  In this work, we consider \textit{quantum unital Otto heat engines}. The
latter refers to the fact that both the unitaries of the adiabatic strokes and
the source of the heat provided to the engine preserve the maximally mixed
state. We show how to compute the cumulants of either the dephased or
undephased engine. For a qubit, we give the analytical expressions of the
averages and variances for arbitrary unitaries and unital channels. We do a
detailed comparative study between the dephased and undephased heat engines.
More precisely, we focus on the effect of the parameters on the average work
and its reliability and efficiency. As a case study of unital channels, we
consider a quantum projective measurement. We show on which basis we should
projectively measure the qubit, either the dephased or undephased heat engine,
to extract higher amounts of work, increase the latter's reliability, and
increase efficiency. Further, we show that non-adiabatic transitions
\textit{are not always detrimental} to thermodynamic quantities. Our results,
we believe, are important for heat engines fueled by \textit{quantum
measurement}.
