Violations of the second law of thermodynamics by a non-completely positive dynamics

We consider a recently proposed model of driven open quantum micro-circuit (Pellegrini F., Phys. Rev. Lett., 107 (2011) 060401) amenable to experimental investigations. We show that such an open quantum system provides a concrete physical instance where we can prove that modeling its time evolution with a dynamics lacking complete positivity conflicts with the second law of thermodynamics.