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arXiv:2610.24542

Comment on "The polarization hierarchy for polynomial optimization over convex bodies, with applications to nonnegative matrix rank"

Julius A. Zeiss

math.OCquant-phSubmitted Oct 17, 2026• Announced Oct 20, 2026
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Abstract

We revisit the feasibility and convergence claims for the polarization hierarchy of Plávala, Ligthart and Gross [Linear Algebra Appl.\ 723 (2025) 15--32]. Counterexamples show that the stated assumption is insufficient for the feasibility characterization and for convergence of the hierarchy with the polarization constraint. We establish corrected results under a stronger algebraic condition and supply a missing compactness argument.

Comments
25 pages, Comment on https://doi.org/10.1016/j.laa.2025.05.019 (arXiv:2406.09506)

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We revisit the feasibility and convergence claims for the polarization hierarchy of Plávala, Ligthart and Gross [Linear Algebra Appl.\ 723 (2025) 15--32]. Counterexamples show that the stated assumption is insufficient for the feasibility characterization and for convergence of the hierarchy with the polarization constraint. We establish corrected results under a stronger algebraic condition and supply a missing compactness argument.
We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the polarization hierarchy , which has previously been introduced for the study of polynomial optimization problems over state spaces of C ⁎ ⁎ -algebras, to convex cones in finite dimensions. If the convex bodies can be characterized by linear or semidefinite programs, then the same is true for our hierarchy. Convergence is proven by relating the problem to a certain de Finetti theorem for general probabilistic theories , which are studied as possible generalizations of quantum mechanics . We apply the method to the problem of nonnegative matrix factorization, and in particular to the nested rectangles problem . A numerical implementation of the third level of the hierarchy is shown to give rise to a very tight approximation for this problem.
Leveraging a recently proposed notion of relative entropy in general probabilistic theories (GPT), we prove a finite de Finetti representation theorem for general convex bodies. We apply this result to address a fundamental question in polynomial optimization: the existence of a convergent outer hierarchy for problems with inequality constraints and analytical convergence guarantees. Our strategy generalizes a quantitative monogamy-of-entanglement argument from quantum theory to arbitrary convex bodies, establishing a uniform upper bound on mutual information in multipartite extensions. This leads to a finite de Finetti theorem and, subsequently, a convergent conic hierarchy for a wide class of polynomial optimization problems subject to both equality and inequality constraints. We further provide a constructive rounding scheme that yields certified interior points with controlled approximation error. As an application, we express the optimal GPT value of a two-player non-local game as a polynomial optimization problem, allowing our techniques to produce approximation schemes with finite convergence guarantees.
Although the quantum state of a many-body system (e.g, a system consisting of several atoms) is described in terms of its degrees of freedom, i.e., a large number of variables representing a point on the compact space of quantum states, there is usually a small number of parameters that can be actually measured experimentally. We show that the problem of deciding whether the underlying quantum state is entangled is equivalent to the problem of deciding whether the set of measured parameters are the moments of some measure defined over the compact space of states of the many-body system. By considering quantum entanglement of n identical atoms we arrive at the truncated moment problem defined for symmetric measures over a product of n copies of unit balls in R^d. We consider moments up to degree 2 since these are most readily available experimentally. We derive necessary and sufficient conditions for belonging to the moment cone, which can be expressed via a linear matrix inequality (LMI) of size at most 2d+2, which is independent of n. We can convert the LMIs into a set of explicit semialgebraic inequalities giving necessary and sufficient conditions for membership in the moment cone, and show that the two conditions approach each other in the limit of large n. The inequalities are derived via considering the dual cone of nonnegative polynomials, and its sum-of-squares (SOS) relaxation. We show that the SOS relaxation of the dual cone is asymptotically exact, and using symmetry reduction techniques (J. Pure Appl. Algebra 192, no. 1-3, 95, arXiv:1205.3102), it can be written as a small linear matrix inequality of size at most 2d+2, which is independent of n. For the cone of nonnegative polynomials we also prove an analogue of the half degree principle for globally nonnegative symmetric polynomials (J. Math. Anal. Appl. 284, no. 1, 174, J. Pure Appl. Algebra 216, no. 4, 850).