Geometric Phases and Symmetric Quantum Measurements

The question of the Symmetric Informationally Complete quantum measurements (SICs for short) is at once modern, having entered the physics literature in 2004, and possessed of a classic feel, since it is brief to state yet connects in unexpected ways to disparate areas of mathematics. SICs provide a new way to talk about and calculate probabilities in quantum physics, and they have the potential to furnish new insight into the fundamentals of the subject itself. On the applied side, they are optimal measurements for certain tasks in quantum information processing. I aim to investigate the details of certain quantities in SIC constructions, the so-called triple products, which can be interpreted as geometric phases akin to the Berry phase, and which moreover have a deep significance in algebraic number theory, as well as a connection to quantum thermodynamics and Jarzynski relations, by way of the Kirkwood--Dirac quasiprobability. My primary focus is a proposal known as the 3d conjecture, a result that, if proven, would simplify the numerical search for SICs, reducing the complexity of the search problem from quadratic in the dimension to linear. Secondary goals are to use a close study of SIC geometric phases to deepen our understanding of quantum thermodynamics, and to improve upon existing work that reconstructs the mathematical formalism of quantum theory from physical principles.