A note on entanglement entropy and quantum geometry

Norbert Bodendorfer

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A note on entanglement entropy and quantum geometry

Norbert Bodendorfer

General Relativity and Quantum Cosmology · Submitted 5 Feb 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We recall existing work on this issue and explain how recent work on generalising these computations to arbitrary spacetime dimensions D+1>2 supports this point of view and makes the duality between entanglement entropy and the entropy computed from counting boundary states manifest.

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It has been argued that the entropy which one is computing in the isolated horizon framework of loop quantum gravity is closely related to the entanglement entropy of the gravitational field and that the calculation performed is not restricted to horizons. We recall existing work on this issue and explain how recent work on generalising these computations to arbitrary spacetime dimensions D+1>2 supports this point of view and makes the duality between entanglement entropy and the entropy computed from counting boundary states manifest. In a certain semiclassical regime in 3+1 dimensions, this entropy is given by the Bekenstein-Hawking formula.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	It has been argued that the entropy which one is computing in the isolated horizon framework of loop quantum gravity is closely related to the entanglement entropy of the gravitational field and that the calculation performed is not restricted to horizons.	background 1.00	0.05	0.35	hedged possibility 0.55
s1	We recall existing work on this issue and explain how recent work on generalising these computations to arbitrary spacetime dimensions D+1>2 supports this point of view and makes the duality between entanglement entropy and the entropy computed from counting boundary states manifest.	contribution 0.68	0.24	0.43	asserted 0.98
s2	In a certain semiclassical regime in 3+1 dimensions, this entropy is given by the Bekenstein-Hawking formula.	result 0.70	0.17	0.15	asserted 0.99

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Original abstract

It has been argued that the entropy which one is computing in the isolated horizon framework of loop quantum gravity is closely related to the entanglement entropy of the gravitational field and that the calculation performed is not restricted to horizons. We recall existing work on this issue and explain how recent work on generalising these computations to arbitrary spacetime dimensions D+1>2 supports this point of view and makes the duality between entanglement entropy and the entropy computed from counting boundary states manifest. In a certain semiclassical regime in 3+1 dimensions, this entropy is given by the Bekenstein-Hawking formula.

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12 arXiv papers cite this in 18 sentences

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Equation [ref] is structurally very similar to the isolated horizon boundary condition [ref] F_bc^i = - c βE^a i ϵ_abc, used in the context of black hole in loop quantum gravity[In computations of black hole entropy following [ref] , one is actually not imposing [ref], but only that the curvature in [ref] derives from the connection that one uses in the boundary symplectic structure [ref] .

Some notes on the Kodama state, maximal symmetry, and the isolated horizon boundary condition · The isolated horizon boundary condition · 0.90

Cites it as support 1 paper

Quanta of area always are accompanied by quanta of entanglement [ref] .

An elementary introduction to loop quantum gravity · Arguments for canonical loop quantum gravity · 0.97

Cites it as background 6 papers

In this subsection, we will briefly introduce the higher-dimensional connection variables, and review the entropy results [ref] that arise from them.

The Wald entropy formula and loop quantum gravity · Entropy in higher-dimensional LQG · 0.86
Before the analysis of the full amplitude $A()$ in Section [ref], Sections [ref]-[ref] make a modification of the amplitude by imposing weakly the parallel restriction to internal $ξ_f$'s, and applies the modified amplitude to the study of entanglement entropy in LQG (see e.g. [ref] for some existing studies of entanglement entropy in LQG).

Semiclassical Behavior of Spinfoam Amplitude with Small Spins and Entanglement Entropy · Introduction · 0.98
Effort has also been made to calculate the entanglement entropy of an arbitrary boundary by LQG [ref] .

Entropy of black holes with arbitrary shapes in loop quantum gravity · Introduction · 0.82

What it cites · 104 sentences without a confident label

Builds on 2 papers

* The results of this paper extend to Lanczos-Lovelock gravity by the results of [ref] when the boundary is restricted to be a non-rotating isolated horizon.

The Wald entropy formula and loop quantum gravity · Comments · 0.90
This argument generalises to higher dimensions by taking the appropriate wedge products of $R^0$, see [ref] .

New Variables for Classical and Quantum Gravity in all Dimensions V. Isolated Horizon Boundary Degrees of Freedom · Further constraints from gauge invariance · 0.95

Uses the method or tool of 1 paper

One only imposes a maximally commuting subset [ref] of the off-diagonal simplicity constraints, which has a maximally commuting subset of the angles [ref] as observables and generates (or gauge fixes) only gauge transformations which leave this subset of [ref] invariant.

On the Implementation of the Canonical Quantum Simplicity Constraint · Further constraints from gauge invariance · 0.90

Compares with 4 papers

This entropy agrees with the one calculated from the isolated horizon computations [ref] up to corrections resulting from gauge invariance constraints which we will discuss in section [ref].

Black hole entropy and SU(2) Chern-Simons theory · Entropy from entanglement and boundary Hilbert spaces · 0.99
This entropy agrees with the one calculated from the isolated horizon computations [ref] up to corrections resulting from gauge invariance constraints which we will discuss in section [ref].

Quantum Geometry of Isolated Horizons and Black Hole Entropy · Entropy from entanglement and boundary Hilbert spaces · 0.99
Furthermore, the computation was shown to be analogous to the entanglement entropy computation performed in [ref] .

Entanglement Entropy in Loop Quantum Gravity · Conclusion · 0.94

Cites as support 1 paper

This provides further evidence for the conjecture stated in [ref] that entanglement entropy is a probe for the architecture of spacetime.

On the Architecture of Spacetime Geometry · Comments · 1.00

Cites as background 26 papers

While this paper doesn't provide (long) technical calculations as e.g. its companions [ref] , it focuses on some conceptual questions left open there which have, to the best of the author's knowledge, not been addressed in a comprehensive fashion elsewhere in the literature.

New Variables for Classical and Quantum Gravity in all Dimensions V. Isolated Horizon Boundary Degrees of Freedom · Introduction · 0.80
While this paper doesn't provide (long) technical calculations as e.g. its companions [ref] , it focuses on some conceptual questions left open there which have, to the best of the author's knowledge, not been addressed in a comprehensive fashion elsewhere in the literature.

Black hole entropy from loop quantum gravity in higher dimensions · Introduction · 0.80
While this paper doesn't provide (long) technical calculations as e.g. its companions [ref] , it focuses on some conceptual questions left open there which have, to the best of the author's knowledge, not been addressed in a comprehensive fashion elsewhere in the literature.

The Wald entropy formula and loop quantum gravity · Introduction · 0.80

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