Non-Commutative Sum-of-Squares

Updated 11 November 2025

Non-commutative sum-of-squares (nc-SOS) is a positivity certificate for symmetric polynomials in non-commuting variables, generalizing the classical SOS paradigm to free *-algebras.
It utilizes decompositions into sums of Hermitian squares and foundational results like Helton's theorem to ensure positivity under all operator substitutions.
The approach leverages Gram-matrix representations and semidefinite programming hierarchies to address optimization challenges in quantum information, control theory, and free real algebraic geometry.

A non-commutative sum-of-squares (nc-SOS) is a positivity certificate for a symmetric (Hermitian) polynomial in non-commuting variables, generalizing the classical SOS paradigm to free *-algebras and operator-theoretic settings. The central paradigm is: a Hermitian non-commutative polynomial is positive semidefinite under all operator substitutions if and only if it admits a decomposition as a sum of Hermitian squares (SOHS). This fundamental property, first established by Helton and extended to matrix- and operator-valued cases, underpins a rapidly developing theory with ramifications for free real algebraic geometry, quantum information, optimization, matrix analysis, and theoretical computer science.

1. Algebraic Foundations of Non-Commutative SOS

Let $R = \mathbb{R}\langle x_1,\dots,x_n\rangle$ denote the real free *-algebra generated by non-commuting symbols $x_i$ , with the involution $*$ acting by $(x_{i_1}\dots x_{i_k})^* = x_{i_k}\dots x_{i_1}$ and $x_i^* = x_i$ . The set of all words (monomials) forms a basis of $R$ .

A polynomial $f \in R$ is called Hermitian (or symmetric) if $f^* = f$ . For a tuple $X = (X_1,\dots,X_n)$ of self-adjoint operators (finite or infinite-dimensional), $f(X)$ is defined via non-commutative functional calculus. The positivity problem is: for $f=f^*$ , does $f(X)\succeq0$ for all operator substitutions $X$ ?

The SOS cone $\Sigma$ consists of all finite sums $f = \sum_{j} p_j^* p_j$ for $p_j \in R$ ; elements are manifestly positive under operator evaluation. The foundational result of Helton and its generalizations establish:

For scalar-valued $f$ , $f(X)\succeq0$ always $\iff$ $f$ is a sum of Hermitian squares.
For matrix- or operator-valued $f$ , analogous single-square factorizations can be obtained under broad conditions (Jindal et al., 9 Nov 2025).

2. Positivstellensätze and Convexity in the Free Algebra

A non-commutative Positivstellensatz gives explicit certificates for positivity on matrix convex semialgebraic sets:

If $L(x) = I_\ell - \sum_{i=1}^{g} A_i x_i$ is a monic linear pencil and $D_L$ its free spectrahedral domain, then for any symmetric $p(x)$ of degree $\leq 2d+1$ ,

$p(X) \succeq 0$ for all $X \in D_L$ $\iff$ $p(x)$ admits a weighted SOS representation:

$p(x) = s(x)^* s(x) + \sum_j f_j(x)^* L(x) f_j(x),$

where $\deg s, \deg f_j \le \frac{1}{2} \deg p$ (Helton et al., 2011).

This theorem is "perfect": degree bounds are optimal and no Archimedean preordering or higher-degree certificates are needed (in contrast to the commutative theory). The proof uses separation via the Hahn-Banach theorem and a GNS construction, exploiting the linearity and monicity of $L(x)$ .

An immediate structural consequence is that in the free setting, matrix convex polynomials of degree $>2$ do not exist: if $p$ is quasi-convex (i.e., all sets $\{X:A-p(X)\succeq0\}$ are convex for all $A\succ0$ ), then either $p$ is matrix-convex and degree at most $2$, or $-p$ is an SOHS. This dichotomy sharply contrasts with classical convex polynomials (Balasubramanian et al., 2012).

3. Gram-Matrix Representations and Gram-like Extensions

Given a symmetric $f \in \mathbb{R}\langle X \rangle$ of degree $2d$, the classical Gram-matrix method constructs a vector $v$ of all monomials of degree $\le d$ and writes $f = v^* G v$ for a unique real symmetric Gram matrix $G$ .

$f$ is SOS $\iff$ $G$ is positive semidefinite.
Operator- and matrix-valued cases are handled analogously, with $G$ taking values in operator algebras (Jindal et al., 9 Nov 2025).

To facilitate efficient representations and completion problems, the concept of Gram-like matrices is introduced (Mukherjee et al., 21 Jan 2025):

For $f = W^* G W$ where $W$ is any tuple of monomials ( $|W| \leq$ ambient basis size) and $G$ is symmetric (or PSD), $G$ is a Gram-like matrix for $f$ .
These allow flexibility in basis selection and are central to extension and completion techniques.

An important application is SOHS extension: given a polynomial with a partial SOHS and additional monomials, it is always possible (under mild conditions) to extend to a full SOHS so that the Gram-like block for the original SOHS part is preserved as a principal submatrix; the process relies on block matrix constructions and Sylvester's criterion. Completion problems—when can a partial Gram-like matrix be extended to a PSD matrix—are characterized exactly by chordality of the specification graph or 2-regularity of the associated projective variety.

4. Optimization and the Non-Commutative SOS Hierarchy

The non-commutative SOS (nc-SOS) hierarchy extends commutative polynomial optimization and the moment-SOS approach to operator scenarios (Bhardwaj et al., 2021, Hastings, 2022):

The optimization variable is a moment matrix $M^{(d)}$ , indexed by words of degree $\leq d/2$ , whose entries are pseudo-expectations $E[X_\alpha^* X_\beta]$ .
The matrix $M^{(d)}$ $M^{(d)}$ must satisfy:
- $M \succeq 0$
- Linear constraints derived from the operator algebra (commutation/anticommutation relations)
- (Optionally) Additional PSD localizing constraints from operator inequalities.
SDP duality yields SOS certificates: $H - \lambda = \sum_a B_a^* B_a$ for Hamiltonian $H$ , with $B_a$ operator polynomials of degree $\leq d/2$ .

Truncations of the moment-SOS SDP hierarchy yield a sequence of lower bounds converging monotonically to the true ground-state value under Archimedean-type conditions.

In quantum contexts, the hierarchy recovers well-known approximations (e.g., $2$-RDM and $3$-RDM relaxations in quantum chemistry). Notably, for certain quartic-fermion systems, degree-4 (2-RDM) fails while a specific degree-6 fragment (mirroring cubic constraints from SYK model analysis) precisely recovers second-order perturbation theory (Hastings, 2022). Successive levels of relaxation reproduce higher orders under conjectured universality for fermionic systems.

5. Operator-Valued SOS, Factorization, and Structural Properties

For operator-valued noncommutative polynomials with Hilbert space coefficients, single-square factorizations generalize the classical Helton theorem (Jindal et al., 9 Nov 2025):

If $p$ is positive on all self-adjoint (or unitary) operator tuples, $p = r^* r$ for some $r$ in the same free *-algebra and operator coefficient space.
The proof proceeds by contradiction: assuming $p$ is positive but not SOS, the Hahn–Banach theorem provides a separating linear functional, which is realized via a finite-rank noncommutative Hankel matrix. The associated GNS construction yields an explicit representation leading to a contradiction of positivity.
The sos cone is closed in the weak operator topology, allowing compactness arguments and extraction of Gram-matrix limits.
The Fock space machinery supplies canonical operator tuples for extracting coefficients and bounding norm growth of Gram matrices.

These factorizations extend to noncommutative trigonometric polynomials and underlie positivity certificates for operator-valued observables in systems theory, quantum information, and free probability.

6. Explicit SOS Certificates for Symmetric and Structured Polynomials

A noncommutative generalization of Hunter's positivity theorem exhibits sharp and minimal SOS representations for complete homogeneous symmetric polynomials (CHS) in the non-commutative setting (Garcia et al., 16 Mar 2025):

The nc CHS polynomial $H_{2d}(x_1,...,x_n)$ is a sum of exactly $\binom{n-1+d}{d}$ hermitian squares, and this count is minimal.
The proof employs a symmetrization map (lifting commutative monomials to their fully symmetrized nc counterparts), Gram-matrix factorization via simplex integration, and Schur complements to identify sharp operator inequalities:

$H_{2d}(X_1,...,X_n) \succeq \mu_{n,d} (X_1^{2d} + \cdots + X_n^{2d})$

with optimal $\mu_{n,d}$ .

This result establishes that not all commutative positivity transfers to the noncommutative setting: certain symmetric nc polynomials lack positivity, so the CHS case is exceptional.
nc-SOS certificates in this setting have implications for free real algebraic geometry and operator inequalities.

7. Computational and Algorithmic Aspects

The nc-SOS framework poses significant computational challenges due to the combinatorial explosion in word basis size:

For $n$ variables and truncation degree $d$ , the Gram/moment matrices scale as $s(d,n) = 1 + n + n^2 + ... + n^d$ .
Exploiting involution symmetry, Hankel structure, and sparsity can reduce problem size, but scalability remains a challenge. Techniques include correlative and term-sparsity decompositions.
For small to moderate $n$ and $d$ (e.g., $n \leq 5$ , $d \leq 4$ ), SDP solvers such as TSSOS, NCSOStools, NCAlgebra, and YALMIP are effective, but complexity is exponential in $d$ .
In special cases (e.g., constraints of the form $R - \sum x_i^2$ ), structural properties allow for first-order and spectral algorithms.

Applications of nc-SOS span operator inequalities, optimization over noncommutative (quantum) variables, control and systems theory, and quantum information, with increasingly fine-grained certificates possible as computational capabilities advance.

The non-commutative sum-of-squares paradigm unifies positivity, convexity, and optimization in free *-algebras. Its theoretical strength—most notably in the optimality and structural tightness of Positivstellensätze and the universality of SOS representations—provides a rigorous algebraic and algorithmic toolkit for a variety of operator, quantum, and matrix-analytic domains. The interplay between algebraic certificate complexity, Gram matrix theory, and feasible relaxations positions nc-SOS as a central object of paper in modern real algebraic geometry and quantum optimization (Helton et al., 2011, Balasubramanian et al., 2012, Bhardwaj et al., 2021, Hastings, 2022, Mukherjee et al., 21 Jan 2025, Garcia et al., 16 Mar 2025, Jindal et al., 9 Nov 2025).