Strassen's support functionals coincide with the quantum functionals

arXiv:2601.21553v2 Announce Type: replace-cross Abstract: Strassen's asymptotic spectrum offers a framework for analyzing the complexity of tensors. It has found applications in diverse areas, from computer science to additive combinatorics and quantum information. A long-standing open problem, dating back to 1991, asks whether Strassen's support functionals are universal spectral points, that is, points in the asymptotic spectrum of tensors. In this paper, we answer this question in the affirmative for tensors over complex numbers by proving that the support functionals coincide with the quantum functionals - universal spectral points that are defined via entropy optimization on entanglement polytopes. We obtain this result as a special case of a general minimax formula for convex optimization on entanglement polytopes (and more general moment polytopes). Our formula can be interpreted as a very general "classical-quantum correspondence" analogous to the well-known relation between the Shannon entropy and the von Neumann entropy. Its proof is based on a recent Fenchel-type duality theorem on Hadamard manifolds due to Hirai. In addition to settling Strassen's question, our results yield a unified and simpler approach to several other tensor invariants, including the asymptotic slice rank, the symmetric quantum functional, the $G$-stable rank, and the non-commutative rank.

Abstract:Strassen's asymptotic spectrum offers a framework for analyzing the complexity of tensors. It has found applications in diverse areas, from computer science to additive combinatorics and quantum information. A long-standing open problem, dating back to 1991, asks whether Strassen's support functionals are universal spectral points, that is, points in the asymptotic spectrum of tensors. In this paper, we answer this question in the affirmative for tensors over complex numbers by proving that the support functionals coincide with the quantum functionals - universal spectral points that are defined via entropy optimization on entanglement polytopes. We obtain this result as a special case of a general minimax formula for convex optimization on entanglement polytopes (and more general moment polytopes). Our formula can be interpreted as a very general "classical-quantum correspondence" analogous to the well-known relation between the Shannon entropy and the von Neumann entropy. Its proof is based on a recent Fenchel-type duality theorem on Hadamard manifolds due to Hirai. In addition to settling Strassen's question, our results yield a unified and simpler approach to several other tensor invariants, including the asymptotic slice rank, the symmetric quantum functional, the $G$-stable rank, and the non-commutative rank.

Computational Complexity (cs.CC); Optimization and Control (math.OC); Quantum Physics (quant-ph)

arXiv:2601.21553 [cs.CC]

(or arXiv:2601.21553v2 [cs.CC] for this version)

https://doi.org/10.48550/arXiv.2601.21553 Focus to learn more arXiv-issued DOI via DataCite

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Strassen's support functionals coincide with the quantum functionals

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