  The wave-like nature of electrons leads to the existence of upper bounds on
the thermoelectric response of nanostructured devices [R. S. Whitney, Phys.
Rev. Lett. 112, 130601 (2014); Phys. Rev. B 91, 115425 (2015)]. This
fundamental result, not present in classical thermodynamics, was demonstrated
exploiting a two-terminal device modelled by non-linear scattering theory. In
the present paper, we consider non-linear quantum transport through the same
type of device working both as thermal machine and as refrigerator. For both
operations, starting from charge and heat current expressions, we provide
analytic quantum bounds for power exchanged, thermal currents and device
efficiencies. For this purpose, we adopt a transmission function that maximizes
the engine efficiency for given power output. For the optimal boxcar- or theta
function-transmission shapes, we provide in a tutorial way an explicit
deduction of the quantum bound expressions reported in the above cited papers.
