We present a theoretical framework to tackle quantum non-Markovian dynamics based on a microscopic collision model (CM), where the bath consists of a large collection of initially uncorrelated ancillas. Unlike standard memoryless CMs, we endow the bath with memory by introducing interancillary collisions between next system-ancilla interactions. Our model interpolates between a fully Markovian dynamics and the continuous interaction of the system with a single ancilla, i.e., a strongly non-Markovian process. We show that in the continuous limit one can derive a general master equation, which, while keeping such features, is guaranteed to describe an unconditionally completely positive and trace-preserving dynamics. We apply our theory to an atom in a dissipative cavity for a Lorentzian spectral density of bath modes, a dynamics which can be exactly solved. The predicted evolution shows a significant improvement in approaching the exact solution with respect to two well-known memory-kernel master equations.

It is known that a reliable geometric quantifier of discord-like correlations can be built by employing the so-called trace distance, which is used to measure how far the state under investigation is from the closest 'classical-quantum' state. To date, the explicit calculation of this indicator for two qubits has only been accomplished for states where the reduced density matrix of the measured party is maximally mixed, a class that includes Bell-diagonal states. Here, we first reduce the required optimization for a general two-qubit state to the minimization of an explicit two-variable function. Using this framework, we show that the minimum can be analytically worked out in a number of relevant cases, including quantum-classical and X states. This provides an explicit and compact expression for the trace distance discord of an arbitrary state belonging to either of these important classes of density matrices.