# Transformation and Uncertainty. Some Thoughts on Quantum Probability Theory, Quantum Statistics, and Natural Bundles - Mathematics > Differential Geometry

Transformation and Uncertainty. Some Thoughts on Quantum Probability Theory, Quantum Statistics, and Natural Bundles - Mathematics > Differential Geometry - Download this document for free, or read online. Document in PDF available to download.

Abstract: This PHD thesis is concerned with uncertainty relations in quantumprobability theory, state estimation in quantum stochastics, and naturalbundles in differential geometry. After some comments on the nature andnecessity of decoherence in open systems and its absence in closed ones, weprove sharp, state-independent inequalities reflecting the Heisenbergprinciple, the necessity of decoherence and the impossibility of perfect jointmeasurement. These bounds are used to judge how far a particular measurement isremoved from the optimal one. We do this for a qubit interacting with thequantized EM field, continually probed using homodyne detection. We calculateto which extent this joint measurement is optimal. We then propose a two-stepstrategy to determine the possibly mixed state of n identically preparedqubits, and prove that it is asymptotically optimal in a local minimax sense,using `Quantum Local Asymptotic Normality- for qubits. We propose a physicalimplementation of QLAN, based on interaction with the quantized EM field. Indifferential geometry, a bundle is called `natural- if diffeomorphisms of thebase lift to automorphisms of the bundle. We slightly extend this notion to`infinitesimal naturality-, requiring only vector fields infinitesimaldiffeomorphisms to lift. We classify these bundles. Physical fields thattransform under infinitesimal space-time transformations must be described interms of infinitesimally natural bundles. All spin structures areinfinitesimally natural. Our framework thus encompasses Fermionic fields, forwhich the notion of a natural bundle is too restrictive. Interestingly,generalized spin structures e.g. spin-c structures are not alwaysinfinitesimally natural. We classify the ones that are. Depending on the gaugegroup at hand, this can significantly reduce the number of allowed space-timetopologies.

Author: ** Bas Janssens**

Source: https://arxiv.org/