Free Spectrahedra: Matrix Convex Sets

• Free spectrahedra are dimension-free analogues of classical convex spectrahedra defined by LMIs that exhibit matrix convexity and free semialgebraic properties.
• They bridge operator theory, noncommutative algebraic geometry, and quantum information with explicit examples like the free cube, ball, and simplex.
• Their structure encompasses duality, nuanced extreme point notions, mapping rigidity, and applications in semidefinite programming and quantum measurement.

Free spectrahedra are dimension-free analogues of classical convex spectrahedra, defined by linear matrix inequalities (LMIs) in noncommuting variables. They form the foundational class of matrix convex sets in free analysis, with deep connections to operator theory, noncommutative real algebraic geometry, free semialgebraic convexity, optimization, and quantum information theory. Their structure theory encompasses classification results, duality and representation theorems, automorphism and mapping rigidity, and a nuanced hierarchy of extreme-point notions.

1. Definition and Fundamental Properties

A free spectrahedron is the solution set to a monic linear matrix inequality in all matrix sizes:

Given $g,d \in \mathbb{N}$ and a tuple $$A = (A_1, \ldots, A_g) \in \mathrm{Herm}_d(\mathbb{C})^g$$, the associated monic linear pencil is

$$L_A(X) = I_d \otimes I_n + \sum_{i=1}^g A_i \otimes X_i$$

for $$X = (X_1, \ldots, X_g) \in \mathrm{Herm}_n(\mathbb{C})^g$$.

The $n$th level of the free spectrahedron is

$$\mathcal{D}_A(n) = \{ X \in \mathrm{Herm}_n(\mathbb{C})^g : L_A(X) \succeq 0 \},$$

where $\succeq 0$ denotes positive semidefiniteness. The full graded set

$$\mathcal{D}_A = \bigsqcup_{n=1}^\infty \mathcal{D}_A(n)$$

is called the free spectrahedron defined by $A$ (Kriel, 2016).

Key properties:

• Matrix convexity: $\mathcal{D}_A$ is closed under matrix convex combinations; that is, closed under simultaneous compressions and direct sums.
• Free semialgebraicity: Each level is cut out by a finite-dimensional LMI, making free spectrahedra exactly the matrix convex, "free basic open semialgebraic" sets (Kriel, 2016, Evert et al., 2016).
• Level-1 section: The set $\mathcal{D}_A(1)$ recovers the classical (commutative) spectrahedron in $\mathbb{R}^g$.
• Examples: The free cube $\{X \mid \|X_i\| \le 1\}$, the free ball $\{X \mid \sum_i X_i^2 \preceq I\}$, and the free simplex $\{ X \mid X_i \succeq 0,~ \sum_i X_i \preceq I \}$ are all free spectrahedra.

2. Classification and Semialgebraic Structure

The Helton–McCullough theorem (Kriel, 2016) gives a full characterization: any matrix convex, free semialgebraic set that is an open set at $0$ is precisely a (possibly finite intersection of) free spectrahedron(s). In particular:

• If a self-adjoint noncommutative polynomial matrix $p$ satisfies $p(0)=I$ and

$\mathcal{S} = \{ X : p(X) \succ 0 \}$

is matrix convex, then $\mathcal{S}$ is the interior of a free spectrahedron.

Circular and Reinhardt free spectrahedra form crucial subclasses, characterized algebraically and via combinatorial data:

• Circular free spectrahedra: Invariant under $X_j \mapsto e^{it} X_j$, have defining pencils reducible to block superdiagonal form (Evert et al., 2016). Free circular spectrahedra (invariant under simultaneous unitary conjugation of all variables) further restrict to block structures with only two blocks.
• Reinhardt spectrahedra: Invariant under arbitrary coordinate-wise toral actions, correspond to colored directed graphs with Reinhardt-neutral edge colorings, providing a graph-theoretic classification (McCullough et al., 2020).

3. Extreme Points and Kreĭn–Milman-Type Theorems

Free spectrahedra, as matrix convex sets, support a richer set of extremality notions:

• Euclidean extreme points: Usual convex hull extremity at a fixed level.
• Matrix extreme points: Nontrivial matrix convex decompositions must be size-trivial and unitarily equivalent.
• Absolute/free extreme points: Strongest, require that any weakly proper matrix convex decomposition comes from trivial direct summands; coincide with irreducible Arveson boundary points (Evert et al., 2016, Evert et al., 2018).

The general Kreĭn–Milman theorem (Kriel, 2016, Evert et al., 2018):

• Every compact free spectrahedron is the matrix convex hull of its absolute (Arveson) extreme points.
• Matrix exposed points are dense among matrix extreme points; every compact matrix convex set is the closed matrix convex hull of its matrix exposed points.
• Explicit Carathéodory bounds are available for expressing points as matrix convex combinations of absolute extreme points (Evert et al., 2018).

Cases where matrix and free extreme points differ are known: there exist tuples with matrix extreme points that are not free extreme, except in the $2 \times 2$ case, where they always coincide (Epperly et al., 2022).

4. Duality, Projections, and Spectrahedrops

Free spectrahedra have a well-developed duality theory:

• The free polar dual of a free spectrahedron $\mathcal{D}_A$ is itself a free spectrahedron associated to the pencil $I - \sum \Omega_j x_j$ (Helton et al., 2014).

Projections of free spectrahedra (free spectrahedrops) yield new matrix convex sets:

• Spectrahedrops: Projections onto a subset of variables of a free spectrahedron; strictly contain the class of free spectrahedra (Helton et al., 2014, Evert, 2023).
• Spectrahedrops (and their duals) are closed under free polar duality; the polar dual of a spectrahedrop is itself a spectrahedrop.
• Not every free spectrahedrop admits a spanning by its free extreme points, in stark contrast to the case of spectrahedra (Evert et al., 27 Jul 2025, Evert, 2023).

5. Automorphisms, Bianalytic Maps, and Rigidity

Bianalytic (invertible analytic) maps between free spectrahedra are subject to severe rigidity:

• Such maps (after normalization) must be convexotonic: rational maps associated to $g$-dimensional algebraic structures determined by the defining pencils (Helton et al., 2018, Augat et al., 2016, Augat et al., 2017).
• If two free spectrahedra are bianalytically equivalent via a convexotonic map, their pencils must span isomorphic algebras.
• Automorphism groups of ball-like free spectrahedra and their structure constants are classified explicitly.
• For Reinhardt and circular spectrahedra, bianalytic automorphisms are forced to be linear up to coordinate rotations or trivial toral symmetries (McCullough et al., 2020, Evert et al., 2016).

6. Applications and Quantitative Metrics

Free spectrahedra are central to quantum information, operator theory, and noncommutative optimization:

• Inclusion constants quantify relaxations in spectrahedral inclusion problems (e.g., the matrix cube problem) and can be computed in closed form in highly symmetric cases such as products of free simplices (Bluhm et al., 19 Dec 2025).
• Inclusion constants play a role in bounding the white-noise robustness of quantum measurement incompatibility and certifying resource-theoretic thresholds.
• Free spectrahedra model feasible regions for dimension-free semidefinite programs, operator system structure, and compatibility domains in general probabilistic theories (Bluhm et al., 2020).

7. Open Problems and Structural Phenomena

Active research directions and distinctions include:

• Generalization of inclusion constant formulas to arbitrary spectrahedra and higher levels.
• Structural differences between real and complex free spectrahedra: duality closure, extreme point spanning, and spectrahedrop geometry are sensitive to the field of coefficients (Evert et al., 27 Jul 2025, Passer, 2021).
• Classification problems for graph-constrained quantum structures, bianalytic automorphism groups in higher-rank Reinhardt domains, and the structure of spectrahedrops beyond spectrahedra.
• Rigorous understanding of extreme-point structure in Cartesian and direct product settings, and their exploitation in noncommutative polynomial optimization (Bluhm et al., 19 Dec 2025, Epperly et al., 2022).

In summary, free spectrahedra serve as the archetypical subclass of matrix convex sets with profound structural, geometric, and algebraic properties. Their theory synthesizes deep results from dilation theory, noncommutative semialgebraic geometry, operator systems, and quantum information, with an ongoing interplay between abstract classification, explicit computation, and applications to optimization and physics.

Principal references: (Kriel, 2016, Helton et al., 2014, Evert et al., 2016, Evert et al., 2018, Augat et al., 2016, Helton et al., 2018, Bluhm et al., 19 Dec 2025, Epperly et al., 2022, Evert et al., 27 Jul 2025, Evert et al., 2016, McCullough et al., 2020, Passer, 2021, Evert, 2023, Bluhm et al., 2020, Piana, 9 Dec 2025).