Refining ensemble $N$-representability of one-body density matrices from partial information

TL;DR

This paper develops a hierarchy of ensemble N-representability problems for one-body density matrices, introducing relaxations that connect to Horn's problem and yield convex polytopes for ensemble DFT restrictions.

Contribution

It introduces a systematic relaxation of ensemble N-representability problems, linking them to Horn's problem and providing new convex constraints for excited-state density functional theory.

Findings

Relaxed problem related to Horn's problem allows explicit solutions.

Convex relaxation yields a polytope constraining lattice site occupations.

New restrictions improve understanding of ensemble density matrices in quantum systems.

Abstract

The $N$-representability problem places fundamental constraints on reduced density matrices (RDMs) that originate from physical many-fermion quantum states. Motivated by recent developments in functional theories, we introduce a hierarchy of ensemble one-body $N$-representability problems that incorporate partial knowledge of the one-body reduced density matrices (1RDMs) within an ensemble of $N$-fermion states with fixed weights $w_i$. Specifically, we propose a systematic relaxation that reduces the refined problem -- where full 1RDMs are fixed for certain ensemble elements -- to a more tractable form involving only natural occupation number vectors. Remarkably, we show that this relaxed problem is related to a generalization of Horn's problem, enabling an explicit solution by combining its constraints with those of the weighted ensemble $N$-representability conditions. An additional convex relaxation yields a convex polytope that provides physically meaningful restrictions on lattice site occupations in ensemble density functional theory for excited states.