  In recent years we have witnessed a concentrated effort to make sense of
thermodynamics for small-scale systems. One of the main difficulties is to
capture a suitable notion of work that models realistically the purpose of
quantum machines, in an analogous way to the role played, for macroscopic
machines, by the energy stored in the idealisation of a lifted weight. Despite
of several attempts to resolve this issue by putting forward specific models,
these are far from capturing realistically the transitions that a quantum
machine is expected to perform. In this work, we adopt a novel strategy by
considering arbitrary kinds of systems that one can attach to a quantum thermal
machine and seeking for work quantifiers. These are functions that measure the
value of a transition and generalise the concept of work beyond the model of a
lifted weight. We do so by imposing simple operational axioms that any
reasonable work quantifier must fulfil and by deriving from them stringent
mathematical condition with a clear physical interpretation. Our approach
allows us to derive much of the structure of the theory of thermodynamics
without taking as a primitive the definition of work. We can derive, for any
work quantifier, a quantitative second law in the sense of bounding the work
that can be performed using some non-equilibrium resource by the work that is
needed to create it. We also discuss in detail the role of reversibility and
correlations in connection with the second law. Furthermore, we recover the
usual identification of work with energy in degrees of freedom with vanishing
entropy as a particular case of our formalism. Our mathematical results can be
formulated abstractly and are general enough to carry over to other resource
theories than quantum thermodynamics.
