  In this article, we define a new mathematical object, called a pair
$D=\left(A,C\right)$ of anti-commutation matrices (ACMP) based on the
anti-commutation relation $a^\dag_{i}a_{j} + a_{j}a^\dag_{i} = \delta_{ij}$
applied to the scalar product between the many-body wavefunctions. This ACMP
explicitly separates the different levels of correlation. The one-body
correlations are defined by a ACMP $D^0=\left(A^0,C^0\right)$ and the two-body
ones by a set of $n$ ACMPs $D^i=\left(A^i,C^i\right)$ where $n$ is the number
of states. We show that we can have a compact and exact parametrization with
$n^4$ parameters of the two-body reduced density matrix (\TRDM) of any pure or
mixed $N$-body state to determine the ground state energy with a
$\mathcal{O}\left(n^6\right)$ complexity.
