Inequalities for fermion density matrices

A partial trace over the occupation numbers of all but k states in the density matrix of an ensemble with an arbitrary number of single‐particle states is defined as the (reduced) k ‐ state density matrix. This matrix is used to obtain a complete, practical solution to the problem of determining the representability of the diagonal elements of the one‐ and two‐particle (reduced) density matrices. This solution is expressed as a series of linear inequalities involving the density‐matrix elements; the inequalities are identical with those derived previously by Davidson and McCrae by a different method. In addition, our method is used to obtain nonlinear, matrix inequalities on the off ‐ diagonal elements of the density matrices.