• entanglement
• quantum channel
• local noise
• qubit

The entanglement is the one of the fundamental features which distinguish the quantum world from the classical one. Thanks to Bell's theorem, we know that it is a genuinely new effect, having no classical counterpart. From the point of view of the quantum information theory, the entanglement has to be regarded primarily as a computational resource. Indeed, it allows us to perform astonishing tasks (such as quantum teleportation or cryptography) which would be impossible in a world subjected to the classical laws.

However, just like all the physical resources, also the entanglement is subjected to deterioration. Actually, one of the main issues physicists have to face in dealing with quantum computation tasks from an experimental point of view is the control of the noise interfering with non-classical correlations in a bipartite quantum system. Through this thesis, we examine the following experimental setup. Alice and Bob share a pair of entangled systems, but Alice's half of the global system suffers a noise caused by an uncontrolled interaction with an external environment. This open evolution of a quantum system (in a time-discretized approach) is called a quantum channel. Our central goal is to classify the amount of noise introduced by a given quantum channel only by means of its action on the entanglement.

The first instance of this plan is the understanding of those channels that never break the entanglement between Alice and Bob, no matter how weak it is (provided that it exists). These channels are called universal entanglement-preserving. Our first contribution is the rigorous proof that the only universal entanglement-preserving channels are the unitary evolutions, taking place when Alice's subsystem is kept perfectly isolated. This means that even if the interaction of Alice's subsystem with the surrounding world is very feeble, nevertheless it can destroy some form of weak entanglement between Alice and Bob. Conceptually, this result clarifies the context of our investigations.

From the physical point of view, it is natural to consider the repeated applications of a given quantum channel on Alice's half of the global system. For the purpose of classifying this noise by means of the damages it produces on the entanglement, we introduce some functionals (defined on the set of quantum channels), called entanglement--breaking indices. The most important ones are the direct n-index and the filtered N-index. The $n$--index is the minimum number of times we have to apply a given channel in order to produce the complete destruction of the entanglement. On the other hand, Alice could play an active role against the noise repeatedly affecting her subsystem, by choosing to apply some quantum channels (called filters) between consecutive actions of the noise. Then, the filtered N-index is by definition the minimum number of iterations of the noise, such that there is no filtering strategy by which Alice can hope to save her entanglement with Bob.

However, every non-unitary filter creates some entanglement between Alice's subsystem and an external environment, lowering the level of quantum correlations between Alice and Bob. Therefore, we initially make the intuitive conjecture that the optimal filtering strategy is obtained by means of unitary operations only. However, once again the quantum entanglement has a surprise in store for us. Indeed, we provide an explicit counterexample, showing that this conjecture is in general false. Actually, the two-dimensional case seems to exhibit an anomalous behaviour, in the sense that our counterexample (exceptionally) does not work. In this respect, we collect a series of clues pointing out that the conjecture we presented could retain its validity for channels acting on two-dimensional quantum systems (called qubits).

Next, we turn our attention to the study of the those channels (called entanglement-saving) which introduce so few noise in the system, that their direct n-index takes an infinite value. In other words, the complete destruction of the entanglement is never reached, regardless of the number of iterations. But also within the class of entanglement-saving channels, there are still two possibilities. Indeed, in the limit of an infinite number of applications of the channel, the amount of entanglement can tend to zero or remain well above a finite threshold. The latter case corresponds to the so-called asymptotically entanglement-saving channels. One of our main contributions is the complete characterization of these channels. It turns out that a quantum channel is asymptotically entanglement-saving if and only if it admits two non-commuting phase points. A phase point is (by definition) an input matrix whose transformation under the action of the channel is simply the multiplication by a phase.

On the other hand, much effort is devoted to gain understanding of the entanglement-saving channels. Although the intrinsic difficulties of coping with quantum systems of arbitrary dimension, rather surprisingly we achieve our goal almost everywhere (that is, apart from a set of measure zero). Indeed, we find that almost everywhere the entanglement-saving property coincides with the presence of a positive semidefinite fixed point for the channel or for some of its powers. Moreover, it is shown that the restriction we pose is irrelevant for the case of two-level systems (qubits). As a consequence, we completely characterize the entanglement-saving qubit channels. In order to give an operational meaning to our abstract results, we provide also a concrete model and a sequence of operations reproducing it.