Spectral properties of cones of approximately representable reduced density matrices

We introduce a definition of the approximate spectrum of an operator which is useful in reduced density matrix theory. With precise knowledge of $\text{D}$² (the cone of representable reduced 2-densities) our approximate spectrum of any Hermitian, symmetric, one-body operator, A, agrees with the usual spectrum of operators on Fock space. The virtue of our definition is that with only approximate knowledge of $\text{D}$² we can compute the approximate spectrum for A. The approximate spectrum turns out to be a sensitive tool in assessing the quality of cones of approximately representable reduced 2-densities. Using this notion, we are able to point out a dramatic failure of one cone of approximately representable reduced 2-densities often referred to in the literature. In addition we show how reduced density matrices for excited states of systems with two-body interactions can be computed, given $\text{D}$⁴ the cone of reduced 4-densities.