Abstract:We find that the standard relative entropy and the Umegaki entropy are designed for the purpose of inferentially updating probabilities and density matrices, respectively. From the same set of inferentially guided design criteria, both of the previously stated entropies are derived in parallel. This formulates a quantum maximum entropy method for the purpose of inferring density matrices in the absence of complete information.

Abstract. The problem of measurement in quantum mechanics is studied within the Entropic Dynamics framework. We discuss von Neumann and Weak measurements, wavefunction collapse, and Weak Values as examples of bayesian and entropic inference.

We construct and propose the "Bayesian Validation Metric" (BVM) as a general model validation and testing tool. We find the BVM to be capable of representing all of the standard validation metrics (square error, reliability, probability of agreement, frequentist, area, probability density comparison, statistical hypothesis testing, and Bayesian model testing) as special cases and find that it can be used to improve, generalize, or further quantify their uncertainties. Thus, the BVM allows us to assess the similarities and differences between existing validation metrics in a new light.The BVM has the capacity to allow users to invent and select models according to novel validation requirements. We formulate and test a few novel compound validation metrics that improve upon other validation metrics in the literature. Further, we construct the BVM Ratio for the purpose of quantifying model selection under user defined definitions of agreement in the presence or absence of uncertainty. This construction generalizes the Bayesian model testing framework.

The recent article 'Entropic Updating of Probability and Density Matrices' [1] derives and demonstrates the inferential origins of both the standard and quantum relative entropies in unison. Operationally, the standard and quantum relative entropies are shown to be designed for the purpose of inferentially updating probability distributions and density matrices, respectively, when faced with incomplete information. We call the inferential updating procedure for density matrices the 'quantum maximum entropy method'. Standard inference techniques in probability theory can be criticized for lacking concrete physical consequences in physics; but here, because we are updating quantum mechanical density matrices, the quantum maximum entropy method has direct physical and experimental consequences. The present article gives a new derivation of the Quantum Bayes Rule, and some generalizations, using the quantum maximum entropy method while discuss some of the limitations the quantum maximum entropy method puts on the measurement process in Quantum Mechanics.

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