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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 (thiswe call restriction), when $n$ goes to infinity. In this context, VolkerStrassen, striving to understand the complexity of matrix multiplication,introduced in 1986 the asymptotic spectrum of tensors. Essentially, theasymptotic restriction problem for a family of tensors $X$, closed under directsum and tensor product, reduces to finding all maps from $X$ to the reals thatare monotone under restriction, normalised on diagonal tensors, additive underdirect sum and multiplicative under tensor product, which Strassen namedspectral points. Strassen created the support functionals, which are spectralpoints 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 openproblem for more than thirty years. We construct for the first time a family ofnontrivial universal spectral points over the complex numbers, using quantumentropy and covariants: the quantum functionals. In the process we connect theasymptotic spectrum to the quantum marginal problem and to the entanglementpolytope.  To demonstrate the asymptotic spectrum, we reprove (in hindsight) recentresults on the cap set problem by reducing this problem to computing asymptoticspectrum of the reduced polynomial multiplication tensor, a prime example ofStrassen. A better understanding of our universal spectral points constructionmay lead to further progress on related questions. We additionally show thatthe quantum functionals characterise asymptotic slice rank for complex tensors.

Universal points in the asymptotic spectrum of tensors