Generalized Pauli conditions on the spectra of one-electron reduced density matrices of atoms and molecules

TL;DR

This paper investigates the spectra of one-electron reduced density matrices in atoms and molecules, revealing how they relate to generalized Pauli conditions and electron correlation, with implications for understanding quantum entanglement.

Contribution

It introduces a geometric approach to measure the proximity of 1-RDM spectra to generalized Pauli conditions and distinguishes between ground and excited states in terms of pinning behavior.

Findings

Ground states' 1-RDM spectra are pinned to the polytope boundary.

Excited states' 1-RDM spectra are not pinned.

Proximity to the boundary correlates with electron correlation and entanglement.

Abstract

The Pauli exclusion principle requires the spectrum of the occupation numbers of the one-electron reduced density matrix (1-RDM) to be bounded by one and zero. However, for a 1-RDM from a wave function, there exist additional conditions on the spectrum of occupation numbers, known as pure N-representability conditions or generalized Pauli conditions. For atoms and molecules, we measure through a Euclidean-distance metric the proximity of the 1-RDM spectrum to the facets of the convex set (polytope) generated by the generalized Pauli conditions. For the ground state of any spin symmetry, as long as time-reversal symmetry is considered in the definition of the polytope, we find that the 1-RDM's spectrum is pinned to the boundary of the polytope. In contrast, for excited states, we find that the 1-RDM spectrum is not pinned. Proximity of the 1-RDM to the boundary of the polytope provides a measurement and classification of electron correlation and entanglement within the quantum system. For comparison, this distance to the boundary of the generalized Pauli conditions is also compared to the distance to the polytope of the traditional Pauli conditions, and the distance to the nearest 1-RDM spectrum from a Slater determinant. We explain the difference in pinning in the ground- and excited-state 1-RDMs through a connection to the N-representability conditions of the two-electron reduced density matrix.