  We recently presented a constructive solution to the N-representability
problem of the two-electron reduced density matrix (2-RDM)---a systematic
approach to constructing complete conditions to ensure that the 2-RDM
represents a realistic N-electron quantum system [D. A. Mazziotti, Phys. Rev.
Lett. 108, 263002 (2012)]. In this paper we provide additional details and
derive further N-representability conditions on the 2-RDM that follow from the
constructive solution. The resulting conditions can be classified into a
hierarchy of constraints, known as the (2,q)-positivity conditions where the q
indicates their derivation from the nonnegativity of q-body operators. In
addition to the known T1 and T2 conditions, we derive a new class of
(2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity
conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of
(2,6)-positivity conditions. The constraints obtained can be divided into two
general types: (i) lifting conditions, that is conditions which arise from
lifting lower (2,q)-positivity conditions to higher (2,q+1)-positivity
conditions and (ii) pure conditions, that is conditions which cannot be derived
from a simple lifting of the lower conditions. All of the lifting conditions
and the pure (2,q)-positivity conditions for q>3 require tensor decompositions
of the coefficients in the model Hamiltonians. Subsets of the new
N-representability conditions can be employed with the previously known
conditions to achieve polynomially scaling calculations of ground-state
energies and 2-RDMs of many-electron quantum systems even in the presence of
strong electron correlation.
