Motivated by problems in algebraic complexity theory (fast matrix multiplication) and combinatorics (cap sets), we study the asymptotic rank and the asymptotic subrank of tensors. Volker Strassen in 1988 gave a dual description of these parameters in terms of all maps from tensors to the reals that are multiplicative under tensor product, additive under direct sum and monotone under local linear maps.
(1) We explicitly construct infinitely many such maps for complex tensors via entanglement polytopes (moment polytopes), which we call the quantum functionals.
(2) We prove that the minimum over the quantum functionals equals the asymptotic slice-rank, a parameter that was introduced by Terence Tao.
This is joint work with Matthias Christandl and Péter Vrana.