  We investigate fundamental connections between thermodynamics and quantum
information theory. First, we show that the operational framework of thermal
operations is nonequivalent to the framework of Gibbs-preserving maps, and we
comment on this gap. We then introduce a fully information-theoretic framework
generalizing the above by making further abstraction of physical quantities
such as energy. It is technically convenient to work with and reproduces known
results for finite-size quantum thermodynamics. With our framework we may
determine the minimal work cost of implementing any logical process. In the
case of information processing on memory registers with a degenerate
Hamiltonian, the answer is given by the max-entropy, a measure of information
known from quantum information theory. In the general case, we obtain a new
information measure, the "coherent relative entropy", which generalizes both
the conditional entropy and the relative entropy. It satisfies a collection of
properties which justifies its interpretation as an entropy measure and which
connects it to known quantities. We then present how, from our framework,
macroscopic thermodynamics emerges by typicality, after singling out an
appropriate class of thermodynamic states possessing some suitable
reversibility property. A natural thermodynamic potential emerges, dictating
possible state transformations, and whose differential describes the physics of
the system. The textbook thermodynamics of a gas is recovered as well as the
form of the second law relating thermodynamic entropy and heat exchange.
Finally, noting that quantum states are relative to the observer, we see that
the procedure above gives rise to a natural form of coarse-graining in quantum
mechanics: Each observer can consistently apply the formalism of quantum
information according to their own fundamental unit of information.
