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There is now strong evidence of a deep connection between entanglement in quantum gravity and the geometry of spacetime. In this dissertation, we study multiple facets of this connection. We start by quantifying the entanglement of a scalar quantum field theory as a function of the curvature of its background. We then shift our focus to the AdS/CFT duality, and we prove multiple logical relationships between geometric statements in AdS and entropic statements in the CFT. Many of these proofs work in the presence of quantum corrections, and we prove under which geometric conditions entanglement wedge nesting continues to imply the quantum null energy condition (QNEC) when the CFT is on an arbitrary curved background. We also demonstrate that the non-gravitational limit of the quantum focusing conjecture implies the QNEC, given the same geometric conditions. Next, we prove the connection between the boundary of the future of a surface and the null geodesics originating orthogonally from that surface. This theorem is important for proving that the area of holographic screens increases monotonically. Finally, we derive the holographic prescription for computing Renyi entropies of a CFT with the formalism of quantum error-correction. In the process, we provide evidence that the quantum gravity degrees of freedom related to the AdS geometry are maximally-mixed.

There is now strong evidence of a deep connection between entanglement in quantum gravity and the geometry of spacetime. In this dissertation, we study multiple facets of this connection. We start by quantifying the entanglement of a scalar quantum field theory as a function of the curvature of its background. We then shift our focus to the AdS/CFT duality, and we prove multiple logical relationships between geometric statements in AdS and entropic statements in the CFT. Many of these proofs work in the presence of quantum corrections, and we prove under which geometric conditions entanglement wedge nesting continues to imply the quantum null energy condition (QNEC) when the CFT is on an arbitrary curved background. We also demonstrate that the non-gravitational limit of the quantum focusing conjecture implies the QNEC, given the same geometric conditions. Next, we prove the connection between the boundary of the future of a surface and the null geodesics originating orthogonally from that surface. This theorem is important for proving that the area of holographic screens increases monotonically. Finally, we derive the holographic prescription for computing Renyi entropies of a CFT with the formalism of quantum error-correction. In the process, we provide evidence that the quantum gravity degrees of freedom related to the AdS geometry are maximally-mixed.

International Computer Music Conference Proceedings: vol. 2007, (dlps) bbp2372.2007.073, http://hdl.handle.net/2027/spo.bbp2372.2007.073, This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. Please contact mpub-help@umich.edu to use this work in a way not covered by the license.

International Computer Music Conference Proceedings: vol. 2007, (dlps) bbp2372.2007.073, http://hdl.handle.net/2027/spo.bbp2372.2007.073, This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. Please contact mpub-help@umich.edu to use this work in a way not covered by the license.