---
title: Holography of Information
url: https://www.emergentmind.com/topics/holography-of-information-hoi
type: topic
---

# Holography of Information

Holography of Information (HoI) denotes a family of claims about redundancy, reconstruction, and encoding in which information ordinarily attributed to a bulk system is available on, or arbitrarily near, an appropriate boundary. In its strict quantum-gravity usage, HoI states that in a theory of quantum gravity a copy of all the information on a Cauchy slice is also available near the boundary of that slice, and that this redundancy is already present at low energy. In AdS/CFT and related formulations, the same idea is sharpened into the claim that geometry is an emergent organization of quantum information, with entanglement, extremal surfaces, and boundary algebras supplying the microscopic language. The term has also been extended to finite-radius reconstruction, evaporating black-hole models, sectorwise formulations in topology-summed gravity, and several broader programs in thermodynamics, many-body theory, cosmology, and information processing [2210.11066][2506.06595][2410.17316].

## 1. Canonical statement and boundary observables

A canonical formulation of HoI is given by the question whether a suitable boundary Hamiltonian has a nontrivial Dirac bracket with bulk operators. In constrained Hamiltonian language, with Dirac matrix
$$
C_{ij}(x,y)\equiv \{\Phi_i(x),\Phi_j(y)\},
$$
the Dirac bracket is
$$
\{F(x_1),G(x_2)\}_{\text{D.B.}}=\{F(x_1),G(x_2)\}-\int dy_1\,dy_2\; \{F(x_1),\Phi_i(y_1)\}\, C^{-1}_{ij}(y_1,y_2)\, \{\Phi_j(y_2),G(x_2)\}.
$$
On this formulation, HoI holds when a boundary observable probes bulk data through a nonzero Dirac bracket, and fails when the corresponding bracket vanishes [2301.01075].

For linearized massless gravity, the relevant boundary observable is obtained from the Gauss/Hamiltonian constraint. Integrating the constraint over a region \(V\) gives
$$
H_{\partial}=\frac{1}{16\pi G_N}\int_{\partial V} d^{D-2}x\; n^i\!\left(\partial_j h_{ij}-\partial_i h^k{}_k\right).
$$
The inverse Dirac matrix is nonlocal, with momentum-space factors like \(1/\mathbf p^2\), and this nonlocality is the structural reason the boundary Hamiltonian acts on bulk operators. The resulting bracket takes the form
$$
\{H_{\partial},O(z)\}_{\text{D.B.}}=\int_V d^d x\,\{T_{00}(x),O(z)\},
$$
so the boundary Hamiltonian brackets with bulk operators exactly as the integrated matter energy would. In this sense, bulk information is recoverable from asymptotic observables, and split states are absent [2301.01075].

The same analysis yields a sharp contrast in linearized Fierz–Pauli massive gravity. The boundary operator can still be defined from the integrated constraint, but the full constraint matrix is local in momentum space rather than carrying the Gauss-law nonlocal inverse structure of the massless case. The resulting Dirac bracket with a strictly bulk insertion collapses to
$$
\{H_{\partial},O(z)\}_{\text{D.B.}}=0.
$$
The stated consequence is that asymptotic boundary observables cannot extract bulk information, local bulk operators remain hidden, and split states exist. The difference is attributed to diffeomorphism invariance: massless gravity ties physical observables to boundary dressings, whereas massive gravity explicitly breaks that redundancy [2301.01075].

## 2. AdS/CFT formulation and bilocal holography

In AdS/CFT, HoI is translated into boundary CFT language by combining boundary vacuum cyclicity with a bilocal reconstruction of bulk fields. The gravity-side motivation uses the Reeh–Schlieder theorem together with the fact that the energy can be measured from near the boundary, so the vacuum projector belongs to the boundary algebra. The resulting claim is that every observable in the vacuum sector is a boundary observable. The AdS/CFT translation of this claim is carried out explicitly in bilocal holography for the free \(O(N)\) vector model dual to higher-spin gravity in \(AdS_4\) [2210.11066].

The basic collective variable is the bilocal field
$$
\sigma(x^+,x_1^-,x_1,x_2^-,x_2)=\phi^a(x^+,x_1^-,x_1)\phi^a(x^+,x_2^-,x_2),
$$
with the normal-ordered fluctuation
$$
\eta(x^+,x_1^-,x_1,x_2^-,x_2)=:\phi^a(x^+,x_1^-,x_1)\phi^a(x^+,x_2^-,x_2):.
$$
The bilocal-to-bulk map is
$$
X=\frac{p_1^+ x_1+p_2^+ x_2}{p_1^++p_2^+},\qquad
Z=\frac{\sqrt{p_1^+ p_2^+}\,|x_1-x_2|}{p_1^++p_2^+},
$$
together with
$$
P^+=p_1^++p_2^+,\qquad
\theta=2\tan^{-1}\!\left(\sqrt{\frac{p_2^+}{p_1^+}}\right).
$$
Since \(Z\) grows with \(|x_1-x_2|\), small-separation bilocals correspond to near-boundary bulk information, while large-separation bilocals probe deep interior bulk data [2210.11066].

The CFT statement of HoI is then an operator-product statement. Bilocals admit an OPE of the schematic form
$$
\sum_{a=1}^N\phi^a(x^\mu+y^\mu)\phi^a(x^\mu-y^\mu)
=\sum_{\cal O} f_{\phi\phi {\cal O}}\,P_{\cal O}(y^\mu,\partial_x)\,{\cal O}(x),
$$
where \({\cal O}\) runs over local primaries and descendants. Because the relevant Lorentzian OPEs are convergent in the appropriate channels, a bilocal operator representing a deep-bulk excitation can be rewritten in terms of boundary-local operators. The paper emphasizes that bulk reconstruction is “fluid”: the same bulk point can be reached by infinitely many semicircles, so one can choose bilocals to avoid OPE obstructions. In that precise sense, the redundancy of information near the AdS boundary becomes an explicit CFT statement [2210.11066].

## 3. Entanglement, extremal surfaces, and emergent spacetime

A stronger informational interpretation identifies holographic spacetime itself as emergent from quantum information. The starting point is the black-hole entropy relation
$$
S_{BH}=\frac{\mathrm A(\Sigma)}{4G_N},
$$
which suggests that gravity organizes entropy by area rather than volume. In this view, a \(d+1\)-dimensional gravitational theory can be equivalent to a \(d\)-dimensional non-gravitational quantum theory on the boundary, and the deep reason holography works is that geometry encodes entanglement [2506.06595].

The basic quantity is the entanglement entropy of a boundary subregion \(A\),
$$
S_A=-\mathrm{Tr}\,\rho_A\log\rho_A,
$$
with holographic prescription
$$
S_A=\mathrm{Ext}_{\Gamma_A}\left[\frac{\mathrm{Area}(\Gamma_A)}{4G_N}\right].
$$
This is the Ryu–Takayanagi / Hubeny–Rangamani–Takayanagi bridge between quantum information and geometry. The prescription makes the area law natural, gives strong subadditivity and related entropy inequalities a geometric interpretation, and identifies the bulk dual of \(A\) as the entanglement wedge, the region enclosed by \(A\) and \(\Gamma_A\) [2506.06595].

Quantum corrections are incorporated by the quantum extremal surface formula
$$
S_A^{(q)}=\mathrm{Ext}_{\Gamma_A}\left[\frac{\mathrm{Area}(\Gamma_A)}{4G_N}+S^{(G)}_{M_A}\right],
$$
where \(S^{(G)}_{M_A}\) is the bulk entanglement entropy in the entanglement wedge \(M_A\). A further generalization is the island formula,
$$
S_A^{(I)}=\mathrm{Ext}_{\Sigma}\left[\frac{\mathrm{Area}(\Sigma)}{4G_N}+S^{(G)}_{A\cup \Sigma}\right].
$$
This formula is described as crucial for black hole evaporation and the Page curve, with replica wormholes contributing to its derivation, although the Lorentzian real-time mechanism remains incompletely understood [2506.06595].

The same essay gives a microscopic heuristic: each Planck-scale patch of area may be associated with entangled qubits, often heuristically like Bell pairs living on extremal surfaces. Spacetime is then a collective manifestation of a vast entanglement network, and the bulk gravitational field encodes the dynamics of entanglement among qubits. Tensor networks, especially MERA-like networks in which entanglement entropy is estimated by counting bonds crossing a minimal cut, serve as toy models of this geometry-from-circuit structure. A major open problem is the identification of the quantum circuit architecture corresponding to a given holographic spacetime, together with the relation of bulk volume or Wheeler–DeWitt action to quantum complexity [2506.06595].

The informational program is extended beyond AdS by the proposal of pseudo-entropy and time-like entanglement. For initial and final states \(|\Psi_i\rangle\) and \(|\Psi_f\rangle\), one defines
$$
\tau_A=\mathrm{Tr}_{\bar A}\frac{|\Psi_i\rangle\langle\Psi_f|}{\langle \Psi_f|\Psi_i\rangle},
\qquad
S_A^{(p)}=-\mathrm{Tr}\,[\tau_A\log\tau_A].
$$
Because \(\tau_A\) is generally non-Hermitian, pseudo-entropy may be complex. The proposal is that the imaginary part of pseudo-entropy, together with time-like entanglement and partly time-like extremal surfaces, may provide a clue to how the time direction emerges in Lorentzian or cosmological holography, particularly in dS/CFT where central charges and geodesic lengths can become complex [2506.06595].

## 4. Finite-radius and low-dimensional realizations

HoI is not restricted to asymptotic infinity. One finite-distance formulation asks whether information inside a ball of finite radius \(R\) in flat spacetime can already be reconstructed from observables on the boundary of that ball. The setting is \((3+1)\)-dimensional Minkowski space with linearised quantum gravity minimally coupled to a free Klein–Gordon scalar, on the \(t=0\) Cauchy slice. States are unitary excitations of the vacuum built from normal-ordered products of \(\phi\) and \(\pi\), with a small perturbative parameter \(\gamma\), and a crucial assumption is that the state has no support in a small neighborhood of the boundary where the measurements are made [2410.17316].

The boundary observable is the Hamiltonian inside the ball,
$$
H_R(t)=\int_{|\vec x|\le R} d^3x\, \mathcal H(t,\vec x),
\qquad
\mathcal H=:\tfrac12\big(\pi^2+(\nabla\phi)^2+m^2\phi^2\big):.
$$
Because the theory is coupled to linearised gravity, this finite-volume Hamiltonian can be rewritten as a boundary term on the sphere \(r=R\). The central result is that boundary correlators with an insertion of the finite-radius Hamiltonian determine the same support functions \(f_n\) as the bulk correlators do. For the most general unitaries containing both \(\phi\) and \(\pi\) insertions, the result is established to first order in \(\gamma\); for the infinite subclass generated only by \(\phi\)-insertions or only by \(\pi\)-insertions, the result is perturbatively exact to all orders in \(\gamma\). The same framework treats massive and massless scalar fields simultaneously [2410.17316].

A second explicit realization appears in the CGHS model coupled to a massless scalar. Here the scalar decomposes into chiral components,
$$
f(x^\pm)=f^+(x^+)+f^-(x^-),
$$
and right future null infinity \(\mathcal I_R^+\) is naturally associated with the right-moving sector \(f^-(x^-)\). The reduced radiative phase space on \(\mathcal I_R^+\) has symplectic form
$$
\Omega\big|_{\mathcal I^+_R}
=
\frac{1}{2}\int_{\mathcal{I}^+_R} dx^-\; \bigl[\delta_1 f^-\;\partial_-(\delta_2 f^-)-\delta_2 f^-\;\partial_-(\delta_1 f^-)\bigr],
$$
and the right Hamiltonian is
$$
H_R=\frac12 \int_{\mathcal I_R^+} dx^-\,T_{--}.
$$
Adding the Bondi mass at all cuts to the radiative algebra yields a gravitationally completed asymptotic algebra, and under the usual HoI assumptions the paper proves
$$
A_{\rm grav}(I_\epsilon)=B(\mathcal H_{f^-}),
\qquad
I_\epsilon=(-\infty,-1/\epsilon),
$$
for every \(\epsilon>0\). The quantum state of the right-moving modes can therefore be recovered from an arbitrarily small neighborhood of the past end of \(\mathcal I_R^+\), and the corresponding Page curve is flat [2606.09346].

| Setting | Boundary data | Result |
|---|---|---|
| Finite ball in linearised gravity [2410.17316] | Correlators with \(H_R\) on \(r=R\) | Boundary data determine the same support functions \(f_n\) as bulk correlators |
| CGHS model [2606.09346] | \(A_{\rm grav}(I_\epsilon)\) near \(\mathcal I_R^+\) | \(A_{\rm grav}(I_\epsilon)=B(\mathcal H_{f^-})\) and the right-moving state is recoverable |

These finite-radius and low-dimensional results shift HoI from an asymptotic slogan to controlled reconstruction statements. They also delimit scope: the finite-ball theorem is perturbative and low-energy, while the CGHS result reconstructs the right-moving sector rather than the full two-boundary state [2410.17316][2606.09346].

## 5. Sectorwise holography, islands, and factorization problems

Recent work has emphasized that HoI need not be a global statement on the full topology-summed Hilbert space. If topology-changing saddles generate genuine baby-universe or \(\alpha\)-sector data, then the physical Hilbert space decomposes sectorwise,
$$
\mathcal H_{\rm phys}=\int d\alpha\,\mathcal H_\alpha,
$$
and asymptotic completeness is naturally reformulated as
$$
\overline{\mathcal A_\infty^{(\alpha)}|0_\alpha\rangle}=\mathcal H_\alpha.
$$
On this account, HoI survives only as a fixed-\(\alpha\) statement unless additional input makes the sector label asymptotically accessible, aligns reconstructions across sectors, or removes the sector structure entirely [2605.20294].

This refinement has consequences for black-hole interior reconstruction. In a fixed \(\alpha\)-sector, HoI may still obstruct AMPS factorization and permit a smooth horizon. In an unconditioned topology-summed state, however, the obstruction is not automatic. The Bell-pair diagnostic uses
$$
|\Phi\rangle=\frac{1}{\sqrt d}\sum_{i=1}^d |i\rangle|i\rangle,
\qquad
|\Phi_\alpha\rangle=(\mathbf 1\otimes U_\alpha)|\Phi\rangle,
$$
and evaluates the fidelity of a sector-independent reconstruction by
$$
F(V)=\int d\alpha\, p(\alpha)\,\frac{|\mathrm{Tr}(V^\dagger U_\alpha)|^2}{d^2}.
$$
The conclusion is that sectorwise smoothness does not imply sector-independent smoothness: a single global smooth interior exists only if the \(U_\alpha\) are aligned up to phases, if the observer can resolve \(\alpha\), or if there is no genuine \(\alpha\)-structure. The resulting tension is with the expectation that horizon normalcy should be determined by local semiclassical geometry. If the Baby Universe Hypothesis holds and \(\dim\mathcal H_{\rm BU}=1\), the obstruction disappears [2605.20294].

A different factorization issue appears in the CGHS/RST comparison between HoI and the island formula. In the CGHS analysis, HoI gives a flat Page curve because the gravitationally completed asymptotic algebra already equals \(B(\mathcal H_{f^-})\). The authors then argue that if Hilbert-space factorization between left- and right-moving sectors,
$$
\mathcal H \sim \mathcal H_{f^+}\otimes\mathcal H_{f^-},
$$
holds in the quantum theory, then the island entropy cannot be identified with the von Neumann entropy of any one-sided asymptotic algebra. On this view, the discrepancy between HoI and islands is not exhausted by a choice between radiative and gravitationally completed algebras; it becomes a version of the factorization problem for CGHS/RST models [2606.09346].

These analyses jointly imply that HoI is conditional on representation, gauge structure, and sector decomposition. In that restricted sense, the principle is neither a universal substitute for nonperturbative quantum gravity nor a uniform statement about all boundary algebras.

## 6. Broader extensions and analogical usages

Beyond the strict gravitational principle, “holography of information” and closely related holographic language have been extended to several other domains. Some of these are presented as microscopic realizations of area-scaling information storage, others as analogical or operational encodings on lower-dimensional data.

| Domain | Representative formulation | Core claim |
|---|---|---|
| Thermodynamics [1411.6342] | Partition function on a contour in complex parameter space | Boundary values of \(Z(\beta,\lambda)\) determine thermodynamics inside an analytic region |
| Cosmology [1105.4461] | Landauer + holographic scaling | Information energy can behave like dark energy |
| Gauge boundaries [1511.03525] | Boundary Stückelberg qubits | Information capacity is controlled by the Stückelberg energy gap |
| Many-body criticality [1810.02336] | Gapless modes on a critical level | Area-scaling memory space with “burden of memory” stabilization |
| Screens and computation [1302.3470] | Information from transition probabilities across a screen | Spherical screens encode \(I=2\pi\Delta\varphi\) and imply a finite computation bound |
| Magnonic devices [1401.5133] | Spin-wave interference in magnetic matrices | Information is recovered from distributed interference patterns |
| Topological coding [2011.11040] | Pure braid group generators | Information is represented in a two-dimensional topological formalism |
| Holographic algorithms [1005.1349] | Holant theorem and c-tensor | Global sums are invariant under suitable holographic basis transformations |

In thermodynamic holography, the partition function
$$
Z(\beta,\lambda)=\mathrm{Tr}\,e^{-\beta H(\lambda)}
$$
is treated as an analytic function of a complexified physical parameter, so that Cauchy’s formula reconstructs interior values from boundary data,
$$
Z(\beta,\lambda')=\frac{1}{2\pi i}\oint_{\mathcal C}\frac{Z(\beta,\lambda)}{\lambda-\lambda'}\,d\lambda.
$$
The experimental link is probe-spin coherence,
$$
\langle S_+(t)\rangle=\frac{Z(\beta,\lambda + i t n/\beta)}{Z(\beta,\lambda)},
$$
which directly measures the partition function on a vertical line in the complex plane and therefore permits holographic reconstruction of free-energy differences [1411.6342].

In the Stückelberg formulation, an information boundary forces the introduction of boundary Stückelberg degrees of freedom that act as qubits. Their information capacity is measured by
$$
I\equiv \frac{\hbar}{R\,\Delta},
$$
and for a massless gauge system the gap scales as \(\Delta_{\text{Stückelberg}}=\alpha(R)\hbar/R\), giving
$$
I=\frac{1}{\alpha(R)}.
$$
For gravity, \(\alpha_G(R)\sim L_P^2/R^2\), so the resulting capacity scales as \(R^2/L_P^2\) and saturates the Bekenstein bound. The paper identifies this with maximal holography and proposes a correspondence between boundary Stückelberg modes and Bogoliubov modes of critical many-body systems [1511.03525].

The many-body model of bosons on \(S^d\) gives an explicit microscopic realization of area-law storage. At a critical occupation number of a master mode, a level of gapless modes emerges whose degeneracy scales as
$$
\mathcal N_k\sim (k_d)^{d-1}
\sim \left(\frac{R}{L_k}\right)^{d-1},
$$
and the binary memory space has entropy
$$
S=\mathcal N_k\ln 2.
$$
The distinctive dynamical claim is the “burden of memory”: when the system evolves away from criticality, patterns loaded into the gapless sector increase the energy cost of departure from the holographic state, so heavily loaded memories survive longer and information is off-loaded only by transitions between successive holographic states, with scrambling and entanglement during the transfer [1810.02336].

A cosmological extension combines Landauer’s principle with holographic scaling. The information energy is defined as
$$
E_{\text{info}}=N k_B T\ln 2,
$$
with \(N\propto a^2\) from holographic scaling of information content and, at late times, \(T\propto a\). The resulting information energy density becomes approximately constant for \(z\lesssim 1\), with effective \(w\approx -1\), and is proposed as a model of dark energy with a finite acceleration epoch tied to star formation history [1105.4461].

Other works use holographic language more operationally. A computational-screen analysis defines information as the logarithm of the exponentially small transition probability across a screen, yielding \(I\simeq 2ks\) for a plane and \(I=2\pi\Delta\varphi\) for a spherical screen, and from this derives a bound of about \(10^{121}\) future computations in the visible Universe [1302.3470]. Magnonic holographic memory implements distributed encoding in spin-wave interference patterns within a YIG magnetic matrix, where local junction magnets modulate phases and the stored configuration is reconstructed from the output hologram [1401.5133]. A pure braid group proposal treats classical information as effectively two-dimensional and encodes symbols as pure braid generators \(l_{ij}\), while a proof of the Holant theorem supplies the c-tensor formalism behind holographic algorithms used in information-theoretic counting problems [2011.11040][1005.1349].

Taken together, these broader programs suggest that “holography of information” now names both a specific gravitational redundancy principle and a wider research style in which lower-dimensional, boundary, or collective data suffice to determine a higher-dimensional or more distributed informational structure. In the narrow sense, HoI concerns bulk reconstruction in quantum gravity; in the broad sense, it names a recurrent strategy for encoding global structure in constrained boundary data.

Source: https://www.emergentmind.com/topics/holography-of-information-hoi