  We consider a theory of quantum thermodynamics with multiple conserved
quantities (or charges). To this end, we generalize the seminal results of
Sparaciari et al. [PRA 96:052112, 2017] to the case of multiple, in general
non-commuting charges, for which we formulate a resource theory of
thermodynamics of asymptotically many non-interacting systems. To every state
we associate the vector of its expected charge values and its entropy, forming
the phase diagram of the system. Our fundamental result is the Asymptotic
Equivalence Theorem (AET), which allows us to identify the equivalence classes
of states under asymptotic approximately charge-conserving unitaries with the
points of the phase diagram.
  Using the phase diagram of a system and its bath, we analyze the first and
the second laws of thermodynamics. In particular, we show that to attain the
second law, an asymptotically large bath is necessary. In the case that the
bath is composed of several identical copies of the same elementary bath, we
quantify exactly how large the bath has to be to permit a specified work
transformation of a given system, in terms of the number of copies of the
elementary bath systems per work system (bath rate). If the bath is relatively
small, we show that the analysis requires an extended phase diagram exhibiting
negative entropies. This corresponds to the purely quantum effect that at the
end of the process, system and bath are entangled, thus permitting classically
impossible transformations. For a large bath, system and bath may be left
uncorrelated and we show that the optimal bath rate, as a function of how
tightly the second law is attained, can be expressed in terms of the heat
capacity of the bath.
  Our approach solves a problem from earlier investigations about how to store
the different charges under optimal work extraction protocols in physically
separate batteries.
