Surface/State Correspondence as a Generalized Holography

TL;DR

The paper introduces surface/state correspondence, a generalized holographic framework mapping convex codimension-two surfaces in arbitrary spacetimes to quantum states in an extensive Hilbert space, thereby dispensing with the need for spacetime boundaries. It defines pure or mixed state duals based on topology, derives a holography-like entanglement formula S_A^{\\Sigma} = A(\\gamma_A^{\\Sigma})/(4 G_N) and an effective entropy S_{\\mathrm{eff}}(\\Sigma) = A(\\Sigma)/(4 G_N), and connects these ideas to tensor networks via (c)MERA; it further develops a Fisher information metric G^{(B)}_{uu} linking quantum state distinguishability to bulk geometric data, analyzes it in AdS/CFT, and supports the framework with examples including pure AdS, BH vs Soliton, flat spaces, and de Sitter spaces. The work shows how bulk locality and gravitational dynamics may emerge from entanglement structure in a boundary-free setting, with the Cramer-Rao bound implying suppressed radial fluctuations at large N. Overall, the proposal offers a unifying viewpoint on holography, quantum information, and spacetime emergence, inviting further exploration in tensor-network realizations and non-AdS geometries.

Abstract

We propose a new duality relation between codimension two space-like surfaces in gravitational theories and quantum states in dual Hilbert spaces. This surface/state correspondence largely generalizes the idea of holography such that we do not need to rely on any existence of boundaries in gravitational spacetimes. The present idea is motivated by the recent interpretation of AdS/CFT in terms of the tensor networks so called MERA. Moreover, we study this correspondence from the viewpoint of entanglement entropy and information metric. The Cramer-Rao bound in quantum estimation theory implies that the quantum fluctuations of radial coordinate of the AdS is highly suppressed in the large N limit.

Figures (4)

• Figure 1: Schematic Pictures of Surface/State Correspondence
• Figure 2: Calculations of Entanglement Entropy in Surface/State Correspondence.
• Figure 3: Tensor Network of MERA and Surface/State Correspondence in AdS/CFT. In the standard AdS/CFT correspondence, the CFT state (UV state) is given by $|\Phi_{UV}\rangle$ and is defined by the quantum state realized at the boundary of the above MERA network. The black lines describe the flow of quantum states on discretized lattice points, which are acted by the coarse-graining and (dis)entangler operations. For a given convex closed surface $\Sigma$, we define the corresponding pure quantum state $|\Phi(\Sigma)\rangle$ by contracting the indices of the tensors starting from the UV state $|\Phi_{UV}\rangle$, following the tensor network as $\Sigma$ is homologous to the AdS boundary. Note that we can add a dummy trivial state $|0\rangle$ for each coarse-graining operator so that both $|\Phi(\Sigma)\rangle$ and $|\Phi_{UV}\rangle$ live in the same Hilbert space ${\cal H}_{tot}$. Or equally, we can start from any point inside the region surrounded by $\Sigma$ and expand into $\Sigma$ to eventually find the state $|\Phi(\Sigma)\rangle$. Therefore we can apply this correspondence to any networks even without boundaries.
• Figure 4: Infinitesimal Deformations of Surfaces for Information Metric.