  While quantum measurement theories are built around density matrices and
observables, the laws of thermodynamics are based on processes such as are used
in heat engines and refrigerators. The study of quantum thermodynamics fuses
these two distinct paradigms. In this article, we highlight the usage of
quantum process matrices as a unified language for describing thermodynamic
processes in the quantum regime. We experimentally demonstrate this in the
context of a quantum Maxwell's demon, where two major quantities are commonly
investigated; the average work extraction $\langle W \rangle$ and the efficacy
$\gamma$ which measures how efficiently the feedback operation uses the
obtained information. Using the tool of quantum process matrices, we develop
the optimal feedback protocols for these two quantities and experimentally
investigate them in a superconducting circuit QED setup.
