  A theory of feedback controlled heat transport in quantum systems is
presented. It is based on modelling heat engines as driven multipartite systems
subject to projective quantum measurements and measurement-conditioned unitary
evolutions. The theory unifies various results presented in the previous
literature. Feedback control breaks time reversal invariance. This in turn
results in the fluctuation relation not being obeyed. Its restoration occurs by
an appropriate accounting of the information gain and information use via
measurements and feedback. We further illustrate an experimental proposal for
the realisation of a Maxwell demon using superconducting circuits and single
photon on-chip calorimetry. A two level qubit acts as a trapdoor which,
conditioned on its state is coupled to either a hot resistor or a cold one. The
feedback mechanism alters the temperatures felt by the qubit and can result in
an effective inversion of temperature gradient, where heat flows from cold to
hot thanks to information gain and use
