Quantum thermoelectric transmission functions with minimal current fluctuations

Thermodynamic uncertainty relations (TURs) represent a benchmark result in nonequilibrium physics that allows to place fundamental lower bounds on the noise-to-signal ratio (precision) of currents in nanoscale devices. Originally formulated for classical time-homogeneous Markov processes, these relations, were shown to be violated in thermoelectric engines and photovoltaic devices supporting quantum-coherent transport. However, the extent to which these violations may occur still represents a missing piece of the puzzle. In this work, we provide such answer in a definitive way within the general Landauer-B\"uttiker formalism for noninteracting systems, beyond any perturbative regime, e.g., linear response. In particular, using analytical constrained-optimization techniques, we rigorously demonstrate that the transmission function which maximizes the reliability of thermoelectric devices (i.e., which minimizes the fluctuations of its steady-state currents) for fixed average power and efficiency is a collection of boxcar functions. This allows us to show that TURs can be violated by arbitrarily large amounts, depending on the temperature and chemical potential gradients, thus providing guidelines to the design of optimal devices.