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arXiv:1709.08511 · Physical review. D/Physical review. D.

Intertwiner entanglement on spin networks

Etera R. Livine

Published Jan 24, 2018• 31 citations
arXivHTMLPDF

Abstract

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 $\mathrm{ln}(2j+1)$ term depending on the spin $j$ living on the link between them. This $\mathrm{ln}(2j+1)$ 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.

DOI
10.1103/physrevd.97.026009

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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)\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.
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.
Starting from recent results on the geometric formulation of quantum mechanics, we propose a new information geometric characterization of entanglement for spin network states in the context of quantum gravity. For the simple case of a single-link fixed graph (Wilson line), we detail the construction of a Riemannian Fisher metric tensor and a symplectic structure on the graph Hilbert space, showing how these encode the whole information about separability and entanglement. In particular, the Fisher metric defines an entanglement monotone which provides a notion of distance among states in the Hilbert space. In the maximally entangled gauge-invariant case, the entanglement monotone is proportional to a power of the area of the surface dual to the link thus supporting a connection between entanglement and the (simplicial) geometric properties of spin network states. We further extend such analysis to the study of nonlocal correlations between two nonadjacent regions of a generic spin network graph characterized by the bipartite unfolding of an intertwiner state. Our analysis confirms the interpretation of spin network bonds as a result of entanglement and to regard the same spin network graph as an information graph, whose connectivity encodes, both at the local and nonlocal level, the quantum correlations among its parts. This gives a further connection between entanglement and geometry.
The purpose of this letter is to explore the relation between gauge fields, which are at the base of our understanding of fundamental interactions, and the quantum entanglement. To this end, we investigate the case of SU(2) gauge fields. It is first argued that holonomies of the SU(2) gauge fields are naturally associated with maximally entangled two-particle states. Then, we provide some evidence that the notion of such gauge fields can be deduced from the transformation properties of maximally entangled two-particle states. This new insight unveils a possible relation between gauge fields and spin systems, as well as contributes to understanding of the relation between tensor networks (such as MERA) and spin network states considered in loop quantum gravity. In consequence, our results turn out to be relevant in the context of the emerging Entanglement/Gravity duality.
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.
Abstract Black hole entropy is one of the few windows into the quantum aspects of gravitation, and its study over the years has highlighted the holographic nature of gravity. At the non-perturbative level in quantum gravity, promising explanations are being explored in terms of the entanglement entropy between regions of space. In the context of loop quantum gravity, this translates into an analysis of the correlations between the regions of the spin network states defining the quantum state of the geometry of space. In this paper, we explore a class of states, motivated by results in condensed matter physics, satisfying an area law for entanglement entropy and having non-trivial correlations. We highlight that entanglement comes from holonomy operators acting on loops crossing the boundary of the region.
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).