Fluctuation relation for quantum heat engines and refrigerators

At the very foundation of the second law of thermodynamics lies the fact that no heat engine operating between two reservoirs of temperatures T-C <= T-H can outperform the ideal Carnot engine: < W >/< Q(H)> <= 1 - T-C/T-H. This inequality follows from an exact fluctuation relation involving the nonequilibrium work W and heat exchanged with the hot bath Q(H). In a previous work (Sinitsyn 2011 J. Phys. A: Math. Theor. 44 405001) this fluctuation relation was obtained under the assumption that the heat engine undergoes a stochastic jump process. Here we provide the general quantum derivation, and also extend it to the case of refrigerators, in which case Carnot's statement reads < Q(C)>/vertical bar < W >vertical bar <= (T-H/T-C - 1)(-1).