# A variation principle for ground spaces

## TL;DR

This paper characterizes the lattice structure of ground spaces of hermitian matrices, revealing their geometric properties and connections to quantum marginals and local Hamiltonians.

## Contribution

It introduces a novel characterization of lattice elements and maximal elements of ground spaces using operator cone constraints, advancing the understanding of quantum marginal geometry.

## Key findings

- Lattice elements of ground spaces are characterized by operator cone constraints.
- Maximal lattice elements correspond to specific subspace structures.
- Results link ground space lattices to quantum marginal face lattices.

## Abstract

The ground spaces of a vector space of hermitian matrices, partially ordered by inclusion, form a lattice constructible from top to bottom in terms of intersections of maximal ground spaces. In this paper we characterize the lattice elements and the maximal lattice elements within the set of all subspaces using constraints on operator cones. Our results contribute to the geometry of quantum marginals, as their lattices of exposed faces are isomorphic to the lattices of ground spaces of local Hamiltonians.