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arXiv:2403.18020 · Physical review. D/Physical review. D.

Entanglement entropy of coherent intertwiner in loop quantum gravity

Gaoping Long, Qian Chen, Jinsong Yang

Published Sep 5, 2024• 7 citations
arXivHTMLPDF

Abstract

In this paper, we carry out the entanglement calculations on the coherent intertwiners. We first consider the entanglement introduced by the group averaging of the tensor-product-type intertwiner on a 4-valents vertex. Then, we calculate explicitly the entanglement carried by the gauge-invariant coherent intertwiner with four legs. Our numerical results show that the entanglement can be controlled by the local semiclassical geometry described by the coherent intertwiner. We also extend our analytical calculation to the coherent intertwiners with an arbitrary number of legs. Especially, we apply the previous results to the entanglement of the spin-network state labeled by coherent intertwiners, with the network puncturing a boundary by its vertices. We show that the entanglement of such spin-network state is not only determined by the area of the boundary but also carries a quantum correction controlled by the semiclassical geometry associated to the vertices puncturing the boundary.

DOI
10.1103/physrevd.110.064017

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In this paper, we carry out the entanglement calculations on the coherent intertwiners. We first consider the entanglement introduced by the group averaging of the tensor-product-type intertwiner on a 4-valents vertex. Then, we calculate explicitly the entanglement carried by the gauge-invariant coherent intertwiner with four legs. Our numerical results show that the entanglement can be controlled by the local semiclassical geometry described by the coherent intertwiner. We also extend our analytical calculation to the coherent intertwiners with an arbitrary number of legs. Especially, we apply the previous results to the entanglement of the spin-network state labeled by coherent intertwiners, with the network puncturing a boundary by its vertices. We show that the entanglement of such spin-network state is not only determined by the area of the boundary but also carries a quantum correction controlled by the semiclassical geometry associated to the vertices puncturing the boundary.
In the context of quantum gravity, we clarify entanglement calculations on spin networks: we distinguish the gauge-invariant entanglement between intertwiners located at the nodes and the entanglement between spin states located on the network's links. We compute explicitly these two notions of entanglement between neighboring nodes and show that they are always related to the typical ln(2j+1)\mathrm{ln}(2j+1)ln(2j+1) term depending on the spin jjj living on the link between them. This ln(2j+1)\mathrm{ln}(2j+1)ln(2j+1) contribution comes from looking at nongauge-invariant states; thus we interpret it as gauge breaking and unphysical. In particular, this confirms that pure spin network basis states do not carry any physical entanglement, so that true entanglement and correlations in loop quantum gravity comes from spin or intertwiner superpositions.
The space Inv(j1,j2,j3,j4)(j_1,j_2,j_3,j_4)(j1​,j2​,j3​,j4​) of SU(2)-invariant four-valent tensors, also known as intertwiners, can be understood as the quantum states of a tetrahedron in Euclidean space with fixed areas. In loop quantum gravity, they are states of the smallest ``atom of space" with non-zero volume. At the same time they correspond to four-party tensor product states invariant under global rotations. We consider the multipartite entanglement of states in Inv(j1,j2,j3,j4)(j_1,j_2,j_3,j_4)(j1​,j2​,j3​,j4​) using the recently proposed entropic fill. Numerically evaluating entropic fill in the case of equal spins between 1/21/21/2 and 111111, we find that the distributions of entanglement are very different for intertwiners as compared to generic tensors, and for coherent intertwiners as compared to generic ones. The peak in the distribution seems to be at the highest entanglement for generic intertwiners and at the lowest for generic tensors, but in terms of average entanglement, the roles are switched: average entanglement is highest in arbitrary tensors and lower in intertwiners, at least in the regime of large jjj. We also consider the dependence of entanglement on the geometric data of coherent intertwiners. Coherent intertwiners labeled by a symmetric tetrahedron have maximal entanglement, while minimal entanglement implies a coherent state labeled by a degenerate tetrahedron. Away from these cases, entanglement depends on the geometry in an intricate way.
The non-factorizing nature of the Hilbert space in Loop Quantum Gravity (LQG) due to gauge invariance requires a generalized definition of entanglement entropy. This work employs the framework of von Neumann algebras to investigate the entanglement entropy in LQG. On a graph, the holonomy and flux operators within a region and on the boundary generate a non-factor type I von Neumann algebra, which is used to define the entanglement entropy for LQG states. This algebraic formalism is applied to ``fixed-area states''--superpositions of spin networks associated with a surface with a definite macroscopic area given by the LQG area spectrum. By maximizing the entropy, we derive a geometrical area law where the entanglement entropy is proportional to the area. In addition, we show that bulk entanglement can renormalize the area-law coefficient and produce logarithmic corrections. The results in this paper closely relate to LQG black hole entropy.
The relation between loop quantum gravity (LQG) and tensor networks is explored from the perspectives of the bulk-boundary duality and holographic entanglement entropy. We find that the LQG spin-network states in a space $\mathrm{\ensuremath{\Sigma}}$ with boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$ is an exact holographic mapping similar to the proposal in [X.-L. Qi, Exact holographic mapping and emergent space-time geometry, arXiv:1309.6282]. The tensor network, understood as the boundary quantum state, is the output of the exact holographic mapping emerging from a coarse-graining procedure of spin networks. Furthermore, when a region AAA and its complement A‾\overline{A}A are specified on the boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$, we show that the boundary entanglement entropy S(A)S(A)S(A) of the emergent tensor network satisfies the Ryu-Takayanagi formula in the semiclassical regime, i.e., S(A)S(A)S(A) is proportional to the minimal area of the bulk surface attached to the boundary of AAA in $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$.
For quantum gravity states associated with open spin network graphs, we study how the entanglement entropy of the boundary degrees of freedom (spins on open edges) is affected by the bulk data, specifically by its combinatorial structure and by the quantum correlations among intertwiner degrees of freedom. For a specific assignment of bulk edge spins and slightly entangled intertwiners, we recover the Ryu-Takayanagi formula (with a properly discrete geometric notion of area, thanks to the underlying quantum gravity formalism) and its corrections due to the entanglement entropy of the bulk state. We also show that the presence of a region with highly entangled intertwiners deforms the minimal-area surface, which is then prevented from entering that region when the entanglement entropy of the latter exceeds a certain bound. This entanglement-based mechanism leads thus to the formation of a black hole--like region in the bulk.
We investigate the bipartite entanglement for the boundary states in a simple type of spin networks with dangling edges, in which the two complementary parts are linked by two or more edges. Firstly, the spin entanglement is considered in the absence of the intertwiner entanglement. By virtue of numerical simulations, we find that the entanglement entropy usually depends on the group elements. More importantly, when the intertwiner entanglement is taken into account, we find that it is in general impossible to separate the total entanglement entropy into the contribution from spins on edges and the contribution from intertwiners at vertices. These situations are in contrast to the case when the two vertices are linked by a single edge.