  Recent developments in formal verification have identified approximate
liftings (also known as approximate couplings) as a clean, compositional
abstraction for proving differential privacy. This construction can be defined
in two styles. Earlier definitions require the existence of one or more witness
distributions, while a recent definition by Sato uses universal quantification
over all sets of samples. These notions have each have their own strengths: the
universal version is more general than the existential ones, while existential
liftings are known to satisfy more precise composition principles.
  We propose a novel, existential version of approximate lifting, called
$\star$-lifting, and show that it is equivalent to Sato's construction for
discrete probability measures. Our work unifies all known notions of
approximate lifting, yielding cleaner properties, more general constructions,
and more precise composition theorems for both styles of lifting, enabling
richer proofs of differential privacy. We also clarify the relation between
existing definitions of approximate lifting, and consider more general
approximate liftings based on $f$-divergences.
