Tensor Networks from Kinematic Space

TL;DR

This work reframes the relationship between tensor networks and holographic geometry by identifying the MERA network with the kinematic space of CFT intervals, rather than with a discretized bulk geometry. Using Crofton-type measures and conditional mutual information, it shows that MERA’s causal structure and cut-counting reproduce key entanglement-geometric quantities, including differential entropy and geodesic-length relations, in both the vacuum AdS$_3$/CFT$_2$ setup and the thermofield double/BTZ context. The authors establish a precise dictionary linking MERA’s inclusive/exclusive causal cones to kinematic-space causality, with conditional mutual information serving as a discrete volume counting isometries within causal diamonds. They further develop a quotient MERA description that captures entwinement and horizon-crossing geodesics in the two-sided black hole, suggesting that excited states can be handled via a generalized compression network where geometry emerges from information-theoretic constraints. Overall, the paper provides a unified framework where entanglement structure and geometry arise from a discrete, causally constrained network, with potential extensions to excited states and beyond AdS$_3$/CFT$_2$.

Abstract

We point out that the MERA network for the ground state of a 1+1-dimensional conformal field theory has the same structural features as kinematic space---the geometry of CFT intervals. In holographic theories kinematic space becomes identified with the space of bulk geodesics studied in integral geometry. We argue that in these settings MERA is best viewed as a discretization of the space of bulk geodesics rather than of the bulk geometry itself. As a test of this kinematic proposal, we compare the MERA representation of the thermofield-double state with the space of geodesics in the two-sided BTZ geometry, obtaining a detailed agreement which includes the entwinement sector. We discuss how the kinematic proposal can be extended to excited states by generalizing MERA to a broader class of compression networks.

Figures (17)

• Figure 1: MERA naturally lives on half of two-dimensional de Sitter space, the kinematic space for an equal-time slice of AdS$_3$.
• Figure 2: The kinematic coordinates $\alpha$ and $\theta$ correspond to the half-opening angle of the geodesic and the angular location of its center-point repectively. Geodesics in the hyperbolic plane are mapped to points on kinematic space.
• Figure 3: Volumes of causal diamonds in kinematic space compute conditional mutual informations of triples of contiguous intervals. As a special case, causal diamonds with one vertex on the boundary compute mutual informations of adjacent intervals.
• Figure 4: Examples of tensor networks. (a) A featureless tensor network composed of a single tensor. This can prepare a generic state, as in eq. (\ref{['genwavefn']}). (b) A tensor network composed of a chain of tensors contracted together (a matrix product state). (c) The unitary (resp. isometric) character of the disentanglers and isometries in MERA means that these tensors cancel out when contracted with their hermitian conjugates.
• Figure 5: The MERA lattices for states on a line and a circle.