Violations of the second law of thermodynamics by a non completely   positive dynamics
  We consider a recently proposed model of driven open quantum microcircuit [F. Pellegrini et al., Phys. Rev. Lett. 107, 060401 (2011)] amenable to experimental investigations. We show that such an open quantum system provides a concrete physical instance where to prove that modeling its time-evolution with a dynamics lacking complete positivity conflicts with the second law of thermodynamics. 
G. Argentieri, F. Benatti, R. Floreanini, M. Pezzutto
quant-ph cond-mat.other