Structure of Fermionic Density Matrices: Complete N-representability Conditions

TL;DR

This paper provides a complete set of conditions for the N-representability problem, enabling accurate characterization of two-electron reduced density matrices for N-electron systems.

Contribution

It introduces a hierarchy of N-representability conditions derived from the bipolar theorem and tensor decompositions, completing previous incomplete criteria.

Findings

Existing classical conditions D, Q, G, T1, T2 are incorporated

New conditions are derived and naturally appear in the hierarchy

Some conditions are computationally feasible for strongly correlated systems

Abstract

We present a constructive solution to the N-representability problem---a full characterization of the conditions for constraining the two-electron reduced density matrix (2-RDM) to represent an N-electron density matrix. Previously known conditions, while rigorous, were incomplete. Here we derive a hierarchy of constraints built upon (i) the bipolar theorem and (ii) tensor decompositions of model Hamiltonians. Existing conditions D, Q, G, T1, and T2, known classical conditions, and new conditions appear naturally. Subsets of the conditions are amenable to polynomial-time computations of strongly correlated systems.