  We investigate the evolution of open quantum systems in the presence of
initial correlations with an environment. Here the standard formalism of
describing evolution by completely positive trace preserving (CPTP) quantum
operations can fail and non-completely positive (non-CP) maps may be observed.
  A new classification of correlations between a system and environment using
quantum discord is explored. However, we find quantum discord is not a
symmetric quantity between exchange of systems and this leads to ambiguity in
classifications - states which are both quantum and classically correlated
depending on the order of the two systems.
  State preparation in quantum process tomography is investigated with regard
to non-CP maps. In SQPT the preparation procedure can influence the
complete-positivity of the reconstructed quantum operation if our system is
initially correlated with an environment. We examine a recently proposed
preparation procedures using projective measurements, and propose our own
protocol that uses a single measurement followed by unitary rotations. The
former can give rise to non-CP evolution while the later will always give rise
to a CP map. State preparation in AAPT was found always to give rise to CP
evolution.
  We examine the effect of statistical noise in process tomography and find it
can result in the identification of a non-CP when the evolution should be CP.
The variance of the distribution for reconstructed processes is found to be
inversely proportional to the number of copies of a state used to perform
tomography. Finally, we detail an experiment using currently available linear
optics QC devices to demonstrate non-CP maps arising in SQPT.
