Matrix convex sets over the Euclidean ball and polar duals of real free spectrahedra

Abstract

We show that the free spectrahedron determined by universal anticommuting self-adjoint unitaries is not equal to the minimal matrix convex set over the ball in dimension three or higher. This example, as well as other matrix convex sets over the ball, then provides context for structure results on the extreme points of coordinate projections. In particular, we show that the free polar dual of a real free spectrahedron is rarely the projection of a real free spectrahedron, contrasting a prior result of Helton, Klep, and McCullough over the complexes. We use this to show that spanning results for free spectrahedra that are closed under complex conjugation do not extend to free spectrahedrops that meet the same assumption. These results further clarify the role of the coefficient field.

arXiv comment25 pages. Version 2 has minor updates and typo corrections. To appear in Linear Algebra and Its Applications

References in corpus (15)

• Separability criterion and inseparable mixed states with positive partial transposition
• The noncommutative Choquet boundary
• The Choquet boundary of an operator system
• The matricial relaxation of a linear matrix inequality
• Minimal and maximal matrix convex sets
• Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization
• Extreme points of matrix convex sets, free spectrahedra and dilation theory
• Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets
• The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra
• An elementary introduction to the geometry of quantum states with pictures
• Compressions of compact tuples
• Shape, Scale, and Minimality of Matrix Ranges
• Matrix range characterizations of operator system properties
• Completely Bounded Norms of $k$-positive Maps
• Extreme points of matrix convex sets and their spanning properties