Strassen's support functionals coincide with the quantum functionals

Technical Summary: Strassen's Support Functionals Coincide with Quantum Functionals

Problem Statement
The paper addresses a fundamental open problem in the theory of the asymptotic spectrum of tensors, originally posed by Strassen in 1991. Strassen's framework characterizes the asymptotic restriction between tensors (a relation capturing the complexity of multilinear problems like matrix multiplication) via "spectral points"—functionals that are monotone, multiplicative, additive, and normalized. While Strassen introduced a class of candidate spectral points known as support functionals ($\zeta_\theta$), defined via entropy optimization over the support polytope of a tensor, it remained unknown whether these functionals are universal spectral points (i.e., valid for the family of all $d$-tensors).

Concurrently, Christandl, Vrana, and Zuiddam (2017) constructed a family of universal spectral points called quantum functionals ($F_\theta$), defined via entropy optimization over the entanglement polytope (or moment polytope). Although it was known that $F_\theta(t) \leq \zeta_\theta(t)$, the precise relationship between these two quantities was unresolved. The central question is whether Strassen's support functionals coincide with the quantum functionals, thereby establishing them as universal spectral points.

Methodology
The authors resolve this question by proving a general minimax formula for convex optimization on moment polytopes. The proof strategy relies on the following components:

1. Hadamard Manifold Duality: The core technical tool is a recent Fenchel-type duality theorem for convex optimization on Hadamard manifolds (complete, simply-connected Riemannian manifolds with non-positive curvature) due to Hirai [Hir25]. The authors apply this to the manifold $PD = PD(n_1) \times \dots \times PD(n_d)$, the product of spaces of positive definite Hermitian matrices equipped with the affine-invariant Riemannian metric.
2. Kempf–Ness Function: They utilize the Kempf–Ness function, $f_v(x) = \log \langle v, x \cdot v \rangle$, associated with a tensor $v$. The differential of this function at the identity corresponds to the moment map, and its image determines the moment polytope $\Delta(v)$.
3. Gradient Optimization Minimax: The authors establish a general minimax theorem (Theorem 1.3/4.4) stating that for any convex, symmetric, lower semi-continuous function $F$, the minimum of $F$ over the moment polytope $\Delta(t)$ is equal to the maximum over all basis changes $g \in GL$ of the minimum of $F$ over the support polytope $\Omega(g \cdot t)$:
   $$\min_{p \in \Delta(t)} F(p) = \max_{g \in GL} \min_{p \in \Omega(g \cdot t)} F(p)$$
4. Specialization: By choosing $F(p) = -\sum \theta_i H(p_i)$ (where $H$ is Shannon entropy), the left-hand side becomes the quantum functional $F_\theta(t)$, and the right-hand side becomes the support functional $\zeta_\theta(t)$.

Key Contributions and Results

• Equality of Functionals (Theorem 1.1): The primary result is the proof that for every tensor $t$ and every probability distribution $\theta$, the quantum functional and Strassen's support functional are equal:
  $$F_\theta(t) = \zeta_\theta(t)$$
  This confirms that Strassen's support functionals are indeed universal spectral points.
• Simplified Proof of Universality: The result provides a new, direct proof that quantum functionals are universal spectral points. It bypasses the complex invariant-theoretic machinery used in the original proof by Christandl et al., replacing it with elementary convex-analytic methods derived from the minimax formula.
• Connection to Asymptotic Slice Rank (Corollary 1.2): The paper establishes a direct link between the asymptotic slice rank of tensors and the asymptotic vertex cover number of hypergraphs. Specifically, the asymptotic slice rank $\widetilde{SR}(t)$ can be computed as the minimum asymptotic vertex cover number over all basis changes:
  $$\widetilde{SR}(t) = \min_{g \in GL} \widetilde{\tau}(H_{g \cdot t})$$
  This generalizes previous results known only for free tensors.
• General Minimax Framework (Theorem 1.3 & 1.4): The authors provide a general minimax formula applicable to any convex, symmetric, lower semi-continuous function on moment polytopes associated with rational representations of $GL$. This framework unifies various tensor parameters, including the non-commutative rank, G-stable rank, and symmetric quantum functionals.
• Symmetric Quantum Functionals (Section 5.1): The paper extends the minimax description to symmetric quantum functionals (defined for the diagonal action of $GL(n)$), providing a new direct proof of their sub-multiplicativity.

Significance
The paper claims significance in several areas:

1. Resolution of a Long-Standing Open Problem: It settles the 35-year-old question regarding the universality of Strassen's support functionals.
2. Unification of Concepts: It reveals a deep structural connection between the support polytope (a combinatorial object related to the tensor's non-zero entries) and the entanglement/moment polytope (a geometric object related to quantum marginals), showing they yield identical values for entropy-based optimization.
3. Methodological Advancement: By leveraging Hirai's duality on Hadamard manifolds, the paper offers a powerful new tool for analyzing tensor invariants, potentially simplifying future proofs in algebraic complexity and quantum information theory.
4. Computational Implications: The connection to the asymptotic vertex cover number offers a new perspective on computing the asymptotic slice rank, linking algebraic complexity to hypergraph combinatorics.

The authors note that while the results are stated for $GL$ actions, the proofs are general and may extend to other groups if Hirai's conjecture on strong duality for all Hadamard manifolds holds true. They also highlight that the minimax formula could serve as a foundation for constructing new spectral points or developing efficient algorithms for computing tensor invariants.