Fejér--Riesz factorization for positive noncommutative trigonometric polynomials

TL;DR

This paper extends the classical Fejér--Riesz factorization to positive matrix-valued noncommutative trigonometric polynomials defined on a mixed domain $\mathscr W\times\mathfrak Y$, where $\mathscr W$ is a free semigroup or a free product of copies of $\mathbb Z_2$ and $\mathfrak Y$ is discrete. The authors prove that, under strict positivity across all unitary representations, such a polynomial $A$ factors as $A=B^*B$ with $B$ analytic and with a degree bound in the $\mathscr W$-variables that is optimal, while providing a robust Positivstellensatz-type certificate in the trivial-$\mathfrak Y$ case. Two independent methodological contributions support the main results: a PSD Parrott theorem with group-valued entries and degree-controlled PSD matrix completion results, which are then tied to unitary representations via a realization framework and nested truncation arguments. The work yields degree-bounded sos certificates for Bell-type inequalities and sharp counterexamples demonstrating the necessity of the positivity hypotheses in certain noncommutative groups. Overall, the paper provides new factorization tools and a principled approach to noncommutative positivity with concrete algebraic and quantum-information implications.

Abstract

We prove a Fejér-Riesz type factorization for positive matrix-valued noncommutative trigonometric polynomials on $\mathscr{W}\times\mathfrak{Y}$, where $\mathscr{W}$ is either the free semigroup $\langle x \rangle_g$ or the free product group $\mathbb{Z}_2^{g}$, and $\mathfrak{Y}$ is a discrete group. More precisely, using the shortlex order, if $A$ has degree at most $w$ in the $\mathscr{W}$ variables and is uniformly strictly positive on all unitary representations of $\mathscr{W}\times\mathfrak{Y}$, then $A=B^{*}B$ with $B$ analytic and of $\mathscr{W}$-degree at most $w$; this degree bound is optimal, and strict positivity is essential. As an application, we obtain degree-bounded sums-of-squares certificates for Bell-type inequalities in $\mathbb{C}[\mathbb{Z}_2^{*g}\times \mathbb{Z}_2^{*h}]$ from quantum information theory. In the special case $\mathscr{W}=\mathbb{Z}^h$ we recover, in the matrix-valued setting, the classical commutative multivariable Fejér-Riesz factorization. For trivial $\mathfrak{Y}$ we obtain a ``perfect'' group-algebra Positivstellensatz on $\mathbb{Z}_2^{*g}$ that does not require strict positivity; this result is sharp, as demonstrated by counterexamples in $\mathbb{Z}_2*\mathbb{Z}_3$ and $\mathbb{Z}_3^{*2}$. To establish our main results two novel ingredients of independent interest are developed: (a) a positive-semidefinite Parrott theorem with entries given by functions on a group; and (b) solutions to positive semidefinite matrix completion problems for $\langle x \rangle_g$ or the free product group $\mathbb{Z}_2^{*g}$ indexed by words in $\mathscr{W}$ of length $\le w$.

Theorems & Definitions (56)

• Theorem 1.1: Noncommutative Fejér--Riesz theorem
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6: Group algebra Positivstellensatz for $\mathbb{Z}^{*{\tt{g}}}_2$
• Remark 1.7
• Remark 1.8
• Remark 1.9
• Remark 1.10