Learning the Alpha-bits of Black Holes

TL;DR

This paper formalizes how bulk operators in AdS/CFT containing black holes can be reconstructed on the boundary in a state-dependent, yet highly controlled, manner using universal subspace quantum error correction. It introduces alpha-bits as the boundary-encodable fraction of bulk information and proves that, for regions bounded by two extremal surfaces, the boundary region encodes the alpha-bits of the interposed bulk region with α=(𝔄₂−𝔄₁)/𝔄₀; its results show reconstructions are necessarily approximate for large code spaces and relate the phenomena to Hawking radiation and existing interior-operator proposals. Through analyses of BTZ black holes, tensor-network models, and a concrete space of black-hole microstates, the authors demonstrate how alpha-bits emerge in AdS/CFT and provide both holographic and information-theoretic bounds on reconstruction accuracy, culminating in the perspective that black holes realize explicit, capacity-achieving alpha-bit codes. The work clarifies when and how state dependence enters entanglement wedges and boundary reconstructions, linking quantum error correction, holography, and black-hole information in a tightly coherent framework with broad implications for quantum gravity and information theory.

Abstract

When the bulk geometry in AdS/CFT contains a black hole, the boundary reconstruction of a given bulk operator will often necessarily depend on the choice of black hole microstate, an example of state dependence. As a result, whether a given bulk operator can be reconstructed on the boundary at all can depend on whether the black hole is described by a pure state or thermal ensemble. We refine this dichotomy, demonstrating that the same boundary operator can often be used for large subspaces of black hole microstates, corresponding to a constant fraction $α$ of the black hole entropy. In the Schrodinger picture, the boundary subregion encodes the $α$-bits (a concept from quantum information) of a bulk region containing the black hole and bounded by extremal surfaces. These results have important consequences for the structure of AdS/CFT and for quantum information. Firstly, they imply that the bulk reconstruction is necessarily only approximate and allow us to place non-perturbative lower bounds on the error when doing so. Second, they provide a simple and tractable limit in which the entanglement wedge is state-dependent, but in a highly controlled way. Although the state dependence of operators comes from ordinary quantum error correction, there are clear connections to the Papadodimas-Raju proposal for understanding operators behind black hole horizons. In tensor network toy models of AdS/CFT, we see how state dependence arises from the bulk operator being `pushed' through the black hole itself. Finally, we show that black holes provide the first `explicit' examples of capacity-achieving $α$-bit codes. Unintuitively, Hawking radiation always reveals the $α$-bits of a black hole as soon as possible. In an appendix, we apply a result from the quantum information literature to prove that entanglement wedge reconstruction can be made exact to all orders in $1/N$.

Figures (10)

• Figure 1: A black hole with horizon area $\mathcal{A}_0$ in AdS-space. The boundary is separated into two regions, $A$ and $\bar{A}$ with shared boundary $\partial A$. There are two important bulk minimal surfaces with boundary $\partial A$. The minimal surface homologous to $\bar{A}$ has area $\mathcal{A}_1$, while the minimal surface homologous to $A$ has area $\mathcal{A}_2$.
• Figure 2: Alice has a quantum state $\ket{\psi} \in \mathcal{H}_{\tilde{A}}$ consisting of $n$ qubits, for some large $n$. She adds a few qubits in a fixed state $\ket{0}$ (embedding $\mathcal{H}_{\tilde{A}}$ as a subspace of a slightly larger Hilbert space $\mathcal{H}_A$), applies a Haar random unitary $U$ and then sends slightly more than half of the qubits to Bob, thowing the rest away. We say that Alice has sent the zerobits of the state $\ket{\psi}\in \mathcal{H}_{\tilde{A}}$ to Bob.
• Figure 3: A simple two model of an evaporating black hole that incorporates thermodynamic irreversibility. At each time step, two qubits are released from the black hole as Hawking radiation and then a random isometry is applied to the black hole that adds a single qubit. This model is an example of a random tensor network and hence obeys a version of the Ryu-Takayanagi formula.
• Figure 4: Ryu-Takayanagi surfaces for empty AdS-space and a black hole in AdS. In each case, the boundary is separated into two regions, $A$ and $\bar{A}$. In empty AdS, the boundary between these regions is also the boundary of a single minimal surface through the bulk, with (divergent) area $\mathcal{A}$, which is known as the Ryu-Takayanagi or RT surface. This minimal surface separates the bulk into two regions, $a$ and $\bar{a}$, which we refer to as the entanglement wedges of $A$ and $\bar{A}$ respectively. When a black hole with horizon area $\mathcal{A}_0$ is introduced to the bulk, it creates infinitely many locally minimal surfaces with the same boundary as $A$ and $\bar{A}$. In particular we are interested in the minimal surface homologous to $\bar{A}$ with area $\mathcal{A}_1$ and the minimal surface homologous to $A$ with area $\mathcal{A}_2$. They divide the bulk into three regions $a$, $\bar{a}$ and $a'$, where $a'$ lies between the minimal surfaces and contains the black hole. For the thermal or canonical ensemble, regions $a$ and $\bar{a}$ form the entanglement wedges of $A$ and $\bar{A}$ respectively.
• Figure 5: The black hole is now entangled with a reference system $R$. If the entanglement between the black hole and $R$ has entropy greater than $\frac{\mathcal{A}_2 - \mathcal{A}_1}{4 G_N}$, the entanglement wedge of $\bar{A} \cup R$ is $\bar{a} \cup a' \cup R$. Otherwise the entanglement wedge of $\bar{A} \cup R$ is only $\bar{a} \cup R$.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2