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.