We explore aspects of perturbative quantum field theory towards the goal of illuminating features of quantum gravity. In Chapter 1, we begin with the color-kinematic duality, a surprising relationship between gauge theoretic and kinematic properties of scattering amplitudes that, via the double-copy property, leads to a deep connection between gauge theory and gravity. We investigate the relationship between the color-kinematic duality and amplitude relations in ABJM theory and find that the duality is satisfied eight points without associated amplitude relations present in all other known instances of the duality.

In Chapter 2, we derive a basic consistency check for AdS3/CFT2, a tractable example of the well-known AdS/CFT duality. We show that, under certain mild assumptions on the light spectrum of the CFT, CFT correlators of light operators match those computed in AdS perturbatively in 1/N: in a black hole background for high temperatures, and in thermal AdS for low temperatures.

Next we turn to entanglement entropy, an information-theoretic quantity that in the CFT may encode dual AdS geometry. Time-dependent entanglement entropy has been studied for generic excited states. However, localized unitary operators in particular are in correspondence with Hamiltonian perturbations, and are basic building blocks of local excitations. In Chapter 3, we detail these operators' locality properties as well as the differences between excited states created by Hermitian operators and those created with localized unitary operators.

In Chapter 4, we initiate the perturbative exploration of entanglement entropy with a time-dependent Hamiltonian, computing past first order in perturbation theory. We find a universal structure of entanglement propagation: interactions entangle unentangled excitations according to entanglement diagrams that track the flow of entanglement. We provide diagrammatic tools to simplify computations of loop entanglement diagrams.