Extreme Points of Spectrahedra

TL;DR

This paper characterizes the extreme points of spectrahedra, providing conditions for extremality, low-rank structure in finite constraints, and applications to correlation matrices, with implications in statistics and quantum mechanics.

Contribution

It introduces a new characterization of extreme points of spectrahedra, including a rank-based criterion and a simple description of the elliptope in finite dimensions.

Findings

Extreme points characterized by density conditions in trace-class operators.

Low-rank structure of extreme points under finite constraints.

Simple Hadamard product characterization of correlation matrices.

Abstract

We consider the problem of characterizing extreme points of the convex set of positive linear operators on a possibly infinite-dimensional Hilbert space under linear constraints. We show that even perturbations of points in such sets admit what resembles a Douglas factorization. Using this result, we prove that an operator is extreme iff a corresponding set of linear operators is dense in the space of trace-class self-adjoint operators with range contained in the closure of the range of that operator. If the number of constraints is finite, we show that the extreme point must be of low-rank relative to the number of constraints and derive a purely rank-based characterization of the extreme points. In the finite-dimensional setting, our results lead to a remarkably simple characterization of the elliptope, that is, the set of correlation matrices, in terms of the Hadamard product which allows us to characterize the set of matrices which constitute the equality case of the Hadamard rank inequality when the involved matrices are equal and positive semi-definite. We illustrate the importance of our results using examples from statistics and quantum mechanics.