The face lattice of the set of reduced density matrices and its coatoms

Abstract

The lattice of faces of the convex set of reduced density matrices is essential for the construction of the information projection to a hierarchical model. The lattice of faces is also important in quantum state tomography. Yet, the description and computation of these faces is elusive in the simplest examples. Here, we study the face lattice of the set of two-body reduced density matrices: We show that the three-qubit lattice has no elements of rank seven and that it has a family of coatoms of rank five. This contrasts with the three-bit lattice, where every coatom has rank six. We discovered the coatoms of rank five using a novel experimental method, which employs convex duality, semidefinite programming, and algebra. We also discuss nonexposed points for three and six qubits. Using frustration-free Hamiltonians, we provide a new characterization of probability distributions that factor.