Quantum Surface and Intertwiner Dynamics in Loop Quantum Gravity

Feller, Alexandre; Livine, Etera R.

doi:10.48550/arxiv.1703.01156

Cited by 7 publications

(16 citation statements)

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“…• The distinction between intertwiner entanglement and spin state entanglement invites to revisit the use of entanglement entropy for black holes in loop quantum gravity [26,27]. Indeed, the standard point of view is that the black hole boundary cuts spin network edges so that the horizon is a collection of spin states -SU(2) punctures-defining a quantum surface [28,29]. So as explained in [27], all the entanglement entropy between the black hole interior and exterior is usually taken as the sum of the ln(2j + 1) contributions coming from all the edges crossing the horizon.…”

Section: Discussionmentioning

confidence: 99%

Intertwiner entanglement on spin networks

Livine

2018

Phys. Rev. D

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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) term depending on the spin j living on the link between them. This ln(2j + 1) contribution comes from looking at non-gauge 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.In the background independent framework of quantum gravity, quantum states of geometry are defined up to diffeomorphisms with no reference to any coordinate systems or special choice of metric. Then distances and curvature, which define the geometry, are reconstructed from the interaction between subsystems, best quantified by the correlation and entanglement between those subsystems. This perspective sets the field of quantum information at the heart of research in quantum gravity, with essential roles to play for entanglement, decoherence and quantum localization in probing quantum states of geometries and thinking about the quantum-to-classical transition for the space-time geometry.The loop quantum gravity framework provides a local definition of quantum states of space geometry as spin networks. They are collections of quantum states invariant under 3d rotations linked to each other by rotations encoding the change of frame from one state to the next, thus forming a network of frame transformations defining the quantum 3d space. These spin networks can further be understood as the quantization of discrete "twisted" geometries [1][2][3], where the 3d space is made of quantized flat chunks of volume glued together. Then the spin networks' evolution is described by spinfoam path integrals, built from topological quantum field theory (TQFT) [4][5][6][7]. This quantum geometry concept is ultimately meant to replace the paradigm of classical smooth manifold to describe the space-time geometry.Spin networks describe the geometry at the Planck scale. They need to be coarse-grain in order to derive the structure of quantum geometries at larger scales. The issue is that the definition of spin networks is ultra-local, with algebraic structures associated to each node of the network without any overall constraint (at the kinematical level), so it is a hard problem to get a dynamical coherent picture of smooth geometry arising semiclassically. The quantum gravity dynamics, implementing the quantum Einstein equations and the diffeomorphism invariance of quantum geometry states, should ul- * Electronic address: etera.livine@ens-lyon.fr timately address this question. However we would still require good observables to probe t...

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Section: Discussionmentioning

confidence: 99%

Intertwiner entanglement on spin networks

Livine

2018

Phys. Rev. D

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“…4, this means considering a graph Γ with open edges e ∈ ∂Γ puncturing the boundary surface. We do not endow the boundary with extra structure, representing the 2d boundary intrinsic geometry as in [24,34,35] or locality on the boundary as in [36], but discuss the minimal boundary structure.…”

Section: A Corners Boundary States and Mapsmentioning

confidence: 99%

“…Thus a coarsegraining procedure trading the bulk states for the boundary state irremediably creates entropy. In particular, endowing the bulk states with a unitary dynamics would naturally lead to a decoherence process (and possibly re-coherence) for the boundary states [36,56].…”

Section: A Bulk State To Boundary Density Matrixmentioning

confidence: 99%

Loop Quantum Gravity's Boundary Maps

Chen,

Livine

2021

Preprint

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In canonical quantum gravity, the presence of spatial boundaries naturally leads to a boundary quantum states, representing quantum boundary conditions for the bulk fields. As a consequence, quantum states of the bulk geometry needs to be upgraded to wave-functions valued in the boundary Hilbert space: the bulk become quantum operator acting on boundary states. We apply this to loop quantum gravity and describe spin networks with 2d boundary as wave-functions mapping bulk holonomies to spin states on the boundary. This sets the bulk-boundary relation in a clear mathematical framework, which allows to define the boundary density matrix induced by a bulk spin network states after tracing out the bulk degrees of freedom. We ask the question of the bulk reconstruction and 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 further perform a first investigation in the algebraic structure of induced boundary density matrices and show how correlations between bulk excitations, i.e. quanta of 3d geometry, get reflected into the boundary density matrix. Contents II. Boundary Density MatrixA. Bulk state to Boundary density matrix B. The closure defect basis and SU(2)-invariance of the boundary density matrix C. Universal bulk reconstruction from the boundary density matrix D. Probing the first layer of the bulk: Bouquets of boundary edges III. Examples: Boundary Density Matrix for Candy Graphs A. The four-qubit candy graph B. The six-qubit candy graph Conclusion & Outlook Acknowledgement References

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“…We consider a bounded region of a spin network state, with the graph puncturing the boundary surface at N points (see e.g. [20] for a discussion on the definition of quantum surfaces in loop quantum gravity). Each puncture carries an arbitrary spin, so that the boundary state lives in the tensor product of N arbitrary spins with the only constraint that the sum of all N boundary spins is an integer:…”

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confidence: 99%

“…A useful decomposition of the boundary Hilbert space is to partition it in terms of the closure defect [6,7,20,21]. Mathematically, this means recoupling all the spins j i carried by the N punctures into a single overall spin s. This spin, living on the intermediate channel, is necessarily an integer 4 :…”

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confidence: 99%

From coarse-graining to holography in loop quantum gravity

Livine¹

2018

EPL

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We discuss the relation between coarse-graining and the holographic principle in the framework of loop quantum gravity and ask the following question: when we coarse-grain arbitrary spin network states of quantum geometry, are we integrating out physical degrees of freedom or gauge degrees of freedom? Focusing on how bulk spin network states for bounded regions of space are projected onto boundary states, we show that all possible boundary states can be recovered from bulk spin networks with a single vertex in the bulk and a single internal loop attached to it. This partial reconstruction of the bulk from the boundary leads us to the idea of realizing the Hamiltonian constraints at the quantum level as a gauge equivalence reducing arbitrary spin network states to one-loop bulk states. This proposal of "dynamics through coarse-graining" would lead to a one-toone map between equivalence classes of physical states under gauge transformations and boundary states, thus defining holographic dynamics for loop quantum gravity.Loop quantum gravity sets up a non-perturbative framework for quantum gravity, with evolving quantum state of geometries and area and volume operators with quantized spectra at the Planck scale (for reviews of both the basic formalism and recents developments, see [1][2][3][4]). It faces a triptych of interlaced issues: the coarse-graining of quantum geometry states from the Planck scale to larger scales, the definition of quantum dynamics consistent with the holographic principle and the implementation of (discretized) diffeomorphism at quantum level as the fundamental gauge symmetry of the theory (or, in other words, the implementation of a relativity principle for quantum geometry). These encompass more technical questions, such as anomaly cancellation, a well-behaved continuum limit and the perturbative renormalisation of quantum gravity corrections. In this short letter, we would like to discuss the relation between coarse-graining and holography.First, there is the natural question of whether loop quantum gravity, and more generally any approach to quantum gravity, should be holographic. On top on the area-entropy law for black holes, the related generalized entropy bounds for general relativity and the AdS/CFT correspondence at asymptotic infinity, the insight into holography is directly related to the invariance under diffeomorphism. This symmetry is at the heart of general relativity and is the mathematical translation of the relativity principle. However it is a tough challenge to identify and construct diffeomorphism-invariant observables (especially in pure gravity). Considering a bounded region allows to introduce a boundary, which acts on an anchor: looking for observables invariant under bulk diffeomorphisms leaving the boundary invariant seems to point towards the idea that all physical observables about the bulk geometry could/should be represented as boundary observables. We believe it is crucial, in order to understand better the structure and implications of loop quan- * Electron...

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Quantum Surface and Intertwiner Dynamics in Loop Quantum Gravity

Cited by 7 publications

References 0 publications

Intertwiner entanglement on spin networks

Intertwiner entanglement on spin networks

Loop Quantum Gravity's Boundary Maps

From coarse-graining to holography in loop quantum gravity

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