Structure of Fermionic Density Matrices: Complete $N$-Representability Conditions

We present a constructive solution to the $N$-representability problem: a full characterization of the conditions for constraining the two-electron reduced density matrix to represent an $N$-electron density matrix. Previously known conditions, while rigorous, were incomplete. Here, we derive a hierarchy of constraints built upon (i) the bipolar theorem and (ii) tensor decompositions of model Hamiltonians. Existing conditions $D$, $Q$, $G$, $T1$, and $T2$, known classical conditions, and new conditions appear naturally. Subsets of the conditions are amenable to polynomial-time computations of strongly correlated systems.

Figure 1

The convex set $P_N^{2*}$ of two-body operators that are positive semidefinite in their trace with any $N$-particle density matrix is contained within the convex set $P_N^{3*}$ of analogous three-body operators, which in turn is contained within the set $P_N^{r*}$. Hence, the extreme points of $P_N^{2*}$ can be characterized completely by the convex combination of the extreme points of $P_N^{r*}$, which are given by Eq. (8).