We theoretically investigate fluctuation relations in a classical incomplete measurement process where only partial information is available. The scenario we consider consists of two coupled single-electron boxes where one or both devices can undergo a nonequilibrium transformation according to a chosen protocol. The entropy production of only one of the two boxes is recorded and fluctuation relations for this quantity are put to a test, showing strong modifications whose nature depends upon the specific case study.

We address memory effects and diffusive properties of a continuous-time quantum walk on a one-dimensional percolation lattice affected by spatially correlated random telegraph noise. In particular, by introducing spatially correlated time-dependent fluctuations in nearest-neighbor hopping amplitudes, we describe random domains characterized by global noise. The resulting open dynamics of the walker is then unraveled by an ensemble average over all the noise realizations. Our results show that time-dependent noise assisted by spatial correlations leads to strong memory effects in the walker dynamics and to robust diffusive behavior against the detrimental action of uncorrelated noise. We also show that spatially correlated classical noise enhances localization breaking, thus making a quantum particle spread on longer distances across the lattice.

We investigate the open dynamics of an atomic impurity embedded in a one-dimensional Bose-Hubbard lattice. We derive the reduced evolution equation for the impurity and show that the Bose-Hubbard lattice behaves as a tunable engineered environment allowing one to simulate both Markovian and non-Markovian dynamics in a controlled and experimentally realizable way. We demonstrate that the presence or absence of memory effects is a signature of the nature of the excitations induced by the impurity, being delocalized or localized in the two limiting cases of a superfluid and Mott insulator, respectively. Furthermore, our findings show how the excitations supported in the two phases can be characterized as information carriers.

We address some key conditions under which many-body lattice models, intended mainly as simulated condensed matter systems, can be investigated via immersed, fully controllable quantum objects, namely quantum probes. First, we present a protocol that, for a certain class of many-body systems, allows for full momentum resolved spectroscopy using one single probe. Furthermore, we demonstrate how one can extract the two-point correlations using two entangled probes. We apply our theoretical proposal to two well-known exactly solvable lattice models, a 1D Kitaev chain and 2D superfluid Bose-Hubbard model, and show its accuracy as well as its robustness against external noise.