  Accurately describing work extraction from a quantum system is a central
objective for the extension of thermodynamics to individual quantum systems.
The concepts of work and heat are surprisingly subtle when generalizations are
made to arbitrary quantum states. We formulate an operational thermodynamics
suitable for application to an open quantum system undergoing quantum evolution
under a general quantum process by which we mean a completely-positive and
trace-preserving map. We derive an operational first law of thermodynamics for
such processes and show consistency with the second law. We show that heat,
from the first law, is positive when the input state of the map majorises the
output state. Moreover, the change in entropy is also positive for the same
majorisation condition. This makes a strong connection between the two
operational laws of thermodynamics.
