Abstract
The AdS/CFT correspondence realises the holographic principle where information in the bulk of a space is encoded at its border. We are yet a long way from a full mathematical construction of AdS/CFT, but toy models in the form of holographic quantum error correcting codes (HQECC) have replicated some interesting features of the correspondence. In this work we construct new HQECCs built from random stabilizer tensors that describe a duality between models encompassing local Hamiltonians whilst exactly obeying the Ryu-Takayanagi entropy formula for all boundary regions. We also obtain complementary recovery of local bulk operators for any boundary bipartition. Existing HQECCs have been shown to exhibit these properties individually, whereas our mathematically rigorous toy models capture these features of AdS/CFT simultaneously, advancing further towards a complete construction of holographic duality.
Highlights
The AdS/conformal field theory (CFT) correspondence realises the holographic principle where information in the bulk of a space is encoded at its border
Perhaps even to models that have at least some features of bulk gravitational physics, given the non-trivial interplay between symmetries and locality involved in AdS/CFT — and in gravitational physics more generally. [5] showed this was not the case: they gave a tensor network toy model that was able to map any local bulk Hamiltonian to local Hamiltonian on the boundary, whilst reproducing all the same features of AdS/CFT as the original HaPPY code [4]
It was known that the HaPPY code does not exactly reproduce the correct entropy scaling encapsulated in the Ryu-Takayanagi formula from AdS/CFT. [6] improved on the original construction by showing that random tensor networks were able to reproduce the Ryu-Takayanagi entropy scaling exactly
Summary
Previous work has established various holographic quantum error correcting codes (HQECC) based on tensor network structures as toy models of the AdS/CFT correspondence [4–11]. The main triumph of this model is that the entanglement entropy of all boundary regions obey the Ryu-Takayanagi formula with the expected corrections when there is non-trivial quantum entanglement in the bulk This natural likeness between the entanglement structure of high-dimensional random tensor networks and holography might suggest that there is a deeper link between semi-classical gravity and scrambling/chaos. Guided by these results we will work towards constructing a HQECC that simultaneously exhibits both the local Hamiltonian bulk-boundary correspondence of [5] and the Ryu-Takayanagi agreement of [6].
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