrm{simple}}$ of allowed measurements or circuits. Indistinguishability against that class is weaker than small trace or diamond distance and must not be reported as information-theoretic recovery.

First application

Take $W:\mathbb C^3\to\mathbb C^2$ with

$$W|0\rangle=|0\rangle, \qquad W|1\rangle=|1\rangle, \qquad W|2\rangle=\frac{|0\rangle+|1\rangle}{\sqrt2}.$$

The vector $|2\rangle-(|0\rangle+|1\rangle)/\sqrt2$ lies in $\ker W$. On $S=\operatorname{span}\{|0\rangle,|1\rangle\}$ the map is exactly isometric, so every qubit observable is represented. On all of $\mathbb C^3$, the projector $|2\rangle\langle2|$ does not preserve the kernel and has no state-independent encoded action.

This small singular-value calculation is the required model: list the singular values, kernel, recoverable algebra, and subset. A holographic proposal must provide the corresponding data for its effective state family.

Adversarial control

Choose two normalized EFT states whose difference lies in $\ker W$ and demand that a fundamental measurement distinguish them. It cannot. Next choose an operator that moves a kernel vector out of the kernel; two equivalent representatives then predict different encoded outcomes. No decoder or complexity assumption repairs this algebraic inconsistency for an observer allowed that operation.

The strongest surviving claim may be approximate isometry for a restricted simple-state set or computational indistinguishability for a declared circuit class—not a global encoding of all EFT observables.

Regime, evidence ceiling, and handoff

Dimension mismatch is most relevant when an EFT interior state count grows beyond $e^{S_{\mathrm{BH}}}$ or at very late times. The approximation must track $N$, entropy, time, energy, circuit complexity, and nonperturbative $e^{-O(N^2)}$ overlaps. Bulk EFT, string, KK, and gravitational-loop cutoffs still apply; a non-isometric ansatz does not constitute a UV completion.

The evidence ceiling is a proposal with solvable models and quantitative consistency bounds. It is not an established resolution of black-hole interiors. Continue to QEC Evidence, Current Disputes, and Status for the dated comparison and to Code Subspaces, Logical Algebras, and Encoding Maps for the isometric baseline.

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

• Akers, C., Engelhardt, N., Harlow, D., Penington, G., and Vardhan, S. (2022), “The Black Hole Interior from Non-Isometric Codes and Complexity,” preprint. arXiv:2207.06536.
• Antonini, S., Balasubramanian, V., Bao, N., Cao, C., and Chemissany, W. (2025), “Non-Isometry, State Dependence and Holography,” preprint, revised January 2025. arXiv:2411.07296.