Bulk causality from Quantum erasure correction

Abstract

The AdS/CFT correspondence [1] has greatly advanced our understanding of the link between spacetime geometry and entanglement, becoming a strong and remarkable tool for studying strongly-coupled quantum field theories. It has been proposed that the holographic encoding of bulk AdS space [2] in the dual CFT is represented by a quantum error-correcting code that can be implemented by a tensor network. B. Swingle [3] suggested a strong natural connection between entanglement renormalization and holographic duality where a MERA (Multi-Scale-Entanglement-Renormalization-anstaz, initially proposed by G. Vidal [4]) tensor network can be viewed as a skeleton for an emergent holographic space. Swingle noticed that MERA both encodes a CFT state and also resembles a slice of AdS spacetime, being a perfect tool to construct a map between the boundary and the bulk spacetime. In this picture, quantum entanglement in the boundary theory can be interpreted as a building block for the emergent bulk spacetime. Somewhat recently, the AdS/MERA correspondence has gained a lot of interest, providing a deeper insight into the full quantum nature of spacetime [5]. The MERA network shows a close connection between geometry and entanglement and can be described as a quantum state, containing information ordered in a network-like layered structure of isometric tensors about the correlations and properties of the bulk spacetime. The family of tensor networks provide an approximate description of the wave functions with long-range entanglement of the same type described by the ground states of local scale-invariant Hamiltonians. The MERA network serves as a lattice discretization of spatial slices of AdS spacetime. In this picture, the base of the MERA tree lies on the boundary of the AdS slice while the MERA lattice sites will fill out the bulk of the AdS slice. The local bulk operators can be depicted as logical operators on a subspace of states in the CFT and seem to be protected by the multipartite entanglement structure from boundary erasures. The quantum erasure can be reformulated in terms of causality constraints on the bulk spacetime. We study MERA structure as a causal constraint on the flow of information for each computational step in the circuit and providing an optimized evaluation of the expectation value of each local observable in the network. We propose as general Ansatz that the local causal structure of AdS spacetime [6] is protected by nonlocal boundary operators against erasure errors that may act at random unknown locations on the boundary. MERA tensor networks may provide a good tool to study the causal structure of quantum spacetime by taking the length scale as degree of freedom in AdS geometry [7] and using entanglement entropy to define the notion of holographic [8] distance. We use the AdS-Rindler reconstruction in order to provide a toy model for describing global MERA causal structure in the context of the bulk/boundary correspondence. In this way, causality can be seen as a direct consequence of the fact that spacetime geometry in the bulk theory is naturally connected to the entanglement structure of the boundary theory. A quantum error-correcting code [9] can be used as a toy model in order to explain the emergence of local causal connections between events on a uniform lattice tiling of the bulk comparable to AdS curvature scale. While all physical variables of the quantum code will reside on the boundary, all logical operators that preserve the causality conditions will reside in the bulk. The local operators in the bulk will be mapped to nonlocal operators on the boundary, under a correspondence between boundary and bulk regions reminiscent of AdS/CFT. Based on the idea that a CFT with a gravity dual [10] must possess error correcting properties, we propose a model for the causal structure of the spacetime, using a quantum error-correcting code with a tensor network structure of AdS/MERA flavour. The entire tensor network becomes an encoder for the quantum error-correcting code, while the bulk/boundary degrees of freedom can be described as logical/physical degrees of freedom. We recover the skeleton of the causal structure of the bulk from MERA-like boundary operators acting as an error correcting code for the Ads spacetime.