Intertwiner entanglement on spin networks

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 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...

Reflections on the information paradigm in quantum and gravitational physics

“…Most of these have occurred in the context of the spin networks of loop quantum gravity. Beginning with the proposal [84,85] to define emergent geometries therefrom through coarse-graining and correlation strengths, there are now systematic efforts to materialise these ideas [86,87]. However, this proposal faces a great challenge: a proper notion of entanglement and physical correlations in quantum gravity requires a diffeomorphism invariant description of subsystems.…”

Hidden messenger from quantum geometry: Towards information conservation in quantum gravity

“…It is an old story [28] that such a horizon breaks the gauge symmetry and generates topological defects on the horizon. So there seems to be a contradiction as pointed out in [24]. The hidden messegers revealed in this paper provides another way to understand this issue: the black hole horizon with spin networks' punctures indeed breaks the gauge symmetry, but it is so locally on the puncturing edges which, as messengers, inform the preservation of the global gauge invariance.…”

Entanglement in simple spin networks with a boundary

“…By virtue of numerical evaluations we demonstrate that, in general, the entanglement entropy depends on the group elements on edges, which has previously been pointed out in ref. [19]. Our numerical results imply that once the spins on edges are defined, bounds for the spin entanglement should exist.…”