Code Properties from Holographic Geometries

Almheiri, Dong, and Harlow [J. High Energy Phys. 04 (2015) 163.] proposed a highly illuminating connection between the $\mathrm{AdS}/\mathrm{CFT}$ 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 that 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 that are small compared to the AdS curvature radius.

Popular Summary

A deep link might exist between two seemingly disparate but far-reaching ideas in physics: quantum error correction and the holographic principle. Quantum error correction concerns using redundant encoding to protect quantum information from damage, and it has been studied extensively because of its relevance to reliable operation of noisy quantum computers. The holographic principle, meanwhile, states that all information about a volume of space can be encoded on the surface area of that volume, much as a hologram encodes a 3D image on a 2D surface. Recent research suggests that how our physical space is structured may also correspond to redundantly represented information. We further explore this connection between redundant information and geometry.

Specifically, we look at connections between quantum error-correcting codes and the “holographic correspondence,” which asserts that a suitably chosen quantum theory, without gravity, can be precisely equivalent to a theory of quantum gravity in a negatively curved spacetime. We analyze the properties of holographic quantum codes, quantum error-correcting codes that capture the essential features of the holographic correspondence. These codes provide an information-theoretic interpretation for physical notions such as points in space, black holes, and spacetime curvature.

Our work provides a new paradigm for designing quantum error-correction schemes and secret sharing codes, from which we expect many new constructions, and it also clarifies the information-theoretic foundations of the holographic correspondence.

Figure 1

Geometric notions of minimal surface and entanglement wedge. In each diagram, 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 diagram, $B$ is the interior of a Euclidean ellipse; the boundary region $R=R_1\bigsqcup R_2$ 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 $\mathcal{E}[R_1\bigsqcup R_2]$ to be significantly larger than $\mathcal{E}[R_1]\bigsqcup \mathcal{E}[R_2]$. On the right, $B$ is the Poincaré disc, portraying an infinite hyperbolic geometry. The minimal surface is a geodesic in the bulk with end points on $\partial B$.

Figure 2

The left diagram illustrates the thermofield double construction, in which a bulk manifold $B$ with logical boundary $\mathrm{\Lambda }$ is extended to 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 diagrams illustrate that, for a boundary region $R$ contained in the physical boundary $\mathrm{\Phi }$ of $B$, the corresponding minimal surface lies in $B$.

Figure 3

Necessary condition ${\chi }_{\mathrm{\Lambda }}=\mathrm{\Lambda }$ for the interior boundary of a Riemannian manifold $B$ to be identified as a logical system. In both diagrams, the physical Hilbert space $\mathcal{H}$ resides on the exterior boundary $\mathrm{\Phi }$ of $B$, and $\mathrm{\Lambda }$ is the boundary of the punctures in the bulk, which are shaded in black. The green region is the entanglement wedge $\mathcal{E}[\mathrm{\Phi }]$, bounded by $\mathrm{\Phi }$ and the minimal surface ${\chi }_{\mathrm{\Phi }}={\chi }_{\mathrm{\Lambda }}$ separating $\mathrm{\Phi }$ from $\mathrm{\Lambda }$; the gray region is $B\backslash \mathcal{E}[\mathrm{\Phi }]$. For purposes of illustration, we assume the bulk metric is Euclidean. On the left, we have ${\chi }_{\mathrm{\Lambda }}=\mathrm{\Lambda }$, and the interpretation of $\mathrm{\Lambda }$ as a logical system is consistent. On the right, we have ${\chi }_{\mathrm{\Lambda }}\ne \mathrm{\Lambda }$, and two possible reasons for this are illustrated. First, a connected component of $\mathrm{\Lambda }$ may fail to be convex. Second, the union ${\mathrm{\Lambda }}_1$ of several connected components of $\mathrm{\Lambda }$ may be encapsulated by a surface ${\tilde{\mathrm{\Lambda }}}_1$ with smaller area than ${\mathrm{\Lambda }}_1$, in which case the logical system resides on ${\tilde{\mathrm{\Lambda }}}_1$ rather than ${\mathrm{\Lambda }}_1$. The emergence of this new logical system is reminiscent of the merging of small black holes to form a larger black hole.

Figure 4

Two possible geometries for the entanglement wedge $\mathcal{E}[R^{\prime }]$ of a boundary region $R^{\prime }=R_1\bigsqcup R_2$ with two connected components separated by the interval $H$. In the left diagram, the minimal surface is ${\chi }_{R^{\prime }}={\chi }_{R_1}\bigsqcup {\chi }_{R_2}$, and the entanglement wedge is $\mathcal{E}[R^{\prime }]=\mathcal{E}[R_1]\bigsqcup \mathcal{E}[R_2]$. In the right diagram, the minimal surface is ${\chi }_{R^{\prime }}={\chi }_R\bigsqcup {\chi }_H$, where $R=R_{1}H{R}_2$, and the entanglement wedge is $\mathcal{E}[R^{\prime }]=\mathcal{E}[R]\backslash \mathcal{E}[H]$.

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 $R_{\min }$ 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.

Figure 6

A recursive coding network to which Eqs. (a3) and (a4) apply, with logical qubits at the top and physical qubits at the bottom. To derive that the distance is bounded below by $d^{\ell }$ and the price is bounded above by $p^{\ell }$, it suffices to observe that there is a unique “causal” path connecting each physical qubit to each logical qubit. Here, a [[4,2,2]] code (with the encoder drawn as a parallelogram) is concatenated to obtain a [[16,4,4]] code, and a causal path connecting a physical qubit to a logical qubit is highlighted.