Code properties from holographic geometries

TL;DR

This work formalizes a deep link between AdS/CFT holography and operator algebra quantum error correction (OAQEC), introducing distance and price as geometric and algebraic measures of how bulk logical algebras can be protected and reconstructed on the boundary. It develops holography-specific results via the entanglement wedge, punctures, and negative curvature, and reveals uberholography, where bulk information can be supported on fractal boundary regions with a universal fractal dimension $\alpha\approx0.786$. The authors establish a holographic strong quantum Singleton bound, relate local correctability to the quantum Markov condition across different bulk curvatures, and discuss when boundary locality is preserved or broken. Together, these findings sharpen the connection between emergent bulk geometry and boundary entanglement, with potential implications for black hole physics and quantum gravity in AdS-like settings.

Abstract

Almheiri, Dong, and Harlow [arXiv:1411.7041] proposed a highly illuminating connection between the AdS/CFT holographic correspondence and operator algebra quantum error correction (OAQEC). Here we explore this connection further. We derive some general results about OAQEC, as well as results that apply specifically to quantum codes which admit a holographic interpretation. We introduce a new quantity called `price', which characterizes the support of a protected logical system, and find constraints on the price and the distance for logical subalgebras of quantum codes. We show that holographic codes defined on bulk manifolds with asymptotically negative curvature exhibit `uberholography', meaning that a bulk logical algebra can be supported on a boundary region with a fractal structure. We argue that, for holographic codes defined on bulk manifolds with asymptotically flat or positive curvature, the boundary physics must be highly nonlocal, an observation with potential implications for black holes and for quantum gravity in AdS space at distance scales small compared to the AdS curvature radius.

Figures (6)

• Figure 1: This figure illustrates the geometric notions of minimal surface and entanglement wedge. In each pane, we highlight a boundary region $R$ with a crayon stroke; the corresponding minimal surface $\chi_R$ is indicated, and the entanglement wedge ${\mathcal{E}}[R]$ is shaded in green. On the left $B$ is a hyperboloid whose boundary $\partial B$ has two connected components, where $R$ is one of those components (the one on the right). The minimal surface cuts the hyperboloid at its waist, and the entanglement wedge is everything to the right of $\chi_R$. In the central pane $B$ is the interior of a Euclidean ellipse; the boundary region $R= R_1 \sqcup R_2$ has two connected components, and $\chi_R$ also has two connected components. As shown, the connected components of $\chi_R$ need not be homologous to $R_1$ and $R_2$, allowing ${\mathcal{E}}[R_1 \sqcup R_2]$ to be significantly larger than ${\mathcal{E}}[R_1] \sqcup {\mathcal{E}}[R_2]$. On the right $B$ is the Poincaré disc, portraying an infinite hyperbolic geometry. The minimal surface is a geodesic in the bulk with endpoints on $\partial B$.
• Figure 2: The left pane illustrates the thermofield double construction, in which a bulk manifold $B$ with logical boundary $\Lambda$ is extended to a two copies of $B$ with their logical boundaries identified. This doubled manifold $B\tilde{B}$ describes two holographic codes whose logical systems are maximally entangled. The other two panes illustrate that, for a boundary region $R$ contained in the physical boundary $\Phi$ of $B$, the corresponding minimal surface lies in $B$.
• Figure 3: This figure illustrates the necessary condition $\chi_\Lambda= \Lambda$ for the interior boundary of a Riemannian manifold $B$ to be identified as a logical system. In both panes, the physical Hilbert space ${\mathcal{H}}$ resides on the exterior boundary $\Phi$ of $B$, and $\Lambda$ is the boundary of the punctures in the bulk, which are shaded in black. The green region is the entanglement wedge ${\mathcal{E}}[\Phi]$, bounded by $\Phi$ and the minimal surface $\chi_\Phi= \chi_\Lambda$ separating $\Phi$ from $\Lambda$; the gray region is $B \setminus {\mathcal{E}}[\Phi]$. For purposes of illustration we assume the bulk metric is Euclidean. On the left, we have $\chi_\Lambda = \Lambda$ and the interpretation of $\Lambda$ as a logical system is consistent. On the right, we have $\chi_\Lambda \neq \Lambda$ and two possible reasons for this are illustrated. First, a connected component of $\Lambda$ may fail to be convex. Second, the union $\Lambda_1$ of several connected components of $\Lambda$ may be encapsulated by a surface $\tilde{\Lambda}_1$ with smaller area than $\Lambda_1$, in which case the logical system resides on $\tilde{\Lambda}_1$ rather than $\Lambda_1$. The emergence of this new logical system is reminiscent of the merging of small black holes to form a larger black hole.
• Figure 4: This figure illustrates the two possible geometries for the entanglement wedge ${\mathcal{E}}[R']$ of a boundary region $R' = R_1\sqcup R_2$ with two connected components separated by the interval $H$. In the left pane, the minimal surface is $\chi_{R'} = \chi_{R_1} \sqcup \chi_{R_2}$ and the entanglement wedge is ${\mathcal{E}}[R'] = {\mathcal{E}}[R_1] \sqcup {\mathcal{E}}[R_2]$. In the right pane, the minimal surface is $\chi_{R'} = \chi_R \sqcup \chi_H$, where $R= R_1HR_2$, and the entanglement wedge is ${\mathcal{E}}[R'] = {\mathcal{E}}[R] \setminus {\mathcal{E}}[H]$.
• Figure 5: This figure illustrates uberholography for the case of a two-dimensional hyperbolic bulk geometry. The inner logical boundary is contained inside the entanglement wedge, shaded in blue, of a boundary region $R$. By repeatedly punching holes of decreasing size out of this boundary region, we obtain a much smaller region $R_{\rm min}$ whose entanglement wedge still contains the logical boundary. Thus the logical algebra is supported on a fractal boundary set, whose geometry is reminiscent of the Cantor set.

Theorems & Definitions (21)

• Definition 1: correctability
• Theorem 1: criterion for correctability
• Definition 2: correctable subsystem
• Lemma 1: criterion for correctability of a subsystem
• Lemma 2: reconstruction
• Definition 3: distance
• Definition 4: price
• Lemma 3: complementarity
• proof
• Lemma 4: no free lunch