Take a homogeneous two-point function C(r), define the information distance d(r) = −ln C(r), and read it as the geodesic length between boundary points separated by r of an emergent (d+1)-dimensional bulk. Then the decay class of C fixes the curvature class of that bulk. We prove this by an elementary, self-contained computation. Exponential decay yields a flat bulk; power-law, scale-invariant decay yields a bulk of constant negative curvature—hyperbolic, anti–de Sitter. For the stretched-exponential family C(r) = exp(−(r/ξ)^α) one obtains an explicit closed form, which runs continuously from flat at α = 1 to hyperbolic at α → 0, with a scale-dependent negative curvature in between. Only α enters. No holographic duality is assumed, and Einstein's equations are never used; yet the qualitative MERA/holographic correspondence—criticality with hyperbolic geometry—drops out as an explicit theorem. The signature is simple: R = −d(d+1) at criticality, R = 0 when gapped. It can be measured directly on a lattice.
Take a homogeneous two-point function C(r), define the information distance d(r) = −ln C(r), and read it as the geodesic length between boundary points separated by r of an emergent (d+1)-dimensional bulk. Then the decay class of C fixes the curvature class of that bulk. We prove this by an elementary, self-contained computation. Exponential decay yields a flat bulk; power-law, scale-invariant decay yields a bulk of constant negative curvature—hyperbolic, anti–de Sitter. For the stretched-exponential family C(r) = exp(−(r/ξ)^α) one obtains an explicit closed form, which runs continuously from flat at α = 1 to hyperbolic at α → 0, with a scale-dependent negative curvature in between. Only α enters. No holographic duality is assumed, and Einstein's equations are never used; yet the qualitative MERA/holographic correspondence—criticality with hyperbolic geometry—drops out as an explicit theorem. The signature is simple: R = −d(d+1) at criticality, R = 0 when gapped. It can be measured directly on a lattice.
The holographic bound writes a region's entanglement entropy as an area, S=A/4DA². Read as a length, DA is the area carried per bit of entanglement, bounded below by the Planck length. This note argues that DA is not a rewriting of the entropy but a different object with two powers the entropy as a number does not have: it is commensurable across domains, and it is geometric. On the first, the filling fraction f=DA²/L² places a quantum critical chain, a black hole, and a single-bit Landauer erasure on one DA axis, where the three entropies — entanglement, Bekenstein–Hawking, classical — are not directly comparable. On the second, the scaling of the entanglement length builds an emergent geometry: for a one-dimensional critical system DA forces, through Ryu–Takayanagi, an emergent AdS of constant negative curvature, while a gapped system gives a flat interior. We verify the geometric content numerically through the entanglement c-function, which is flat to 10⁻⁴–10⁻⁶ for critical chains (scale invariance, constant curvature) and decays to zero for a gapped chain (flat interior). The running rate β equals the central charge as an identity of the Calabrese–Cardy formula, not a fit; free-fermion calculations show β→c/6 as the finite-size correction vanishes from L=64 to L=4096. A second condition, the log-fit quality R², is mandatory: a volume-law state forges a spurious β that only R² exposes. The diagnostic becomes a geometry classifier with four classes — critical (hyperbolic, β>0), gapped (flat, β→0), glassy, and thermal — each carrying a clean representative, the glassy class realized by a genuine interacting disorder model (many-body localization in the random-field Heisenberg chain, which sweeps from a volume-law ergodic interior to an area-law localized interior). We state the honest boundaries: the central-charge reading does not extend to higher dimensions, to dynamics, or to topological order, and the cross-domain DA axis is an organizing coordinate, not a claim that the three systems are one physics. The contribution is a unifying compression–geometry perspective in which DA, as a holographic length, is the axis — not a new force, and not a renaming of the entropy.
A holographic code assigns a state-dependent entropy remainder to every boundary region, but such remainders need not be the extremal lengths of any one bulk metric. We study this common-source question for the proto-area of a calibrated recovery, the entropy of a boundary region minus that of the matter recovered from it. At zeroth order the reference values must match the lengths of one reference geometry in a fixed renormalization scheme, and this matching also determines the entropy-to-length factor. On a graph with a fixed minimum-cut chamber the data derivatives must lie in the image of the cut-incidence matrix, whose left null vectors are witnesses. On the hyperbolic disk the first variation of renormalized length is the rank-two geodesic X-ray transform, with closed range in an adapted data space, and the geodesic displacement, fixed by a forced Jacobi equation, enters at second order. The resulting normal Hessian gives a gauge-invariant criterion, necessary and sufficient, for a regular pair of response coefficients to come from one metric two-jet. For the constant-curvature family generated by a boundary reparametrization the Hessian has a nonzero rank-four mode, so the image of the length map is curved. For a fixed recovery the proto-area Hessian is minus the Bogoliubov-Kubo-Mori information lost in recovery, which the Petz map decomposes orthogonally at a product reference, and which, on a finite-dimensional space of logical directions with positive minimum retained fraction, is at most a sharp multiple of the retained information. Geometric realization requires this loss to reproduce the normal length Hessian on every interval. Two codes show that exact regional area operators can fail to sew into one graph geometry, and that a matched reference with vanishing first response can fail at second order. All criteria refer to one recovery branch and one extremal-surface branch.
A central insight of holography is that spacetime geometry emerges from the quantum entanglement of a boundary theory. This work proposes and validates a precise, local manifestation of this principle: in a 1+1D boundary conformal field theory (CFT) dual to an asymptotically bulk, the Ricci scalar curvature is proportional to the negative second derivative of the entanglement entropy density. This hypothesis, motivated by the Ryu-Takayanagi formula and the geodesic deviation equation, posits that local fluctuations in boundary entanglement directly encode local bulk curvature. The relation is tested numerically using a tensor network toy model inspired by the Multi-scale Entanglement Renormalization Ansatz (MERA). Localized suppressions of boundary entanglement, analogous to removing disentanglers in MERA, are shown to induce localized positive curvature spikes in the emergent bulk geometry. The results provide direct computational evidence for the entanglement-curvature correspondence, reinforcing the paradigm that spacetime geometry is built from quantum entanglement.
This project aims to construct a rigorous, computable, and testable discrete holographic duality model within the CDGM framework. Based on the discretization of the {5,4} tiling of the two-dimensional hyperbolic plane, the bulk network is exactly mapped to a multi-scale entanglement renormalization ansatz (MERA) tensor network on the boundary, with explicit constructions of isometry tensors and disentanglers. Anticipated results include: the entanglement entropy of a boundary region satisfies the discrete RT formula with a relative error less than 0.5% in the thermodynamic limit; the boundary two-point correlation function exhibits power-law decay with a scaling exponent consistent with a free scalar conformal field theory. This preregistration establishes a timestamped record of the research plan; full derivations and numerical details will be presented in subsequent publications.
Abstract We analyze boundary spin correlation functions of the hyperbolic-lattice Ising model from the holographic point of view. Using the corner-transfer-matrix renormalization-group (CTMRG) method, we demonstrate that the boundary correlation function exhibits power-law decay with quasiperiodic oscillation, while the bulk correlation function always decays exponentially. On the basis of the geometric relation between the bulk correlation path and distance along the outer edge boundary, we find that scaling dimensions for the boundary correlation function can be well explained by a combination of the bulk correlation length and background curvatures inherent to the hyperbolic lattice. We also investigate the cutoff effect of the bond dimension in CTMRG, revealing that the long-distance behavior of the boundary spin correlation is accurately described even with a small bond dimension. In contrast, the short-distance behavior rapidly loses its accuracy.
In recent years, tensor network states have emerged as a very useful conceptual and simulation framework to study quantum many-body systems at low energies. In this paper, we describe a particular way in which any given tensor network can be viewed as a representation of two different quantum many-body states. The two quantum many-body states are said to correspond to each other by means of the tensor network. We apply this ``tensor network state correspondence''---a correspondence between quantum many-body states mediated by tensor networks as we describe---to the multi-scale entanglement renormalization ansatz (MERA) representation of ground states of one dimensional (1D) quantum many-body systems. Since the MERA is a 2D hyperbolic tensor network (the extra dimension is identified as the length scale of the 1D system), the two quantum many-body states obtained from the MERA, via tensor network state correspondence, are seen to live in the bulk and on the boundary of a discrete hyperbolic geometry. The bulk state so obtained from a MERA exhibits interesting features, some of which caricature known features of the holographic correspondence of String theory. We show how (i) the bulk state admits a description in terms of ``holographic screens'', (ii) the conformal field theory data associated with a critical ground state can be obtained from the corresponding bulk state, in particular, how pointlike boundary operators are identified with extended bulk operators. (iii) We also present numerical results to illustrate that bulk states, dual to ground states of several critical spin chains, have exponentially decaying correlations, and that the bulk correlation length generally decreases with increase in central charge for these spin chains.