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The black hole interior from non-isometric codes and complexity

Chris Akers, Netta Engelhardt, Daniel Harlow, Geoff Penington, Shreya Vardhan

TL;DR

This work proposes that black hole interiors can be realized through non-isometric holographic codes protected by computational complexity, reconciling the apparent abundance of interior degrees of freedom with a finite fundamental Hilbert space. By integrating measure concentration, Weingarten calculus, and quantum extremal surface ideas, the authors construct soluble models in which sub-exponential observables preserve inner products and allow state-specific reconstructions of interior operators in a manner consistent with entanglement wedge reconstruction. The dynamical versions of the model reproduce the Page curve and Hayden–Preskill scrambling, while coarse-graining via simple entropy and Python’s lunch concepts provides a geometric and complexity-theoretic picture of what interior information is accessible. The framework harmonizes unitary evaporation, island formation, and interior measurements within a non-local holographic map, offering a coherent alternative to traditional isometric encodings and highlighting the role of complexity in gravitational effective field theory’s validity. Overall, the paper advances a concrete, calculable route to understanding black hole interior emergence and information transfer without relying on Euclidean gravity alone.

Abstract

Quantum error correction has given us a natural language for the emergence of spacetime, but the black hole interior poses a challenge for this framework: at late times the apparent number of interior degrees of freedom in effective field theory can vastly exceed the true number of fundamental degrees of freedom, so there can be no isometric (i.e. inner-product preserving) encoding of the former into the latter. In this paper we explain how quantum error correction nonetheless can be used to explain the emergence of the black hole interior, via the idea of "non-isometric codes protected by computational complexity". We show that many previous ideas, such as the existence of a large number of "null states", a breakdown of effective field theory for operations of exponential complexity, the quantum extremal surface calculation of the Page curve, post-selection, "state-dependent/state-specific" operator reconstruction, and the "simple entropy" approach to complexity coarse-graining, all fit naturally into this framework, and we illustrate all of these phenomena simultaneously in a soluble model.

The black hole interior from non-isometric codes and complexity

TL;DR

This work proposes that black hole interiors can be realized through non-isometric holographic codes protected by computational complexity, reconciling the apparent abundance of interior degrees of freedom with a finite fundamental Hilbert space. By integrating measure concentration, Weingarten calculus, and quantum extremal surface ideas, the authors construct soluble models in which sub-exponential observables preserve inner products and allow state-specific reconstructions of interior operators in a manner consistent with entanglement wedge reconstruction. The dynamical versions of the model reproduce the Page curve and Hayden–Preskill scrambling, while coarse-graining via simple entropy and Python’s lunch concepts provides a geometric and complexity-theoretic picture of what interior information is accessible. The framework harmonizes unitary evaporation, island formation, and interior measurements within a non-local holographic map, offering a coherent alternative to traditional isometric encodings and highlighting the role of complexity in gravitational effective field theory’s validity. Overall, the paper advances a concrete, calculable route to understanding black hole interior emergence and information transfer without relying on Euclidean gravity alone.

Abstract

Quantum error correction has given us a natural language for the emergence of spacetime, but the black hole interior poses a challenge for this framework: at late times the apparent number of interior degrees of freedom in effective field theory can vastly exceed the true number of fundamental degrees of freedom, so there can be no isometric (i.e. inner-product preserving) encoding of the former into the latter. In this paper we explain how quantum error correction nonetheless can be used to explain the emergence of the black hole interior, via the idea of "non-isometric codes protected by computational complexity". We show that many previous ideas, such as the existence of a large number of "null states", a breakdown of effective field theory for operations of exponential complexity, the quantum extremal surface calculation of the Page curve, post-selection, "state-dependent/state-specific" operator reconstruction, and the "simple entropy" approach to complexity coarse-graining, all fit naturally into this framework, and we illustrate all of these phenomena simultaneously in a soluble model.
Paper Structure (46 sections, 25 theorems, 309 equations, 26 figures)

This paper contains 46 sections, 25 theorems, 309 equations, 26 figures.

Table of Contents

  1. Introduction
  2. A Simple Model:
  3. Notation
  4. A model of the interior
  5. Measure concentration and sub-exponential states
  6. Measure concentration
  7. Circuit complexity and overlaps of sub-exponential states
  8. Quantum extremal surfaces and entropy in the Hawking state
  9. Reconstruction of effective field theory operators
  10. Attempts at simple reconstruction
  11. State-specific reconstruction
  12. Entanglement wedge reconstruction
  13. Subspace-dependent reconstruction
  14. Measurement theory for the black hole interior
  15. Review of measurement in quantum mechanics
  16. ...and 31 more sections

Key Result

Lemma 3.1

(Levy) Let $F:\mathbb{S}^d\to \mathbb{R}$ be $\kappa$-Lipschitz, meaning that $|F(x)-F(y)|\leq \kappa |x-y|$ for all $x,y\in \mathbb{S}^d$, with $|x-y|$ being the Euclidean (chordal) distance on $\mathbb{S}^d$. Then in the uniform probability measure on $\mathbb{S}^d$ we have where $\langle F\rangle$ is the expectation value of $F$ and $\mathrm{Pr}[\cdot]$ indicates the probability that "$\cdot$"

Figures (26)

  • Figure 1: The holographic encoding map in AdS/CFT: an approximate isometry $V:\mathcal{H}_{bulk}\to\mathcal{H}_{Boundary}$ maps the effective bulk degrees of freedom to the fundamental boundary degrees of freedom.
  • Figure 2: Subregion duality in AdS/CFT: given a boundary subregion $B$, its minimal quantum extremal surface $X_B^{\rm min}$ divides the bulk space into the entanglement wedge $b$ of $B$ and the entanglement wedge $\overline{b}$ of $\overline{B}$, and all bulk observables in $b$ (or $\overline{b}$) on this low-energy subspace can be represented as operators in $B$ (or $\overline{B}$).
  • Figure 3: Holography for a black hole $B$ radiating into a reservoir $R$. In the effective picture there are left- and right-moving modes $\ell$ and $r$ on a "nice slice" Cauchy surface, and the holographic map $V:\mathcal{H}_\ell\otimes\mathcal{H}_r\to\mathcal{H}_B$ tells us how these interior modes are encoded in the fundamental degrees of freedom. We will always take $V$ to be a linear map, but in general it cannot be an isometry.
  • Figure 4: Our model holographic map $V$: a Haar-typical unitary $U$ acts on the effective field theory degrees of freedom $\ell$ and $r$ in the black hole interior, together with some fixed additional state $|\psi_0\rangle_f$, and then post-selects on some auxilliary degrees of freedom $P$, resulting in a state of the fundamental degrees of freedom $B$.
  • Figure 5: Computing the typical overlap between $V|\psi_1\rangle$ and $V|\psi_2\rangle$. By equation \ref{['Uresults']} the unitary integral connects the indices going into the top/bottom of $U$ to those going into the bottom/top of $U^\dagger$, and then divides by the total dimensionality $|P||B|$. This factor of the dimensionality is cancelled by the factor of $|P|$ in $V^\dagger V$ and a factor of $|B|$ coming from a $B$ index loop.
  • ...and 21 more figures

Theorems & Definitions (44)

  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.1
  • Theorem 5.1
  • Theorem 5.2
  • Lemma 6.1
  • proof
  • Lemma 8.1
  • proof
  • Theorem 11.1
  • ...and 34 more