Weighted slice rank and a minimax correspondence to Strassen's spectra

TL;DR

This paper introduces weighted slice rank and establishes a minimax correspondence with quantum functionals, connecting tensor rank notions to Strassen's spectra and extending their applicability across different fields.

Contribution

It develops a novel weighted slice rank and links it to quantum functionals through a minimax theorem, broadening the understanding of tensor ranks and Strassen's spectra.

Findings

Weighted slice rank generalizes tensor rank notions.

A minimax correspondence links weighted slice rank to quantum functionals.

Quantum functionals are extended to all fields, including finite fields.

Abstract

Structural and computational understanding of tensors is the driving force behind faster matrix multiplication algorithms, the unraveling of quantum entanglement, and the breakthrough on the cap set problem. Strassen's asymptotic spectra program (FOCS 1986) characterizes optimal matrix multiplication algorithms through monotone functionals. Our work advances and makes novel connections among two recent developments in the study of tensors, namely - the slice rank of tensors, a notion of rank for tensors that emerged from the resolution of the cap set problem (Ann. of Math. 2017), - and the quantum functionals of tensors (STOC 2018), monotone functionals defined as optimizations over moment polytopes. More precisely, we introduce an extension of slice rank that we call weighted slice rank and we develop a minimax correspondence between the asymptotic weighted slice rank and the quantum functionals. Weighted slice rank encapsulates different notions of bipartiteness of quantum entanglement. The correspondence allows us to give a rank-type characterization of the quantum functionals. Moreover, whereas the original definition of the quantum functionals only works over the complex numbers, this new characterization can be extended to all fields. Thereby, in addition to gaining deeper understanding of Strassen's theory for the complex numbers, we obtain a proposal for quantum functionals over other fields. The finite field case is crucial for combinatorial and algorithmic problems where the field can be optimized over.