Informationally Complete Measurements and Optimal Representations of Quantum Theory

Minimal informationally complete quantum measurements (MICs) furnish probabilistic representations of quantum theory. These representations cleanly present the Born rule as an additional constraint in probabilistic decision theory, a perspective advanced by QBism. Because of this, their structure illuminates important ways in which quantum theory differs from classical physics. MICs have, however, so far received relatively little attention. In this dissertation, we investigate some of their general properties and relations to other topics in quantum information. A special type of MIC called a symmetric informationally complete measurement makes repeated appearances as the optimal or extremal solution in distinct settings, signifying they play a significant foundational role. Once the general structure of MICs is more fully explicated, we speculate that the representation will have unique advantages analogous to the phase space and path integral formulations. On the conceptual side, the reasons for QBism continue to grow. Most recently, extensions to the Wigner\u27s friend paradox have threatened the consistency of many interpretations. QBism\u27s resolution is uniquely simple and powerful, further strengthening the evidence for this interpretation