  Practical measures of electronic decoherence, called distilled purities, that
are applicable to many-body systems are introduced. While usual measures of
electronic decoherence such as the purity employ the full $N$-particle density
matrix which is generally unavailable, the distilled purities are based on the
$r$-body reduced density matrices ($r$-RDMs) which are more accessible
quantities. The $r$-body distilled purities are derivative quantities of the
previously introduced $r$-body reduced purities [I. Franco and H. Appel, J.
Chem. Phys. 139, 094109 (2013)] that measure the non-idempotency of the
$r$-RDMs. Specifically, the distilled purities exploit the structure of the
reduced purities to extract coherences between Slater determinants with integer
occupations defined by a given single-particle basis that compose an electronic
state. In this way, the distilled purities offer a practical platform to
quantify coherences in a given basis that can be used to analyze the quantum
dynamics of many-electron systems. Exact expressions for the one-body and
two-body distilled purities are presented and the utility of the approach is
exemplified via an analysis of the dynamics of oligo-acetylene as described by
the Su-Schrieffer-Heeger Hamiltonian. Last, the advantages and limitations of
the purity, reduced purity and distilled purity as measures of electronic
coherence are discussed.
