  This PHD thesis is concerned with uncertainty relations in quantum
probability theory, state estimation in quantum stochastics, and natural
bundles in differential geometry. After some comments on the nature and
necessity of decoherence in open systems and its absence in closed ones, we
prove sharp, state-independent inequalities reflecting the Heisenberg
principle, the necessity of decoherence and the impossibility of perfect joint
measurement. These bounds are used to judge how far a particular measurement is
removed from the optimal one. We do this for a qubit interacting with the
quantized EM field, continually probed using homodyne detection. We calculate
to which extent this joint measurement is optimal. We then propose a two-step
strategy to determine the (possibly mixed) state of n identically prepared
qubits, and prove that it is asymptotically optimal in a local minimax sense,
using `Quantum Local Asymptotic Normality' for qubits. We propose a physical
implementation of QLAN, based on interaction with the quantized EM field. In
differential geometry, a bundle is called `natural' if diffeomorphisms of the
base lift to automorphisms of the bundle. We slightly extend this notion to
`infinitesimal naturality', requiring only vector fields (infinitesimal
diffeomorphisms) to lift. We classify these bundles. Physical fields that
transform under (infinitesimal) space-time transformations must be described in
terms of (infinitesimally) natural bundles. All spin structures are
infinitesimally natural. Our framework thus encompasses Fermionic fields, for
which the notion of a natural bundle is too restrictive. Interestingly,
generalized spin structures (e.g. spin-c structures) are not always
infinitesimally natural. We classify the ones that are. Depending on the gauge
group at hand, this can significantly reduce the number of allowed space-time
topologies.
