The tracial Hahn–Banach theorem, polar duals, matrix convex sets, and projections of free spectrahedra

Abstract

This article investigates matrix convex sets. It introduces tracial analogs, which we call contractively tracial convex sets. Critical in both contexts are completely positive (cp) maps. While unital cp maps tie into matrix convex sets, trace preserving cp (CPTP) maps tie into contractively tracial sets. CPTP maps are sometimes called quantum channels and are central to quantum information theory. Free convexity is intimately connected with Linear Matrix Inequalities (LMIs) $L(x) = A\_0 + A\_1 x\_1 + \cdots + A\_g x\_g \succeq 0$ and their matrix convex solution sets ${ X : L(X \succeq 0}$, called free spectrahedra. The Effros–Winkler Hahn–Banach Separation Theorem for matrix convex sets states that matrix convex sets are solution sets of LMIs with operator coefficients. Motivated in part by cp interpolation problems, we develop the foundations of convex analysis and duality in the tracial setting, including tracial analogs of the Effros–Winkler Theorem. The projection of a free spectrahedron in $g+h$ variables to $g$ variables is a matrix convex set called a free spectrahedrop. As a class, free spectrahedrops are more general than free spectrahedra, but at the same time more tractable than general matrix convex sets. Moreover, many matrix convex sets can be approximated from above by free spectrahedrops. Here a number of fundamental results for spectrahedrops and their polar duals are established. For example, the free polar dual of a free spectrahedrop is again a free spectrahedrop. We also give a Positivstellensatz for free polynomials that are positive on a free spectrahedrop.