The Inferential Design of Entropy and its Application to Quantum Measurements

This thesis synthesizes probability and entropic inference with Quantum Mechanics 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 entropy are completed in tandem following the same inferential principles and design criteria. This provides the first design derivation of the quantum relative entropy while also reducing the number of required design criteria to two. The result of Caticha and Giffin [7,8] and Giffin's thesis [9] is invaluable; it shows that the standard (relative) maximum entropy method is a "universal method of inference" - it is able to process information simultaneously that neither a Bayesian nor a standard maximum entropy method can process alone while being able to reproduce the results of both. An analogous conclusion is reached in this thesis, the quantum maximum entropy method derived in [1] is the "universal method of density matrix inference". This was accomplished by deriving, from the quantum maximum entropy method, a Quantum Bayes Rule [2] and generalizations that cannot be obtained from a Quantum Bayes or a quantum (von Neumann) maximum entropy method [10] alone while being able to reproduce the results of both. The expanded results of [2] more-or-less follow the main results of [7-9] in structure, but instead use density matrices rather than probability distributions. As the quantum maximum entropy method only uses the standard quantum mechanical formalism, the quantum maximum entropy method derived here may be appended to the standard quantum mechanical formalism and remove collapse as a required postulate, in agreement with [11,12]. The second part of this thesis revolves around the foundational theory of Quantum Mechanics called Entropic Dynamics [13]. Entropic Dynamics uses the standard maximum entropy method and information geometry to reformulate the Schrodinger Equation and Quantum Mechanics as a theory of inference. We derive a density matrix formalism of Quantum Mechanics using the standard maximum entropy method in Entropic Dynamics, and thus, the quantum maximum entropy method [1] may be adopted by Entropic Dynamics wholesale. This implies that indeed the standard maximum entropy method is the "universal method of inference" and that the quantum maximum entropy method holds the more specialized title of the "universal method of density matrix inference" as it does not provide an intrinsic mechanism for the unitary evolution of pure states. Entropic Dynamics is different than most interpretations of Quantum Mechanics because, rather than appending an interpretation to Quantum Mechanics, it states its interpretation, "that particles have definite yet unknown positions and that entropic probability updating works", and only then does it derive Quantum Mechanics from these assumptions. This radical shift in interpretation allows one to solve the quantum measurement problem [14] (which was extended to included von Neumann and weak measurements in [3]) and address quantum no-go theorems [4]. Crucial to understanding why these apparent paradoxes pose no issue in Entropic Dynamics is understanding the foundation of inference that Entropic Dynamics is built upon. In particular, when it comes to measurement in Entropic Dynamics, we are able to divvy-up variables (observables, ect.) into two classes: they are the ontic beables [15], which are the positions of particles, and the epistemic inferables [3] that in principle are inferred from detections of position and are therefore not predisposed to be part of the ontology. The fact that observables other than position are inferables in Entropic Dynamics, and that the positions of particles are the ontic beables, allows the Entropic Dynamics formulation of Quantum Mechanics to not be ruled out by the following pertinent no-go theorems [4]: no psi-epistemic [16], Bell-Kochen-Specker [17-19], and Bell's inequality [20]. Entropic Dynamics is found to be viable theory of Quantum Mechanics.