  Thermodynamics connects our knowledge of the world to our capability to
manipulate and thus to control it. This crucial role of control is exemplified
by the third law of thermodynamics, Nernst's unattainability principle, which
states that infinite resources are required to cool a system to absolute zero
temperature. But what are these resources and how should they be utilized? And
how does this relate to Landauer's principle that famously connects information
and thermodynamics? We answer these questions by providing a framework for
identifying the resources that enable the creation of pure quantum states. We
show that perfect cooling is possible with Landauer energy cost given infinite
time or control complexity. However, such optimal protocols require complex
unitaries generated by an external work source. Restricting to unitaries that
can be run solely via a heat engine, we derive a novel Carnot-Landauer limit,
along with protocols for its saturation. This generalizes Landauer's principle
to a fully thermodynamic setting, leading to a unification with the third law
and emphasizes the importance of control in quantum thermodynamics.
