Fluctuation theorems set the cornerstone for describing the statistical properties of classical non-equilibrium thermodynamics based on the principle of microscopic reversibility.
We show that the fluctuation theorems, which are normally considered for thermodynamic processes, can be a powerful tool to study the detailed statistics of quantum systems as well as the effect of coherence transfer through an arbitrary quantum process.
To this end, we establish the symmetry relation between transition probabilities under a given quantum channel and its reverse channel defined by the Petz recovery map. In our framework, the notion of entropy production is extended to the quantum regime by allowing it to have a complex-value. The imaginary part of entropy production plays a crucial role in characterising the evolution of coherence through the quantum channel, as well as deriving the second law from the quantum fluctuation theorem.
Our results can be applied to understand the dissipation and fluctuation of various quantum resources, including quantum free energy, asymmetry, and entanglement in a unified framework. We also discuss how these fluctuating quantum quantities, including the imaginary entropy production, can be experimentally observed.