Strongly correlated quantum systems are not easily described with conventional quantum chemistry formalism because the number of non-negligible configurations grows exponen- tially with the number of orbitals actively participating in the correlation.

In this lecture we will introduce the concept of reduced density matrices for systems of identical fermions and comment on their relevance to problems in quantum chemistry and physics, especially the description of strongly correlated quantum systems. We will discuss Coulsons challenge in which Coulson highlighted the potential advantages of a direct calculation of the two-electron reduced density matrix without the many-electron wave function and cautioned against the difficulty of ensuring that the two-electron reduced density matrix represents an N-electron quantum system, known as the N-representability problem. We will present recent advances for the direct calculation of the two-electron reduced density matrix including the implementation of N-representability conditions by semidefinite programming. Two-electron reduced density matrix (2-RDM) methods can accurately approximate strong electron correlation in molecules and materials at a computational cost that grows non-exponentially with system size [5]. In an application we will treat a quantum chemical system with sextillion (1021) quantum degrees of freedom to reveal the important role of quantum entanglement in its oxidation and reduction.