  We present two projects concerning the main part of my PhD work. In the first
one we study quantum channels, which are the most general operations mapping
quantum states into quantum states, from the point of view of their
divisibility properties. We introduced tools to test if a given quantum channel
can be implemented by a process described by a Lindblad master equation. This
in turn defines channels that can be divided in such a way that they form a
one-parameter semigroup, thus introducing the most restricted studied
divisibility type of this work. Using our results, together with the study of
other types of divisibility that can be found in the literature, we
characterized the space of qubit quantum channels. We found interesting results
connecting the concept of entanglement-breaking channel and infinitesimal
divisibility. Additionally we proved that infinitely divisible channels are
equivalent to the ones that are implementable by one-parameter semigroups,
opening this question for more general channel spaces. In the second project we
study the functional forms of one-mode Gaussian quantum channels in the
position state representation, beyond Gaussian functional forms. We perform a
black-box characterization using complete positivity and trace preserving
conditions, and report the existence of two subsets that do not have a
functional Gaussian form. The study covers as particular limit the case of
singular channels, thus connecting our results with the known classification
scheme based on canonical forms. Our full characterization of Gaussian channels
without Gaussian functional form is completed by showing how Gaussian states
are transformed under these operations, and by deriving the conditions for the
existence of master equations for the non-singular cases.
