Strassen's support functionals coincide with the quantum functionals

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.

💡 Research Summary

The paper resolves a long‑standing open problem dating back to Strassen’s 1991 work: whether Strassen’s support functionals are universal spectral points in the asymptotic spectrum of tensors. Strassen introduced the support functional ζ θ(t) for a probability distribution θ∈Θ, defined as a minimisation over GL‑transformations of the support polytope Ω(g·t) followed by a weighted Shannon‑entropy of its marginal distributions. Independently, Christandl, Vrana and Zuiddam (2018) defined the quantum functional F θ(t) as a maximisation of the same weighted entropy over the entanglement (or moment) polytope Δ(t), which is invariant under GL. While it was known that F θ(t) ≤ ζ θ(t) and that equality holds for the non‑generic class of free (or oblique) tensors, the general relationship remained unknown.

The authors prove that for every tensor t and every θ, the two quantities coincide:  F θ(t) = ζ θ(t). Consequently, the support functionals are indeed universal spectral points, providing a new, direct proof that the quantum functionals are universal as well.

The core of the proof is a general minimax theorem (Theorem 1.3) for convex optimisation over moment polytopes. For any convex, symmetric, lower‑semicontinuous function F on ℝ^{n₁}×⋯×ℝ^{n_d}, they show  min_{p∈Δ(t)} F(p) = max_{g∈GL} min_{p∈Ω(g·t)} F(p). Specialising F(p)=−∑i θ_i H(p_i) yields the equality of the support and quantum functionals. This theorem is derived from a powerful Fenchel‑type duality result on Hadamard manifolds due to Hirai (2025). The authors work on the product manifold PD = PD(n₁)×⋯×PD(n_d) of positive‑definite Hermitian matrices, equipped with the affine‑invariant metric. They consider the Kempf–Ness function f_t(x)=log⟨t, x·t⟩, which is geodesically convex, and whose differential at the identity maps onto the moment polytope Δ(t). Hirai’s theorem provides strong duality between the primal problem inf{x∈PD} Q(df(x)) and a dual problem involving the convex conjugate of Q. By choosing Q to encode the symmetric function F, the dual problem becomes exactly the maximisation over GL of the support‑polytope minimisation, establishing the minimax identity.

Beyond the main equivalence, the paper derives several notable corollaries. The asymptotic slice rank f_SR(t)=lim_{n→∞}SR(t^{⊗n})^{1/n} can be expressed as  f_SR(t)=min_{θ∈Θ} F θ(t), while the asymptotic vertex‑cover number ˜τ(H_t) of the d‑uniform hypergraph associated with t satisfies  ˜τ(H_t)=min_{θ∈Θ} ζ θ(t). Combining these with the equality of the functionals yields a new formula:  f_SR(t)=min_{g∈GL} ˜τ(H_{g·t}), extending a result previously known only for free tensors. This links a purely algebraic invariant (slice rank) with a combinatorial hypergraph parameter (vertex cover), illustrating the breadth of the minimax framework.

The authors also discuss extensions to other tensor invariants such as non‑commutative rank, G‑stable rank, and the symmetric quantum functional, all of which fit into the same convex‑analytic paradigm. They note that if the moment polytope can be accessed (via vertex enumeration, half‑space description, or membership oracles), the minimisation of any convex F can be performed with standard convex‑optimization algorithms. Moreover, Hirai’s Q‑gradient flow provides a natural continuous‑time algorithm for approximating the optimum.

In summary, the paper leverages recent advances in convex analysis on non‑positively curved manifolds to bridge two previously distinct tensor invariants. By establishing a universal minimax identity, it not only settles Strassen’s 35‑year‑old question but also opens a systematic pathway for constructing new universal spectral points and for computing a wide range of tensor parameters through convex‑optimization techniques.