  We investigate the connection between recent results in quantum
thermodynamics and fluctuation relations by adopting a fully quantum mechanical
description of thermodynamics. By including a work system whose energy is
allowed to fluctuate, we derive a set of equalities which all thermodynamical
transitions have to satisfy. This extends the condition for maps to be
Gibbs-preserving to the case of fluctuating work, providing a more general
characterisation of maps commonly used in the information theoretic approach to
thermodynamics. For final states, block diagonal in the energy basis, this set
of equalities are necessary and sufficient conditions for a thermodynamical
state transition to be possible. The conditions serves as a parent equation
which can be used to derive a number of results. These include writing the
second law of thermodynamics as an equality featuring a fine-grained notion of
the free energy. It also yields a generalisation of the Jarzynski fluctuation
theorem which holds for arbitrary initial states, and under the most general
manipulations allowed by the laws of quantum mechanics. Furthermore, we show
that each of these relations can be seen as the quasi-classical limit of three
fully quantum identities. This allows us to consider the free energy as an
operator, and allows one to obtain more general and fully quantum fluctuation
relations from the information theoretic approach to quantum thermodynamics.
