  We use continuous weak measurements of a driven superconducting qubit to
experimentally study the information dynamics of a quantum Maxwell's demon. We
show how information gained by a demon who can track single quantum
trajectories of the qubit can be converted into work using quantum coherent
feedback. We verify the validity of a quantum fluctuation theorem with feedback
by utilizing information obtained along single trajectories. We demonstrate, in
particular, that quantum backaction can lead to a loss of information in
imperfect measurements. We furthermore probe the transition between information
gain and loss by varying the initial purity of the qubit.
