CastroAlvaredo, O. & Doyon, B. (2013). Entanglement in permutation symmetric states, fractal dimensions, and geometric quantum mechanics. Journal of Statistical Mechanics: Theory and Experiment, 2013, P02016 . doi: 10.1088/17425468/2013/02/P02016

PDF
Download (855kB)  Preview 
Abstract
We study the von Neumann and Rényi bipartite entanglement entropies in the thermodynamic limit of manybody quantum states with spins sites that possess full symmetry under exchange of sites. It turns out that there is essentially a onetoone correspondence between such thermodynamic states and probability measures on CP2s. Let a measure be supported on a set of possibly fractal real dimension d with respect to the Study–Fubini metric of CP2s. Let m be the number of sites in a subsystem of the bipartition. We give evidence that in the limit m → ∞, the entanglement entropy diverges like (d/2)logm. Further, if the measure is supported on a submanifold of CP2s and can be described by a density f with respect to the metric induced by the Study–Fubini metric, we give evidence that the correction term is simply related to the entropy associated with f: the geometric entropy of geometric quantum mechanics. This extends results obtained by the authors in a recent letter where the spin$\frac{1}{2}$ case was considered. Here we provide more examples as well as detailed accounts of the ideas and computations leading to these general results. For special choices of the state in the spins situation, we recover the scaling behaviour previously observed by Popkov et al, showing that their result is but a special case of a more general scaling law.
Item Type:  Article 

Subjects:  Q Science > QA Mathematics Q Science > QC Physics 
Divisions:  School of Engineering & Mathematical Sciences > Department of Mathematical Science 
URI:  http://openaccess.city.ac.uk/id/eprint/2335 
Actions (login required)
View Item 
Downloads
Downloads per month over past year