  The study of thermal operations allows one to investigate the ultimate
possibilities of quantum states and of nanoscale thermal machines. Whilst
fairly general, these results typically do not apply to continuous variable
systems and do not take into account that, in many practically relevant
settings, system-environment interactions are effectively bilinear. Here we
tackle these issues by focusing on Gaussian quantum states and channels. We
provide a complete characterisation of the most general Gaussian thermal
operation acting on an arbitrary number of bosonic modes, which turn out to be
all embeddable in a Markovian dynamics, and derive necessary and sufficient
conditions for state transformations under such operations in the single-mode
case, encompassing states with nonzero coherence in the energy eigenbasis
(i.e., squeezed states). Our analysis leads to a no-go result for the
technologically relevant task of algorithmic cooling: We show that it is
impossible to reduce the entropy of a system coupled to a Gaussian environment
below its own or the environmental temperature, by means of a sequence of
Gaussian thermal operations interspersed by arbitrary (even non-Gaussian)
unitaries. These findings establish fundamental constraints on the usefulness
of Gaussian resources for quantum thermodynamic processes.
