  The observation of quantum phenomena often necessitates sufficiently pure
states, a requirement that can be challenging to achieve. In this study, our
goal is to prepare a non-classical state originating from a mixed state,
utilizing dynamics that preserve the initial low purity of the state. We
generate a quantum superposition of displaced thermal states within a microwave
cavity using only unitary interactions with a transmon qubit. We measure the
Wigner functions of these ``hot'' Schr\"odinger cat states for an initial
purity as low as 0.06. This corresponds to a cavity mode temperature of up to
1.8 Kelvin, sixty times hotter than the cavity's physical environment. Our
realization of highly mixed quantum superposition states could be implemented
with other continuous-variable systems e.g. nanomechanical oscillators, for
which ground-state cooling remains challenging.
