Aspects of the relation between the black hole entropy and the entanglement entropy.

The relationship between Bekenstein-Hawking and entanglement entropy is one of the most intriguing problems of black hole and theoretical physics. Although both entropies have the same geometric characterization (they scale as an area), they have a different origin. The Bekenstein-Hawking entropy has it’s roots in the thermal correlations whereas the entanglement entropy describes the quantum correlations. In this thesis we address this problem from a holographic perspective, making large use of the AdS/CFT correspondence. We first look at the BTZ black hole in which case the black hole horizon and the conical singularity are related to each other. Using the modular transformations of the dual 2D CFT, we first obtain an expression of the holographicentanglement entropy for the Euclidean BTZ black hole, AdS3 vacua and conical singularity and then analyse the behaviour of the leading terms in the expansions of the holographic entanglement entropy for the BTZ black hole and the conical singularity. From these calculations, we have extracted the “signatures” through which entanglement entropy differentiates between the horizon and the conical singularity. When we deal with CFT at a finite temperature then it is well known that, entanglement entropy fails as a measure and hence it is replaced by entanglement negativity which is able to separate the thermal/classical correlations from the quantum correlations thereby capturing “distillable” entanglement. We address the problem of finding a suitable holographic prescription to calculate the entanglement negativity for two adjacent intervals when the CFT is at a finite temperature and in turn dual to a black hole in the bulk. We first propose a conjecture for the holographic entanglement negativity for two adjacent intervals in the AdS3/CFT2 setup and perform calculations in the various cases to support this conjecture. We further push forward the validity of this conjecture by calculating the holographic entanglement negativity for two adjacent subsystems when we have RN-AdS black holes in the bulk which in turn are dual to CFT with a conserved charge.