Universal points in the asymptotic spectrum of tensors

TL;DR

This paper introduces the first nontrivial universal spectral points for the asymptotic spectrum of all tensors, using quantum entropy and covariants, with implications for tensor complexity and related problems.

Contribution

The authors construct the first family of nontrivial universal spectral points for all tensors over the complex numbers, connecting quantum information theory with tensor asymptotics.

Findings

Constructed quantum functionals as universal spectral points.

Reproduced recent results on the cap set problem via asymptotic spectrum.

Characterized asymptotic slice rank using quantum functionals.

Abstract

The asymptotic restriction problem for tensors is to decide, given tensors $s$ and $t$, whether the nth tensor power of $s$ can be obtained from the $(n+o(n))$th tensor power of t by applying linear maps to the tensor legs (this we call restriction), when $n$ goes to infinity. In this context, Volker Strassen, striving to understand the complexity of matrix multiplication, introduced in 1986 the asymptotic spectrum of tensors. Essentially, the asymptotic restriction problem for a family of tensors $X$, closed under direct sum and tensor product, reduces to finding all maps from $X$ to the reals that are monotone under restriction, normalised on diagonal tensors, additive under direct sum and multiplicative under tensor product, which Strassen named spectral points. Strassen created the support functionals, which are spectral points for oblique tensors, a strict subfamily of all tensors. Universal spectral points are spectral points for the family of all tensors. The construction of nontrivial universal spectral points has been an open problem for more than thirty years. We construct for the first time a family of nontrivial universal spectral points over the complex numbers, using quantum entropy and covariants: the quantum functionals. In the process we connect the asymptotic spectrum to the quantum marginal problem and to the entanglement polytope. To demonstrate the asymptotic spectrum, we reprove (in hindsight) recent results on the cap set problem by reducing this problem to computing asymptotic spectrum of the reduced polynomial multiplication tensor, a prime example of Strassen. A better understanding of our universal spectral points construction may lead to further progress on related questions. We additionally show that the quantum functionals characterise asymptotic slice rank for complex tensors.