The theory of open quantum system is one of the most important tools for the development of quantum technologies. Furthermore, the Lindblad (or Gorini-Kossakowski-Sudarshan-Lindblad) Master Equation plays a key role as it is the most general generator of Markovian dynamics in quantum systems. In this paper, we present this equation together with its derivation and methods of resolution. The presentation tries to be as self-contained and simple as possible to be useful to readers with no previous knowledge of this field.

In d-dimensional lattices of coupled quantum harmonic oscillators, we analyze the heat current caused by two thermal baths of different temperatures, which are coupled to opposite ends of the lattice, with a focus on the validity of Fourier's law of heat conduction. We provide analytical solutions of the heat current through the quantum system in the nonequilibrium steady state using the rotating-wave approximation and bath interactions described by a master equation of Lindblad form. The influence of local dephasing in the transition of ballistic to diffusive transport is investigated.

The study of the entanglement properties of systems of N fermions has attracted considerable interest during the last few years. Various separability criteria for pure states of N identical fermions have been recently discussed but, excepting the case of two-fermions systems, these criteria are difficult to implement and of limited value from the practical point of view. Here we advance simple necessary and sufficient separability criteria for pure states of N identical fermions. We found that to be identified as separable a state has to comply with one single identity involving either the purity or the von Neumann entropy of the single-particle reduced density matrix. These criteria, based on the verification of only one identity, are drastically simpler than the criteria discussed in the recent literature. We also derive two inequalities verified respectively by the purity and the entropy of the single particle, reduced density matrix, that lead to natural entanglement measures for N -fermion pure states. Our present considerations are related to some classical results from the Hartree-Fock theory, which are here discussed from a different point of view in order to clarify some important points concerning the separability of fermionic pure states.