Concepts surrounding incompatibility in quantum physics

This thesis concerns how attempting to measure certain fundamental properties
of a physical theory can lead us to new insights. The idea of general probabilistic
models is introduced and developed, in order to provide a way to be able to
consider and compare different theories. A measure of incompatibility of two
observables is defined, through the amount of smearing needed to make them
jointly measureable. This measure is then used to characterise the degree of
incompatibility that exists in a given theory. Quantum theory is then shown
to be as incompatible as any other theory, but only in a very course grained
sense. A related way of measuring the strength of incompatibility of a pair
of observables is then shown to put a bound on a measure of the strength of
non-local correlations. The notion of steering, or remote ensemble preparation,
is then shown to be a sufficient condition for the saturation of that bound.
Examples are considered that demonstrate that the given sufficient notion of
steering is not necessary, and it is proposed how the measures can be modified
in cases where the link does not hold. The idea of formulating measures of
error and disturbance associated with a measurement device is discussed. The
notion of a direct test of error or disturbance is used to analyse current proposed
measures, and show their shortcomings.