  For many-electron systems, the second-order reduced density matrix (2-RDM)
provides sufficient information for characterizing their properties of
interests in physics and chemistry, ranging from total energy, magnetism,
quantum correlation and entanglement to long-range orders. Theoretical
prediction of the structural properties of 2-RDM is an essential endeavor in
quantum chemistry, condensed matter physics and, more recently, in quantum
computation. Since 1960s, enormous progresses have been made in developing
RDM-based electronic structure theories and their large-scale computational
applications in predicting molecular structure and mechanical, electrical and
optical properties of various materials. However, for strongly correlated
systems, such as high-temperature superconductors, transition-metal-based
biological catalysts and complex chemical bonds near dissociation limit,
accurate approximation is still out of reach by currently most sophisticated
approaches. This limitation highlights the elusive structural feature of 2-RDM
that determines quantum correlation in many-electron system. Here, we present a
set of constraints on 2-RDM based on the basic geometric property of Hilbert
space and the commutation relations of operators. Numerical examples are
provided to demonstrate the pronounced violation of these constraints by the
variational 2-RDMs. It is shown that, for a strongly correlated model system,
the constraint violation may be responsible for a considerable portion of the
variational error in ground state energy. Our findings provide new insights
into the structural subtlety of many-electron 2-RDMs.
