  The quasiprobability representation of quantum states addresses two main
concerns, the identification of nonclassical features and the decomposition of
the density operator. While the former aspect is a main focus of current
research, the latter decomposition property has been studied less frequently.
In this contribution, we introduce a method for the generalized expansion of
quantum states in terms of so-called quasistates. In contrast to the
quasiprobability decomposition through nonclassical distributions and
pure-state operators, our technique results in classical probabilities and
nonpositive semidefinite operators, defining the notion of quasistates, that
carry the information about the nonclassical characteristics of the system.
Therefore, our method presents a complementary approach to prominent
quasiprobability representations. We explore the usefulness of our technique
with several examples, including applications for quantum state reconstruction
and the representation of nonclassical light. In addition, using our framework,
we also demonstrate that inseparable quantum correlations can be described in
terms of classical joint probabilities and tensor-product quasistates for an
unambiguous identification of quantum entanglement.
