• entanglement
• positive partial transpose
• quantum channels
• quantum dynamical semigroups

In quantum information theory entanglement represents a fragile and volatile resource, responsible for both the unconventional power of quantum computers and the difficulty in building them.
The aim of this thesis is understanding how the presence of noise in a quantum communication line degrades the entanglement content of a bipartite system during its transmission through it. Our attention is restricted to memory-less, time-omogenous noises, which are modeled by so-called quantum dynamical semigroups (QDS). A QDS is completely determined by its associated Lindbladian operator, or generator of the dynamics. Numerical methods and perturbation theory are used to determine the maximum transmission time associated to a given QDS as a function of the generator parameters. These parameters represent the relative weights of coherent and dissipative dynamics, and are generally under a good degree of
control by the experimentalist.

The first chapter of the thesis is devoted to reviewing the theory of the space of states, with a focus on entangled
states and separability tests.
In the second chapter the discussion continues with quantum channels, which provide the correct tool for describing a quantum communication line and the noise tampering it. The notion of QDS, which arise for instance in the weak-coupling limit and under the Born-Markov approximation, is also introduced in this chapter.
The third chapter presents the class of entanglement-breaking channels, which represent the most detrimental form
form of noise possible when considering the transmission of entanglement.
The review ends with a short account of experimental tests of entanglement restoration protocols. In the fourth chapter the we study the entanglement transmission time, defined as the time instant at which entanglement is completely broken, as a mean of classifying noises with respect to their harmfulness to entanglement. Some study cases are then analyzed in detail. The examples chosen are relevant in the transmission of entangled photon pairs through an optical fiber, and match the experiments outlined at the end of the third chapter.

The entanglement transmission time exhibits the same qualitative behaviour in the cases analyzed, dropping quickly as the QDS parameter ratio increases, but eventually stabilizing after a damped oscillation phase.
Furthermore, in both cases a preferred driving direction was present, along which the entanglement degradation was minimal. The height of the transmission time asymptote depends on the angle between such preferred direction and the driving operator-direction, with driving directions farther from the preferred one generally resulting in smaller entanglement
transmission times. The asymptotic behaviour of the figures of merit analyzed is controlled by the stationary states of the channel.
The form of the entanglement transmission time reflects the form of the eigenvalues of the generator of the dynamics, while the precence of a preferred direction is a consequence of commutation properties of the generator operators.
A natural direction for future research is extending the anlysis to the case of continuous variables system, which has been completely left out from the discussion. In particular, Gaussian quantum channels provide a natural candidate, also in sight of their applications in quantum optics.