Fluctuations and entanglement in open quantum systems

The subject of this thesis is the study of the class of the Open Quantum Systems, i.e. quantum systems which are in weak interaction with an external environment, often acting as a reservoir, and whose evolution is consequently influenced by this interaction. Since the coupling of the system with the reservoir is weak, it makes sense to try to disentangle the dynamics of the system of interest from the global time-evolution. However, a particular attention must be taken in the derivation of the reduced evolution law, which is expressed by a Master Equation. When the reduced dynamics of the system has to be extracted from the dynamics of the compound system, different Master Equations can be obtained by the application of different kinds of Markovian approximations, which in turn give different evolution laws. In order to sort out the disparate often incompatible open quantum dynamics, a leading criterion is to ensure that they do not violate important physical properties that a proper reduced dynamics must fulfill. In primis the positivity requirement, that ensures that any physical initial state is mapped into another physical state at any later time. Secondly, the complete positivity condition, that guarantees that physical consistency is preserved also when dealing with open quantum systems which are entangled with other systems. Usually, complete positivity is justified in terms of the existence of entangled states of the open quantum system coupled to an arbitrary, inert ancilla. Complete positivity avoids the appearance of negative probabilities in the spectrum of the time-evolving states of system plus ancilla. The main bulk of this thesis work is to show that a non completely positive dynamics can lead to violations of the Second Law of Thermodynamics.