# Relating the pure and ensemble density matrix functional

## TL;DR

This paper challenges a key theorem in RDMFT by showing that the ensemble and pure density matrix functionals are related through convexification, revealing that complexity persists when moving from pure to ensemble formulations.

## Contribution

The paper refutes the theorem that pure and ensemble functionals coincide and demonstrates that the ensemble functional is the lower convex envelope of the pure functional, highlighting the transfer of complexity.

## Key findings

- The pure and ensemble functionals do not coincide on their common domain.
- The ensemble functional is the lower convex envelope of the pure functional.
- Pure functional determines the ensemble functional even outside its domain.

## Abstract

A crucial theorem in Reduced Density Matrix Functional Theory (RDMFT) suggests that the universal pure and ensemble functional coincide on their common domain of pure N-representable one-matrices. We refute this by a comprehensive analysis of the geometric picture underlying Levy's constrained search. Moreover, we then show that the ensemble functional follows instead as the lower convex envelop of the pure functional. It is particularly remarkable that the pure functional determines the ensemble functional even outside its own domain of pure N-representable one-matrices. From a general perspective, this demonstrates that relaxing pure RDMFT to ensemble RDMFT does not necessarily circumvent the complexity of the one-body pure N-representability conditions (generalized Pauli constraints). Instead, the complexity may simply be transferred from the underlying space of pure N-representable one-matrices to the structure of the universal one-matrix functional.