Beyond toy models: distilling tensor networks in full AdS/CFT

Abstract

We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large-N CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For "tree" tensor networks (i.e., those that are constructed by discretizing spacetime with nonintersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the "holographic entanglement of purification" conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.

Entanglement wedge cross section from CFT: dynamics of local operator quench

“…However, we can easily check the agreement between the reflected entropy and the odd entanglement entropy (up to prefactor 2) by assuming µ 1. We expect that these two minimizing problems provide the same result 21 .…”

“…It is not necessary that this corresponds to the (minimal) entanglement wedge cross section. 21 In the regime µ ∼ , the area of the "entanglement wedge cross section" could be comparable to area of the two RT surfaces. In such regimes, we potentially have this deviation.…”

Eigenstate thermalization hypothesis and approximate quantum error correction

“…These "holographic codes" are robust against erasure errors on the boundary, and it has been shown that the properties of these codes can be used to explain various strange properties and paradoxes of the correspondence [2]. However, the connection to QECCs has yielded more fruit than the original conjecture in [1] -recent work has applied the tools of error correction to study topics such as the link between entanglement and spacetime [3][4][5] and black holes [6,7]. In this work, we continue in this direction by studying the error correcting properties of CFTs in the AdS/CFT correspondence.…”

Quantum state reduction: Generalized bipartitions from algebras of observables

“…It would be very interesting to directly construct the generalized bipartition for the classical observables in an explicit tensor network model (see e.g. [65]). To probe the complementary nature of the resulting reduced state, we could consider, for example, first restricting to all classical observables, then further reducing to the state given only by the observables supported inside a particular lightcone.…”