In this thesis, we use classical and quantum information theory to probe fundamental questions about living and nonliving physical systems, including developing embryos, conformal field theories, black holes, and holographic dualities. We begin by analyzing how spatially varying concentrations of four proteins in the early stage fruit fly embryo are able to encode enough information to specify a precise body plan for the developed adult fly.

We then transition to studying how information is tied to physics in nonliving systems, beginning with a conjecture on the structure of universal terms in the Renyi entropy in 3 + 1-d CFT. In the following pair of chapters, we use "topological" entanglement in AdS3/CFT 2 duality as a springboard for developing a precise correspondence between Liouville theory and the topological sector of 2+1-d gravity.

Then, in the final chapter, we study the "information flow" among energy scales in a 2-d field theory constructed as a CFT deformed by the irrelevant dimension-4 operator TT¯. We do so by constructing an explicit manifestation of holographic RG. By identifying the TT¯ coupling with a hard cutoff in the bulk, we are able to exactly match the thermodynamics of a "black hole in a box" with the physics of an integrable field theory.

Throughout, our goal is to demonstrate that studying uncertainty with information theory, entanglement, and renormalization group flow allows us to organize the unknowns and thus obtain new methods for constraining, characterizing, and dualizing the system at hand. In the process, we learn fundamental properties we might not have otherwise known to study.