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Strassen's support functionals coincide with the quantum functionals

Keiya Sakabe, Mahmut Levent Doğan, Michael Walter

TL;DR

The paper resolves the long-standing question of whether Strassen's upper support functionals are universal spectral points by proving $F_{\theta}(t)=\zeta^{\theta}(t)$ for all tensors $t$ and all $\theta\in\Theta$, thereby unifying the quantum and Strassen formulations. The authors establish a general minimax formula for convex optimization on moment polytopes (and moment polytopes associated with Hadamard manifolds), leveraging Hirai's Fenchel-type duality to connect gradient optimization on $\mathrm{PD}(n_1)\times\cdots\times\mathrm{PD}(n_d)$ with maximization over GL-orbits and minimization over their associated polytopes. This framework immediately yields new, direct proofs of universality for the quantum functionals and provides a broad toolkit to derive and relate a range of tensor invariants, including the symmetric quantum functional, asymptotic slice rank, G-stable rank, and non-commutative rank, via a common convex-analytic lens. The results have both conceptual significance—clarifying the structure of the asymptotic spectrum—and practical implications for computing and applying tensor invariants in complexity theory and quantum information, with entropic scaling algorithms suggested for efficient computation of $F_{\theta}(t)$.

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 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 other moment polytopes) that has further applications to other tensor parameters, including the asymptotic slice rank. Our proof is based on a recent Fenchel-type duality theorem on Hadamard manifolds due to Hirai.

Strassen's support functionals coincide with the quantum functionals

TL;DR

The paper resolves the long-standing question of whether Strassen's upper support functionals are universal spectral points by proving for all tensors and all , thereby unifying the quantum and Strassen formulations. The authors establish a general minimax formula for convex optimization on moment polytopes (and moment polytopes associated with Hadamard manifolds), leveraging Hirai's Fenchel-type duality to connect gradient optimization on with maximization over GL-orbits and minimization over their associated polytopes. This framework immediately yields new, direct proofs of universality for the quantum functionals and provides a broad toolkit to derive and relate a range of tensor invariants, including the symmetric quantum functional, asymptotic slice rank, G-stable rank, and non-commutative rank, via a common convex-analytic lens. The results have both conceptual significance—clarifying the structure of the asymptotic spectrum—and practical implications for computing and applying tensor invariants in complexity theory and quantum information, with entropic scaling algorithms suggested for efficient computation of .

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 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 other moment polytopes) that has further applications to other tensor parameters, including the asymptotic slice rank. Our proof is based on a recent Fenchel-type duality theorem on Hadamard manifolds due to Hirai.
Paper Structure (16 sections, 23 theorems, 102 equations)

This paper contains 16 sections, 23 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Outlook
  4. Preliminaries
  5. Linear algebra
  6. Hadamard manifolds
  7. Euclidean space:
  8. Positive-definite matrices with the affine-invariant Riemannian metric:
  9. Convex functions and Legendre--Fenchel conjugate
  10. A minimax formula for gradient optimization
  11. A minimax formula for convex optimization on moment polytopes
  12. Applications and connections
  13. Symmetric quantum functionals
  14. Asymptotic slice rank
  15. G-stable rank
  16. ...and 1 more sections

Key Result

Theorem 1.1

For every tensor $t$ and every $\theta\in\Theta$, $F_\theta(t)=\zeta^\theta(t)$. In particular, Strassen's support functional is a universal spectral point in the asymptotic spectrum of tensors.

Theorems & Definitions (47)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3: Minimax formula for moment polytope optimization
  • Theorem 1.4: Minimax formula for gradient optimization
  • Proposition 3.1: Davis1957Lewis1996
  • Theorem 3.2: Asymptotic duality Hirai2025, $\mathop{\mathrm{PD}}\nolimits$ version
  • Proposition 3.3: Asymptotic duality, Euclidean version
  • Theorem 3.4: Detailed version of \ref{['thm:general-theorem-intro']}
  • proof
  • Remark 3.5
  • ...and 37 more