Black holes and entanglement

In this doctoral thesis, we study the physics of black holes and the role of entanglement therein. After a review of Black Hole Information Problem, we discuss a specific resolution in Chapter 2. We argue that the geometry dual to a typical AdS black hole microstate has a smooth horizon and a region spacelike to the exterior. We construct state-dependent operators that describe EFT excitations in the spacelike region. Then, we probe the geometry through thought experiments involving shockwaves. We estimate the corresponding out-of-time-order correlators in the CFT and are thus led to conjecture a purely-CFT constraint on general Wightman correlators in the CFT. We end with a discussion of the equivalence between our thought experiments and quantum teleportation in the CFT. In Chapter 2, we construct the thermofield double state (TFD) for two quantum systems satisfying ETH. It is a special, highly entangled state, believed to describe an eternal AdS black hole in holography. It is obtained as the approximate ground state of a simple, local Hamiltonian which involves weak two-body entanglement of few operators. The approximate ground state has a large overlap with the exact one. We also discuss the finiteness of gap at large N. In Chapter 3, we calculate explicitly the bulk entanglement entropy of a scalar field in AdS3 in a 1-particle excited state for a small subregion. We verify the FLM formula for quantum corrections to entanglement entropy in holography

Modular invariance and entanglement entropy

We study the Rényi and entanglement entropies for free 2d CFT’s at finite temperature and finite size, with emphasis on their properties under modular transformations of the torus. We address the issue of summing over fermion spin structures in the replica trick, and show that the relation between entanglement and thermal entropy determines two different ways to perform this sum in the limits of small and large interval. Both answers are modular covariant, rather than invariant. Our results are compared with those for a free boson at unit radius in the two limits and complete agreement is found, supporting the view that entanglement respects Bose-Fermi duality. We extend our computations to multiple free Dirac fermions having correlated spin structures, dual to free bosons on the Spin(2d) weight lattice