The moment polytope of matrix multiplication is not maximal
Maxim van den Berg, Matthias Christandl, Vladimir Lysikov, Harold Nieuwboer, Michael Walter, Jeroen Zuiddam
TL;DR
The paper addresses how moment polytopes distinguish tensor families, notably showing that the moment polytope of matrix multiplication is not maximal within the Kronecker polytope. By developing a minrank-based obstruction and leveraging SL-stability, it proves separations between $\Delta(\mathsf{U}_c)$ and $\Delta(\mathsf{M}_n)$ for ranges $n^2-n+1<c\le n^2$ and extends these results to iterated matrix multiplication tensors. It also demonstrates that asymptotic restriction does not imply moment-polytope inclusion, with implications for quantum marginals and the efficiency of tensor-network ansatzes beyond PEPS. Additionally, the work yields a new proof of the optimal border subrank bound for matrix multiplication and provides extensions to higher-order tensors, strengthening the connection between invariant theory, representation theory, and algebraic complexity. Collectively, these results clarify limitations of degenerations as a characterization of moment-polytopes and illuminate the expressive power of multipartite entanglement structures in tensor networks switching beyond two-party entanglement.
Abstract
Moment polytopes of tensors, the study of which is deeply rooted in invariant theory, representation theory and symplectic geometry, have found relevance in numerous places, from quantum information (entanglement polytopes) and algebraic complexity theory (GCT program and the complexity of matrix multiplication) to optimization (scaling algorithms). Towards an open problem in algebraic complexity theory, we prove separations between the moment polytopes of matrix multiplication tensors and unit tensors. As a consequence, we find that matrix multiplication moment polytopes are not maximal, i.e. are strictly contained in the corresponding Kronecker polytope. As another consequence, we obtain a no-go result for a natural operational characterization of moment polytope inclusion in terms of asymptotic restriction. We generalize the separation and non-maximality to moment polytopes of iterated matrix multiplication tensors. Our result implies that tensor networks where multipartite entanglement structures beyond two-party entanglement are allowed can go beyond projected entangled-pair states (PEPS) in terms of expressivity. Our proof characterizes membership of uniform points in moment polytopes of tensors, and establishes a connection to polynomial multiplication tensors via the minrank of matrix subspaces. As a result of independent interest, we extend these techniques to obtain a new proof of the optimal border subrank bound for matrix multiplication.