The infinite-dimensional HaPPY code: entanglement wedge reconstruction and dynamics

Abstract

We construct an infinite-dimensional analog of the HaPPY code as a growing series of stabilizer codes defined respective to their Hilbert spaces. The Hilbert spaces are related by isometric maps, which we define explicitly. We construct a Hamiltonian that is compatible with the infinite-dimensional HaPPY code and further study the stabilizer of our code, which has an inherent fractal structure. We use this result to study the dynamics of the code and map a nontrivial bulk Hamiltonian to the boundary. We find that the image of the mapping is scale invariant, but does not create any long-range entanglement in the boundary, therefore failing to reproduce the features of a CFT. This result shows the limits of the HaPPY code as a model of the AdS/CFT correspondence, but also hints that the relevance of quantum error correction in quantum gravity may not be limited to the CFT context.

Conformal Properties of Hyperinvariant Tensor Networks

“…However, entanglement relationships in these models are typically sparse; an averaging procedure is required in order to observe CFT-like behavior. This work stands in contrast to the works [29,32], where nsite correlation functions subsequently were shown to result in either zero or a phase.…”

“…This behavior is not typically seen in a CFTgroundstate, which generally exhibits long-range quantum en-tanglement. The existence of algebraically decaying correlation functions [32] are a necesary condition for simulation of CFT-groundstates. Jahn et.…”

Holographic baby universes: an observable story

“…To us, this is one of the many examples that show that in order to formulate a complete theory of quantum gravity, one should not work directly at the level of the Hilbert spaces, but rather at the level of the C * -algebra of observables. For example, in order to formulate entanglement wedge reconstruction in a consistent way for systems with operator pushing such as the infinite-dimensional HaPPY code, in contrast to state-pushing in simpler tensor network models [30,31], we have shown that a very similar GNS technique has to be employed [14,15]. Similarly, exact and approximate theorems for entanglement wedge reconstruction work much better by reformulating the problem directly in terms of algebras of observables, as argued in [13].…”

Holographic tensor networks from hyperbolic buildings

“…Another interesting aspect of our approach is that the Gromov boundary lives at infinity, and makes it very natural to define an infinite-dimensional limit of holographic tensor networks. This problem has already been touched upon in [15,16], and more generally, the question of describing holographic quantum error-correction in the language of infinitedimensional operator algebras is an active research program [14,16,17,23]. Here we will see that there is an appropriate way to take the limit of our holographic codes such that they give a net of holographic conditional expectations.…”

“…These maps have a few shortcomings, like the fact that they do not create an entangled bulk state. See[15] for another possible choice of Hilbert space maps. However, our maps here have good functorial properties at the level of operators, and will be enough for our purposes.…”