Thermodynamic bound on the Fano factor of a coherent thermoelectric heat engine

We show that for fermionic coherent thermoelectric transport, selecting heat-engine operation yields a thermodynamic uncertainty relation for the charge current that imposes a universal lower limit of $F>1\mathrm{/}2$ on the corresponding Fano factor. We find that violations of the thermodynamic uncertainty relation for classical Markov processes, typically associated with a quantum advantage, are far more restricted for heat engines than what is allowed by a generic thermodynamic process. For bosonic and classical carriers, the minimum Fano factor increases to $F>1$, and the thermodynamic uncertainty relation for classical Markov processes is never violated. We provide numerical evidence that all the obtained bounds are tight and can be saturated by properly designed transmissions.