  Recently, measurement-based quantum thermal machines have drawn more
attention in the field of quantum thermodynamics. However, the previous results
on quantum Otto heat engines were either limited to special unital and
non-unital channels in the bath stages, or a specific driving protocol at the
work strokes and assuming the cycle being time-reversal symmetric i.e.
$V^{\dagger}=U$ (or $V=U$). In this paper, we consider a single spin-1/2
quantum Otto heat engine, by first replacing one of the heat baths by an
arbitrary unital channel and then we give the exact analytical expression of
the characteristic function from which all the cumulants of heat and work
emerge. We prove that under the effect of monitoring, $\nu_{2}>\nu_{1}$ is a
necessary condition for positive work, either for a symmetric or
asymmetric-driven Otto cycle. Furthermore, going beyond the average we show
that the ratio of the fluctuations of work and heat is lower and upper-bounded
when the system is working as a heat engine. However, differently from the
previous results in the literature, we consider the third and fourth cumulants
as well. It is shown that the ratio of the third (fourth) cumulants of work and
heat is not upper-bounded by unity nor lower-bounded by the third (fourth)
power of the efficiency, as is the case for the ratio of fluctuations. Finally,
we consider applying a specific unital map that plays the role of a heat bath
in a coherently superposed, manner and we show the role of the initial
coherence of the control qubit on efficiency, on the average work and its
relative fluctuations.
