Universal points in the asymptotic spectrum of tensors
The asymptotic restriction problem for tensors s and t is to find the smallest ≥ 0 such that the nth tensor power of t can be obtained from the (n + o(n))th tensor power of s by applying linear maps to the tensor legs — this is called restriction — when n goes to infinity. Applications include computing the arithmetic complexity of matrix multiplication in algebraic complexity theory, deciding the feasibility of an asymptotic transformation between pure quantum States via stochastic local operations and classical communication in quantum information theory, bounding the query complexity of certain properties in algebraic property testing, and bounding the size of combinatorial structures like tri-colored sum-free sets in additive combinatorics. Naturally, the asymptotic restriction problem asks for obstructions (think of lower bounds in computational complexity) and constructions (think of fast matrix multiplication algorithms). Strassen showed that for obstructions it is sufficient to consider maps from k-tensors to nonnegative reals, that are monotone under restriction, normalised on diagonal tensors, additive under direct sum and multiplicative under tensor product, named spectral points (SFCS 1986 and J. Reine Angew. Math. 1988). Strassen introduced the support functionals, which are spectral points for oblique tensors, a strict subfamily of all tensors (J. Reine Angew. Math. 1991). On the construction side, an important work is the Coppersmith–Winograd method for tight tensors and tight sets. We present the first nontrivial spectral points for the family of all complex tensors, named quantum functionals. Finding such universal spectral points has been an open problem for thirty years. We use techniques from quantum information theory, invariant theory and moment polytopes. We present comparisons among the support functionals and our quantum functionals, and compute generic values. We relate the functionals to instability from geometric invariant theory, in the spirit of Blasiak et al. (Discrete Anal. 2017). We prove that the quantum functionals are asymptotic upper bounds on slice-rank and multi-slice rank, extending a result of Tao and Sawin.
|Keywords||Asymptotic restriction, Asymptotic spectrum, Cap set problem, Classical communication (slocc), Entanglement monotones, Fast matrix multiplication, Moment polytope, Quantum entropy, Reduced polynomial multiplication, Stochastic local operations, Tensors|
|Conference||Annual ACM SIGACT Symposium on Theory of Computing|
Christandl, M, Vrana, P, & Zuiddam, J. (2018). Universal points in the asymptotic spectrum of tensors. In Proceedings of the Annual ACM Symposium on Theory of Computing (pp. 289–296). doi:10.1145/3188745.3188766