Spectrahedral Shadows and Completely Positive Maps on Real Closed Fields

TL;DR

This paper introduces new methods to identify convex semialgebraic sets that are not spectrahedral shadows, using sums of squares and model theory, and applies these to show certain cones are not spectrahedral shadows.

Contribution

It characterizes when nonnegative polynomials form spectrahedral shadows and proves the cone of copositive matrices for size ≥5 is not a spectrahedral shadow, answering a longstanding question.

Findings

Nonnegative polynomial sets are spectrahedral shadows iff sums of squares conditions are met.

The cone of copositive matrices of size ≥5 is not a spectrahedral shadow.

Spectrahedral shadows must be preserved by all unital $R$-linear completely positive maps.

Abstract

In this article we develop new methods for exhibiting convex semialgebraic sets that are not spectrahedral shadows. We characterize when the set of nonnegative polynomials with a given support is a spectrahedral shadow in terms of sums of squares. As an application of this result we prove that the cone of copositive matrices of size $n\geq5$ is not a spectrahedral shadow, answering a question of Scheiderer. Our arguments are based on the model theoretic observation that any formula defining a spectrahedral shadow must be preserved by every unital $\mathbb{R}$-linear completely positive map $R\to R$ on a real closed field extension $R$ of $\mathbb{R}$.