  Various quantum thermodynamic bounds are shown to stem from a single tighter
and more general inequality, consequence of the operator concavity of the
logarithmic function. Such an inequality, which we call the "thermodynamic
reverse bound", is compactly expressed as a quantum relative entropy, from
which it inherits mathematical properties and meaning. As concrete examples, we
apply our bound to evaluate the thermodynamic length for open processes, the
heat exchange in erasure processes, and the maximal energy outflow in general
quantum evolutions.
