14,650 research outputs found
Fisher Metric, Geometric Entanglement and Spin Networks
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 non-local correlations between two non-adjacent
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 non-local level, the quantum correlations among its parts. This gives
a further connection between entanglement and geometry.Comment: 29 pages, 3 figures, revised version accepted for publicatio
Spin models on random graphs with controlled topologies beyond degree constraints
We study Ising spin models on finitely connected random interaction graphs
which are drawn from an ensemble in which not only the degree distribution
can be chosen arbitrarily, but which allows for further fine-tuning of
the topology via preferential attachment of edges on the basis of an arbitrary
function Q(k,k') of the degrees of the vertices involved. We solve these models
using finite connectivity equilibrium replica theory, within the replica
symmetric ansatz. In our ensemble of graphs, phase diagrams of the spin system
are found to depend no longer only on the chosen degree distribution, but also
on the choice made for Q(k,k'). The increased ability to control interaction
topology in solvable models beyond prescribing only the degree distribution of
the interaction graph enables a more accurate modeling of real-world
interacting particle systems by spin systems on suitably defined random graphs.Comment: 21 pages, 4 figures, submitted to J Phys
Non-coherence of arithmetic hyperbolic lattices
We prove, under the assumption of the virtual fibration conjecture for
arithmetic hyperbolic 3-manifolds, that all arithmetic lattices in O(n,1), n>
4, and different from 7, are non-coherent. We also establish noncoherence of
uniform arithmetic lattices of the simplest type in SU(n,1), n> 1, and of
uniform lattices in SU(2,1) which have infinite abelianization.Comment: 26 pages, 3 figure
- …