ua nt - p h / 03 10 06 4 v 6 8 N ov 2 00 3 Entanglement measure for the universal classes of fractons

We introduce the notion of entanglement measure for the universal classes of fractons as an entanglement between ocuppation-numbers of fractons in the lowest Landau levels and the rest of the many-body system of particles. This definition came as an entropy of the probability distribution à la Shannon. Fractons are charge-flux systems classified in universal classes of particles or quasiparticles labelled by a fractal or Hausdorff dimension defined within the interval 1 < h < 2 and associated with the fractal quantum curves of such objects. They carry rational or irrational values of spin and the spin-statistics connection takes place in this fractal approach to the fractional spin particles. We take into account the fractal von Neumann entropy associated with the fractal distribution function which each universal class of fractons satisfies. We consider the fractional quantum Hall effect-FQHE given that fractons can model Hall states. According to our formulation entanglement between occupaton-numbers in this context increases with the universality classes of the quantum Hall transitions considered as fractal sets of dual topological quantum numbers filling factors. We also introduce a relative entropy of entanglement for fractons. We verify that the Hall states have stronger entanglement between ocuppation-numbers and so we can consider this resource for fracton quantum computing. PACS numbers: 03.65.Ud, 03.65.Yz, 71.10.Pm, 05.30.Pr, 73.43.-f