Dissertação de Mestrado #618 – Rodolfo Reis Soldati – 22/02/2019

Quantum information has a wealth of applications in modern quantum field theory stemming from the holographic paradigm. In this dissertation, we review techniques for computing the entanglement entropy of bosonic quantum fields in flat spacetime and extend them to the Einstein universe with uniform spatial curvature.

In the seminal works due to Sorkin et al and Srednicki, space is discretised, thus regularising the theory and rendering the von Neumann entropy finite. An area law for entanglement entropy is found, configuring it as a viable source of entropy for black holes, as proposed by these authors.

Under a characterisation of regularisation-independent contributions to the area law, we sought curvature corrections to this result. We implement numerical calculations in a lattice, based on a more efficient algorithm relying on the covariance matrix description of Gaussian states. We reproduce analytical results expected to hold in any regularisation, thereby providing additional evidence to their universality.

We proceed with a presentation of the recent approach characterising the entanglement entropy of Gaussian states in terms of Kähler structures due to Bianchi et al. The symplectic geometry of phase space and compatible metric and complex structures therein parametrise the full space of covariance matrices, ultimately allowing for an extension of the algorithm to arbitrary Gaussian states.