  Understanding the structure of quantum correlations in a many-body system is
key to its computational treatment. For fermionic systems, correlations can be
defined as deviations from Slater determinant states. The link between
fermionic correlations and efficient computational physics methods is actively
studied but remains ambiguous. We make progress in establishing this connection
mathematically. In particular, we find a rigorous classification of states
relative to $k$-fermion correlations, which admits a computational physics
interpretation. Correlations are captured by a measure $\omega_k$, a function
of $k$-fermion reduced density matrix that we call twisted purity. Vanishing of
$\omega_k$ for a given $k$ puts the state in a class $G_k$ of correlated
states. Sets $G_k$ are nested in $k$, Slater determinants correspond to $k =
1$. Various physically relevant states are in or close to $G_{k=O(1)}$,
including truncated configuration-interaction states, perturbation series
around Slater determinants, and some nonperturbative eigenstates of the 1D
Hubbard model. For each $k = O(1)$, we give an explicit ansatz with a
polynomial number of parameters that covers all states in $G_k$. Potential
applications of this ansatz and its connections to the coupled-cluster
wavefunction are discussed.
