Reduced Density Matrix

The two-electron reduced density matrix recast as the minimally sufficient object for molecular energetics, the operator on the antisymmetric pair space that fixes every one- and two-body observable, compresses to orbital subspaces, and lies in the convex N-representability cone.

For an -electron state on a one-particle space , the two-electron reduced density matrix (2-RDM) is the Hermitian operator on the pair space obtained by tracing the state down to two particles. As an object it is a positive semidefinite (PSD) element of

the Hermitian operators on the antisymmetric two-particle space. Because a molecular Hamiltonian is two-body, the 2-RDM is minimally sufficient. It fixes the energy and every one- and two-body observable (partial charges, spin densities, bond orders, multipoles, NMR shifts) at any correlation strength, with no wavefunction and no mean-field solve.

Restricting to a subspace , an orbital block with projector , is compression , the 2-RDM of the Fock-space partial trace over the modes outside . Compression is a positive linear map and composes on the nose, .

Property, N-representability is a cone condition

Not every PSD is the 2-RDM of an actual -electron state. The admissible ones form the N-representability cone , convex with QMA-complete membership. The PQGT relaxation approximates it from outside, intersecting the PSD conditions on the two-electron (), two-hole (), particle–hole () and partial three-positivity () matrices. This gives the nesting

Only is a symmetric cone; is a spectrahedron (convex, not self-dual or homogeneous) and is neither. Positivity is Jordan-structured; representability is not.

References

• A. J. Coleman, “Structure of Fermion Density Matrices” (Rev. Mod. Phys., 1963)
• Y.-K. Liu, M. Christandl & F. Verstraete, “Quantum Computational Complexity of the N-representability Problem: QMA Complete” (Phys. Rev. Lett., 2007)
• D. A. Mazziotti, “Variational Two-Electron Reduced-Density-Matrix Theory” (PQGT positivity conditions)