  So-called Thermal Operations seem to describe the most fundamental, and
reasonable, set of operations allowable for state transformations at an ambient
inverse temperature $\beta$. However, a priori, they require experimentalists
to manipulate very complex environments and have control over their internal
degrees of freedom. For this reason, the community has been working on creating
more experimentally-friendly operations. In [Perry et al., Phys. Rev. X 8,
041049] it was shown that for states diagonal in the energy basis, that Thermal
Operations can be performed by so-called Coarse Operations, which need just one
auxiliary qubit, but are otherwise Markovian and classical in spirit. In this
work, by providing an explicit counterexample, we show that this one qubit of
memory is necessary. We also fully characterize the possible transitions that
do not require memory for the system being a qubit. We do this by analyzing
arbitrary control sequences comprising level energy changes and partial
thermalizations in each step.
