Bidirectional holographic codes and sub-AdS locality

TL;DR

The paper introduces the bidirectional holographic code (BHC), a tensor-network construction built from pluperfect tensors that realizes a unitary mapping between boundary and bulk Hilbert spaces while enforcing a bulk gauge invariance and enabling a subspace with emergent locality. It unifies boundary states and bulk geometries, showing how classical geometries satisfy the Ryu-Takayanagi formula and how bulk operators can be encoded or reconstructed on the boundary within causal structures such as the causal and entanglement wedges. The work provides explicit stabilizer-code-based constructions, argues for a random-tensor–like intuition in the large-D limit, and analyzes the growth of allowed bulk configurations to reveal a regime where local QFT-like behavior emerges at sub-AdS scales or even flat space. It also outlines open questions on dynamics, curvature, continuum limits, and how to select boundary Hamiltonians that yield local bulk dynamics, suggesting a rich program for exploring holography with sub-AdS locality.

Abstract

Tensor networks implementing quantum error correcting codes have recently been used to construct toy models of holographic duality explicitly realizing some of the more puzzling features of the AdS/CFT correspondence. These models reproduce the Ryu-Takayanagi entropy formula for boundary intervals, and allow bulk operators to be mapped to the boundary in a redundant fashion. These exactly solvable, explicit models have provided valuable insight but nonetheless suffer from many deficiencies, some of which we attempt to address in this article. We propose a new class of tensor network models that subsume the earlier advances and, in addition, incorporate additional features of holographic duality, including: (1) a holographic interpretation of all boundary states, not just those in a "code" subspace, (2) a set of bulk states playing the role of "classical geometries" which reproduce the Ryu-Takayanagi formula for boundary intervals, (3) a bulk gauge symmetry analogous to diffeomorphism invariance in gravitational theories, (4) emergent bulk locality for sufficiently sparse excitations, and (5) the ability to describe geometry at sub-AdS resolutions or even flat space.

Figures (9)

• Figure 1: (a) The definition of a pluperfect tensor with four in-plane indices (black lines) $\alpha\beta\gamma\delta$ and one bulk index (red vertical line) $I$. (b) Illustration of the three pluperfection conditions. The arrow direction indicates that the tensor is a unitary map (up to renormalization factors) between input arrows and output arrows. The yellow triangle labeled with $I$ stands for fixing the input of the bulk index to one of the $D^2$ states $I=1,2,...,D^2$.
• Figure 2: Two examples of tensor networks made by contracting pluperfect tensors.
• Figure 3: (a) A simple example of a tensor network with three pluperfect tensors. (b) Contraction of the bulk indices which proves that the mapping $M$ is an isometry, after proper normalization.
• Figure 4: (a) The gauge transformation on in-plane indices that is equivalent to a bulk operator $W$. (b) For a simple network, an illustration of why the boundary projection of an arbitrary operator in the interior $M^\dagger OM$ commutes with a boundary ${\rm SU(D)}$ transformation $g_1$.
• Figure 5: (a) and (b) Allowed arrow assignments that define a unitary mapping from the input sites (hollow circles) to the output sites (solid circles). The number on each vertex defines the ordering of the sites. (c) illustrates a disallowed arrow configuration with a closed loop of arrows. With any order assignment, there will be arrows going against the ordering (from larger number to smaller number), so that this network does not define a unitary map from the input sites to the output sites. (d) Arrow drawing on part of a graph with non-positive curvature. $A_1$ and $A_2$ are boundary regions labeled by the hollow circles and the solid circles, respectively. The region $A_1\cup A_2=A$ bounds a geodesic line $\gamma_A$. The arrow drawing in the region between $\gamma_A$ and $A$ (the "entanglement wedge") defines a unitary mapping from $\gamma_A\cup A_1$ to $A_2$, which thus defines an isometry from $\gamma_A$ to $A$.