We propose a new characterization of non-Markovian quantum evolution based on the concept of non-Markovianity degree. It provides an analog of a Schmidt number in the entanglement theory and reveals the formal analogy between quantum evolution and the entanglement theory: Markovian evolution corresponds to a separable state and the non-Markovian one is further characterized by its degree. It enables one to introduce a non-Markovianity witness\char22{}an analog of an entanglement witness, and a family of measures\char22{}an analog of Schmidt coefficients, and finally to characterize maximally non-Markovian evolution being an analog of the maximally entangled state. Our approach allows us to classify the non-Markovianity measures introduced so far in a unified rigorous mathematical framework.

Quantum technologies rely on the ability to coherently transfer information encoded in quantum states along quantum channels. Decoherence induced by the environment sets limits on the efficiency of any quantum-enhanced protocol. Generally, the longer a quantum channel is the worse its capacity is. We show that for non-Markovian quantum channels this is not always true: surprisingly the capacity of a longer channel can be greater than of a shorter one. We introduce a general theoretical framework linking non-Markovianity to the capacities of quantum channels and demonstrate how harnessing non-Markovianity may improve the efficiency of quantum information processing and communication.

We analyze non-Markovian evolution of open quantum systems. It is shown that any dynamical map representing the evolution of such a system may be described either by a nonlocal master equation with a memory kernel or equivalently by an equation which is local in time. These two descriptions are complementary: if one is simple, the other is quite involved, or even singular, and vice versa. The price one pays for the local approach is that the corresponding generator keeps the memory about the starting point ``t0.'' This is the very essence of non-Markovianity. Interestingly, this generator might be highly singular; nevertheless, the corresponding dynamics is perfectly regular. Remarkably, the singularities of the generator may lead to interesting physical phenomena such as the revival of coherence or sudden death and revival of entanglement.

In this paper we present a detailed critical study of several recently proposed non-Markovianity measures. We analyze their properties for single-qubit and two-qubit systems in both pure-dephasing and dissipative scenarios. More specifically we investigate and compare their computability, their physical meaning, their Markovian to non-Markovian crossover, and their additivity properties with respect to the number of qubits. The bottom-up approach that we pursue is aimed at identifying similarities and differences in the behavior of non-Markovianity indicators in several paradigmatic open system models. This, in turn, allows us to infer the leading traits of the variegated phenomenon known as non-Markovian dynamics.