Realisation of quantum entanglement and chaos in gravity

Originating from string theory, the holographic correspondence provides a dictionary to convert a quantum theory on the boundary of the Anti-deSitter space (AdS) into a theory of gravity in the bulk AdS space. In this thesis we will study the intersection of quantum information and quantum gravity, focusing on methods of quantifying quantum entanglement and chaos in gravity via the AdS/CFT holographic correspondence. Entanglement entropy of a bipartite quantum system on the boundary is equivalent to the area of a minimal surface in the bulk. In both the boundary and bulk pictures, entanglement entropy is divergent, meaning it equals to infinity. Hence we need to renormalise the entanglement entropy to obtain a finite quantity. The variation of the entanglement entropy is related to the dynamics of the bulk spacetime via the first law of entanglement entropy. We will first present a way to express the renormalised entanglement entropy in terms of the Euler invariant of the bulk entangling surface and other renormalised curvature invariants. Then we use this expression and independently derived the renormalised version of the first law of entanglement entropy. In particular, we use the Hamiltonian formalism of holographic renormalisation to derive the integral form of the first law of entanglement entropy. Quantum chaos is characterised by the scrambling of information that increases exponentially in time. The rate of the exponential growth, known as the Lyapunov exponent, can be measured via the out-of-time-ordered correlation function (OTOC). In holography, the OTOC becomes the gravitational scattering amplitude of high energy particles. We investigate a possible correction to the Lyapunov exponent by considering the classical stringy effect in the bulk gravitational scattering. Following the semi-classical shock wave calculation of gravitational eikonal scattering, we obtain the classical string transverse oscillation contribution to the eikonal phase. We conclude such correction is negligible in the high energy eikonal limit, hence satisfying the chaos bound.