The Inferential Design of Entropy and its Application to Quantum Measurements.

This thesis synthesizes probability and entropic inference with Quantum Mechanics (QM) and quantum measurement [1-6]. It is shown that the standard and quantum relative entropies are tools designed for the purpose of updating probability distributions and density matrices, respectively [1]. The derivation of the standard and quantum relative entropies are completed in tandem and follow from the same inferential principle - the principle of minimal updating [21,66]. As the quantum maximum entropy method is derived using the standard quantum mechanical formalism, the quantum maximum entropy method may be appended to the standard quantum mechanical formalism and remove collapse as a required postulate, in agreement with [11]. The quantum maximum entropy method is found to be a "universal method of density matrix inference" as it can process information about data and moments simultaneously (giving joint generalized quantum inference solutions), which when processed separately gives the Quantum Bayes Rule [2,39] or a canonical quantum (von Neumann) maximum entropy solution [10], respectively, as special cases. The second part of this thesis revolves around a foundational theory of QM called Entropic Dynamics (ED) [13]. Rather than appending an interpretation to QM, ED states its interpretation, "that particles have definite, yet unknown, positions and that entropic probability updating works" - only then does ED derive QM as an application of inference consistent with these assumptions. This shift in interpretation allows one to solve the quantum measurement problem [3,14] and avoid being ruled out by quantum no-go theorems [4]. Observables are divvied-up into two classes in ED: they are the ontic "beables" [15] (particle position), and the epistemic "inferables" [3], which are not predisposed to be part of the ontology as they are inferred in general from position detections.