Correlated vs. uncorrelated noise acting on a quantum refrigerator

Two qubits form a quantum four-level system. The golden-rule based transition rates between these states are determined by the coupling of the qubits to noise sources. We demonstrate that depending on whether the noise acting on the two qubits is correlated or not, these transitions are governed by different selection rules. In particular, we find that for fully correlated or anticorrelated noise, there is a protected state, and the dynamics of the system depends then on its initialization. For nearly (anti)correlated noise, there is a long time scale determining the temporal evolution of the qubits. We apply our results to a quantum Otto refrigerator based on two qubits coupled to hot and cold baths. Even in the case when the two qubits do not interact with each other, the cooling power of the refrigerator does not scale with the number ($=2$ here) of the qubits when there is strong correlation of noise acting on them.