The reduction of density matrices defined with respect to a finite basis set is considered. A basis set can be introduced into the vector space of Hermitian matrices acting on functions of the coordinates of $p$ particles for each value of $p$. An analogy with the construction of spin eigenfunctions is used to obtain a particular basis set called the reducing basis. Reduction of one of these basis elements is either one to one or maps into the origin. Any element in the preimage of a density matrix can thus be resolved into two components, one uniquely determined by the density matrix and the other arbitrary within a certin subset. A restatement of the $N$-representability problem is given, and two sufficient conditions and one necessary condition for $N$ representability are given in terms of distances and norms.

• Received 28 April 1977

DOI:https://doi.org/10.1103/PhysRevA.17.1257