A variation principle for ground spaces

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.Comment: 18 pages, 2 figures, version v3 has an improved exposition, v4 has a new non-commutative example and catches a glimpse of three qubit