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arXiv:2305.02670 · Physical review. D/Physical review. D.

Curvature from multipartite entanglement in quantum gravity states

Simone Cepollaro, Goffredo Chirco, Gianluca Cuffaro, Vittorio D’Esposito

Published Aug 16, 2023• 4 citations
arXivHTMLPDF

Abstract

We investigate multipartite entanglement in quantum geometry states described by loop quantum gravity spin networks. We focus on states corresponding to bounded regions of 3D spacelike slices of spacetime with nonvanishing intrinsic curvature, realized via spin networks defined on a graph with nontrivial $SU(2)$ holonomies. The presence of intrinsic curvature in the region is encoded---via coarse graining---into tag spins attached to the vertices of the graph. The resulting states are generalized bulk-to-boundary maps defined in an extended boundary Hilbert space comprised by the spin representations carried by the tags and by the uncontracted edges at the boundary. We consider a tripartition of a random quantum geometry state in the extended-boundary space consisting of a bipartite boundary subsystem and a bulk-tags complement, and we propose a measure of logarithmic negativity to study the change in the entanglement phases of the boundary marginal mixed state, while varying the dimension of the bulk curvature environment. In the large spin regime, we find that the typical entanglement negativity is well described by the generalized Page curve of a tripartite random state. In particular, we find area scaling behavior of negativity for small curvature, while for large curvature the negativity vanishes, suggesting an effective thermalization of the boundary. Remarkably, the positive partial transpose character of the mixed boundary state corresponds to a change in the effective topology of the network, with the two boundary subregions becoming disconnected.

DOI
10.1103/physrevd.108.046010

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We investigate multipartite entanglement in quantum geometry states described by loop quantum gravity spin networks. We focus on states corresponding to bounded regions of 3D spacelike slices of spacetime with nonvanishing intrinsic curvature, realized via spin networks defined on a graph with nontrivial SU(2)SU(2)SU(2) holonomies. The presence of intrinsic curvature in the region is encoded---via coarse graining---into tag spins attached to the vertices of the graph. The resulting states are generalized bulk-to-boundary maps defined in an extended boundary Hilbert space comprised by the spin representations carried by the tags and by the uncontracted edges at the boundary. We consider a tripartition of a random quantum geometry state in the extended-boundary space consisting of a bipartite boundary subsystem and a bulk-tags complement, and we propose a measure of logarithmic negativity to study the change in the entanglement phases of the boundary marginal mixed state, while varying the dimension of the bulk curvature environment. In the large spin regime, we find that the typical entanglement negativity is well described by the generalized Page curve of a tripartite random state. In particular, we find area scaling behavior of negativity for small curvature, while for large curvature the negativity vanishes, suggesting an effective thermalization of the boundary. Remarkably, the positive partial transpose character of the mixed boundary state corresponds to a change in the effective topology of the network, with the two boundary subregions becoming disconnected.
We investigate multipartite entanglement for quantum states of 3d space geometry, described via generalised random spin networks with fixed areas, in the context of background independent approaches to quantum gravity. We focus on entanglement negativity as a well defined witness of quantum correlations for mixed states, in our setting describing generic subregions of the boundary of a quantum 3d region of space. In particular, we consider a generic tripartition of the boundary of an open spin network state and we compute the typical Rényi negativity of two boundary subregions A and B immersed in the environment C, explicitly for a set of simple open random spin network states. We use the random character of the spin network to exploit replica and random average techniques to derive the typical Rényi negativty via a classical generalised Ising model correspondence, generally used for random tensor networks in the large bond regime. For trivially correlated random spin network states, with only local entanglement between spins located on the network edges, we find that typical log negativity displays a holographic character, in agreement with the results for random tensor networks, in large spin limit. When non-local bulk entanglement between intertwiners at the vertices is considered the negativity increases, while at the same time the holographic scaling is generally perturbed by the bulk contribution.
We investigate multipartite entanglement for quantum states of 3D3D3D space geometry, described via generalized random spin networks with fixed areas, in the context of background independent approaches to quantum gravity. We focus on entanglement negativity as a well defined witness of quantum correlations for mixed states, in our setting describing generic subregions of the boundary of a quantum 3D3D3D region of space. In particular, we consider a generic tripartition of the boundary of an open spin network state and we compute the typical R\'enyi negativity of two boundary subregions AAA and BBB immersed in the environment CCC, explicitly for a set of simple open random spin network states. We use the random character of the spin network to exploit replica and random average techniques to derive the typical R\'enyi negativty via a classical generalized Ising model correspondence generally used for random tensor networks in the large bond regime. For trivially correlated random spin network states, with only local entanglement between spins located on the network edges, we find that typical log negativity displays a holographic character, in agreement with the results for random tensor networks, in large spin limit. When nonlocal bulk entanglement between intertwiners at the vertices is considered the negativity increases, while at the same time the holographic scaling is generally perturbed by the bulk contribution.
This thesis is dedicated to the study of open spin networks. We formulate quasi-local descriptions of loop quantum gravity. We investigate the coarse-graining procedure via tracing over bulk degrees of freedom, which encodes all that we can know about the quantum state of geometry from probing the boundary. We prove a boundary-to bulk universal reconstruction procedure, to be understood as a purification of the mixed boundary state into a pure bulk state. We then move to define multipartite entanglement in spin networks, and show the computation of entanglement excitation from holonomy operator, which also allows us to glimpse bulk curvature from entanglement. Moreover, by investigating another coarse-graining procedure - via gauge-fixing, which does not trace over any bulk degrees of freedom, we show a new interesting connection between bulk geometry and boundary observables via the dynamics of entanglement. Finally, we define the spin network entanglement between spin sub-networks, which correspond to spatial sub-regions. We then generalize the coarse-graining approach, and prove that the entanglement between spin sub-networks is preserved under the coarse-graining (via gauge-fixing).
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 $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$.
In the long-standing quest to reconcile gravity with quantum mechanics, profound connections have been unveiled between concepts traditionally pertaining to a quantum information theory, such as entanglement, and constitutive features of gravity, like holography. Developing and promoting these connections from the conceptual to the operational level unlock access to a powerful set of tools which can be pivotal toward the formulation of a consistent theory of quantum gravity. Here, we review recent progress on the role and applications of quantum informational methods, in particular tensor networks, for quantum gravity models. We focus on spin network states dual to finite regions of space, represented as entanglement graphs in the group field theory approach to quantum gravity, and illustrate how techniques from random tensor networks can be exploited to investigate their holographic properties. In particular, spin network states can be interpreted as maps from bulk to boundary, whose holographic behavior increases with the inhomogeneity of their geometric data (up to becoming proper quantum channels). The entanglement entropy of boundary states, which are obtained by feeding such maps with suitable bulk states, is then proved to follow a bulk area law with corrections due to the entanglement of the bulk state. We further review how exceeding a certain threshold of bulk entanglement leads to the emergence of a black hole-like region, revealing intriguing perspectives for quantum cosmology.
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.