The second law of quantum thermodynamics as an equality

The traditional second law of thermodynamics says that the average amount of work required to change one state into another while in contact with a heat reservoir,  must be at least as large as the change in free energy of the system. Here, we consider a fine-grained notion of the free energy, and show that in terms of it, the second law can be written as an equality. We also obtain a generalisation of the Jarzynski fluctuation theorem which holds for arbitrary initial states, not just the case of an initial thermal state. These can be derived from a fully quantum parent identity. This allows for more general and fully quantum fluctuation relations from the information theoretic approach to quantum thermodynamics.