Helton–McCullough Theorem: Free Convexity & Spectrahedra

• The Helton–McCullough Theorem rigorously defines free convex semialgebraic sets as closures of free spectrahedra using monic linear matrix pencils.
• It establishes that matrix convex noncommutative functions, when entire, must be quadratic, highlighting a striking rigidity in their structure.
• The theorem underpins advances in semidefinite programming, operator algebras, and optimization, influencing applications in quantum systems and real algebraic geometry.

The Helton–McCullough Theorem is a foundational result in free convexity and noncommutative real algebraic geometry. It rigorously characterizes the structure of matrix convex noncommutative semialgebraic sets, establishes deep rigidity for matrix convex noncommutative functions, and underpins a variety of advances in spectrahedral representation, optimization, and operator systems. There are several theorems often referred to under the Helton–McCullough umbrella; this article systematically develops the main variants, their hypotheses, proofs, and contextual significance.

1. Matrix Convexity and Free Spectrahedra

Let $g \in \mathbb{N}$, and define the free Hermitian tuple space by $$S^g := \bigcup_{n=1}^{\infty}\{ X = (X_1, \ldots, X_g) : X_i \in S_n(\mathbb{C}) \text{ Hermitian} \}$$. A matrix convex set $K \subset S^g$ is a set closed under direct sums and simultaneous unitary conjugations and, crucially, under all matrix convex combinations: for any $A^{(j)} \in K(n_j)$ and isometries $V_j \in \mathbb{C}^{n_j \times N}$ with $\sum_j V_j^* V_j = I_N$, the element $\sum_j V_j^* A^{(j)} V_j$ belongs to $K(N)$.

A (monic) linear matrix pencil is defined as $$L(X) = I_d - L_1 \otimes X_1 - \cdots - L_g \otimes X_g$$ for $L_i \in S_d(\mathbb{C})$. The associated free spectrahedron is $$\mathcal{D}_L = \bigcup_{n=1}^{\infty} \{ X \in S^g(n) : L(X) \succeq 0 \}$$, a prototypical matrix convex set.

Free basic open semialgebraic sets are strict positivity domains of Hermitian noncommutative polynomials, $$F_p := \bigcup_{n=1}^\infty \{ X \in S^g(n) : p(X) \succ 0 \}$$, for $p$ Hermitian and noncommutative. If $p$ is linear, $F_p$ is simply the interior of a free spectrahedron (Kriel, 2016).

2. The Helton–McCullough Theorem: Matrix Convex Free Semialgebraic Sets

Theorem (Helton–McCullough): Let $$p(X) \in S(\mathbb{C}\langle X\rangle)^{s \times s}$$ with $p(0) = I_s$, and suppose $F_p$ is matrix convex. Then there exists a monic linear pencil $L(X) = I_D - \sum_{i=1}^g L_i \otimes X_i$ such that

$$F_p = \{ X \in S^g(n) : L(X) \succ 0 \}, \qquad \mathcal{D}_L = \overline{F_p}.$$

One can ensure $D \leq \dim (\text{span of monomials in } p) - 1$; in particular, $D < \infty$.

Interpretation: The closure of any matrix-convex free basic open semialgebraic set is a free spectrahedron. This is the noncommutative analog of spectrahedral representability for convex semialgebraic sets in the commutative setting, but more rigid: every such free convex set is exactly a spectrahedron, without requiring projections (Kriel, 2016).

3. Rigidity of Matrix Convex Noncommutative Functions

A parallel rigidity phenomenon, often also called the Helton–McCullough Theorem, arises for entire matrix convex noncommutative functions:

Let $F(x_1, \ldots, x_g)$ be a Hermitian nc function, real entire (analytic on nc-open sets of self-adjoint tuples), and matrix convex in $x$. Then $F$ is necessarily a noncommutative polynomial of degree at most two: $F(x) = F_0 + L(x) + Q(x),$ with $L(x)$ linear and $Q(x)$ homogeneous quadratic. Higher-degree nc polynomials cannot be globally matrix convex (Helton et al., 2015).

The proof leverages reduction to one-variable slices, application of a Kraus–Donoghue–Bhatia classification of matrix-convex analytic functions (which must be quadratic), and powerseries expansion for all higher monomials vanishing.

The principle generalizes: for Hermitian nc functions $F(a, x)$ that are analytic and matrix convex in $x$ (with $a$ acting as parameters), $F$ is quadratic in $x$ uniformly for all $a$. This establishes "quadratic rigidity" of matrix convexity in the free setting (Helton et al., 2015).

4. Symmetric Determinantal Representation Variant

A distinct deterministic variant due to Helton–McCullough–Vinnikov, often cited as the Helton–McCullough Theorem in convex algebraic geometry, asserts:

For any real multivariate polynomial $p(x_1, \ldots, x_n)$, there exists $m$ and real symmetric matrices $A_0, A_1, \ldots, A_n$ such that

$p(x) = \det(A_0 + x_1 A_1 + \cdots + x_n A_n),$

i.e., $p(x)$ has a symmetric determinantal representation via a linear matrix pencil (Stefan et al., 2021).

Stefan–Welters' recent proof proceeds by successive product substitutions, Schur complement arguments, and induction, highlighting that no heavy machinery beyond linear algebra is required for the general result. This representation is fundamental to semidefinite representability theory, providing the algebraic scaffolding for the relationship between general polynomial inequalities and LMIs.

5. Applications in Operator Algebras and Positivstellensätze

The Helton–McCullough framework extends naturally to Positivstellensätze and duality in operator algebras. The Helton–McCullough Positivstellensatz (Klep et al., 2024) states: For the free $*$-algebra $R\langle x \rangle$ and quadratic module $M(g_1, \dots, g_k)$ generated by Hermitian constraint polynomials $g_j$, if $M(g_1, \dots, g_k)$ is Archimedean and $f$ is self-adjoint with $f(X) \succeq 0$ whenever all $g_j(X) \succeq 0$, then $f = \sigma_0 + \sum_j \sigma_j g_j$ for sums-of-squares $\sigma_0, \sigma_j$.

This result is the backbone of modern hierarchies for noncommutative polynomial optimization, such as the NPA hierarchy and its extensions, providing certificates for semidefinite programming and spectral bounds for noncommutative polynomials (Klep et al., 2024).

6. Separation, Duality, and Bipolar Theorems

The proof of the spectral characterization theorem fundamentally relies on Effros–Winkler separation (matrix Hahn–Banach-type results). The separation principle asserts that points outside a closed matrix convex set are separated by monic pencils, leading to the closure characterization in terms of free spectrahedra (Kriel, 2016).

On the map-duality side, the Helton–Klep–McCullough tracial bipolar theorem provides a complete dual characterization for tracial convex hulls of families of completely positive maps between operator systems, identifying the double tracial polar as the $\tau$-closed tracial convex hull. This connects noncommutative convexity, operator system dualities, and convex geometry in the setting of completely positive maps (Kian, 17 Nov 2025).

7. Examples, Methods, and Generalizations

• Free Matrix Cube: $F = \{ X : I - X_i^2 \succ 0 \}$ is a matrix-convex free basic open set; its closure is the free spectrahedron $S = \{ X : I - X_i^2 \succeq 0 \}$, both captured by a single monic pencil (Kriel, 2016).
• Butterfly (Kraus/Helton–McCullough–Vinnikov) Realization: Any analytic matrix-convex nc function $f$ on a free spectrahedron admits a realization formula combining a linear part and a state-space operator-gain feedback term, extending classical integral representations in commutative convexity to noncommutative analytic functions (Pascoe et al., 2019).
• Matrix Farkas Theorem: The Helton–Klep–McCullough sum-of-squares (SOS) + pencil decomposition for linear matrix polynomials extends Farkas' lemma to matrix settings, underlining the necessity of bounded spectrahedral domains and, for general pencils, monicity and strict positivity for the target (Zalar, 2010).

Table: Main Variants of the Helton–McCullough Theorem

Variant	Statement	Reference
Matrix-convex sets	Every matrix-convex free basic open semialgebraic set is (closure of) a free spectrahedron	(Kriel, 2016)
Function rigidity	Any entire matrix-convex nc function is quadratic	(Helton et al., 2015)
Determinantal representation	Every real polynomial $p(x)$ has a symmetric determinantal form via an LMI	(Stefan et al., 2021)
Positivstellensatz	Noncommutative quadratic module certificates for positivity on free semialgebraic domains	(Klep et al., 2024)
Matrix Farkas Lemma	SOS+pencil representation under boundedness and monicity (refinements for diagonals, etc.)	(Zalar, 2010)

8. Significance and Context

The Helton–McCullough theorems form the basis for the modern structural theory of convexity in the free (noncommutative) setting. They:

• Establish that free convex semialgebraic sets are precisely spectrahedra, deepening the analogy with commutative convex algebraic geometry but in a more rigid regime.
• Impose severe algebraic restrictions on matrix-convex noncommutative functions (must be quadratic if entire), with strong implications for system theory and noncommutative optimization.
• Motivate semidefinite programming as the canonical relaxation for operator-theoretic and quantum problems involving noncommutative polynomial inequalities.
• Underpin duality, separation, and realization results for completely positive maps and operator systems.

These results are instrumental in noncommutative semidefinite programming, spectral bounds for quantum systems, free real algebraic geometry, and the theory of operator algebras. The ongoing refinement of auxiliary hypotheses, explicit bounds, and analytic extension mechanisms continues to inform new methods in convex optimization and quantum information theory.

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References:

(Kriel, 2016, Helton et al., 2015, Stefan et al., 2021, Klep et al., 2024, Zalar, 2010, Kian, 17 Nov 2025, Pascoe et al., 2019)