ON-LINE QLunch: Vladimir Lysikov

Structural and computational understanding of tensors and restrictions between tensors is the driving force behind faster matrix multiplication algorithms, the unraveling of quantum entanglement, and the breakthrough on the cap set problem in combinatorics.

I will describe new connection between two recent developments in the study of tensors: the slice rank of tensors and the quantum functionals.

The slice rank of tensors is an important notion of rank for tensors that emerged from the resolution of the cap set problem.

The quantum functionals are monotone functionals defined for tensors over complex numbers as optimizations over moment polytopes.

The quantum functionals are important in the context of Strassen's asymptotic spectra program, which characterizes optimal restrictions between tensors through monotone functionals.

The correspondence allows to give a rank-type characterization of the quantum functionals, which we conjecture to hold over arbitrary fields.

The finite field case is crucial for combinatorial and algorithmic problems where the field can be optimized over.

This is a joint work with Matthias Christandl and Jeroen Zuiddam (Courant Institute, New York University)