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The study of thermal operations allows one to investigate the ultimatepossibilities of quantum states and of nanoscale thermal machines. Whilstfairly general, these results typically do not apply to continuous variablesystems and do not take into account that, in many practically relevantsettings, system-environment interactions are effectively bilinear. Here wetackle these issues by focusing on Gaussian quantum states and channels. Weprovide a complete characterisation of the most general Gaussian thermaloperation acting on an arbitrary number of bosonic modes, which turn out to beall embeddable in a Markovian dynamics, and derive necessary and sufficientconditions for state transformations under such operations in the single-modecase, encompassing states with nonzero coherence in the energy eigenbasis(i.e., squeezed states). Our analysis leads to a no-go result for thetechnologically relevant task of algorithmic cooling: We show that it isimpossible to reduce the entropy of a system coupled to a Gaussian environmentbelow its own or the environmental temperature, by means of a sequence ofGaussian thermal operations interspersed by arbitrary (even non-Gaussian)unitaries. These findings establish fundamental constraints on the usefulnessof Gaussian resources for quantum thermodynamic processes.

Gaussian Thermal Operations and the Limits of Algorithmic Cooling