Weighted slice rank and a minimax correspondence to Strassen's spectra
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 ranktype 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 DOI:
 10.48550/arXiv.2012.14412
 arXiv:
 arXiv:2012.14412
 Bibcode:
 2020arXiv201214412C
 Keywords:

 Computer Science  Computational Complexity;
 Mathematics  Representation Theory;
 Quantum Physics;
 15A69;
 15A72;
 14L24;
 49N15;
 68Q17
 EPrint:
 29pp, final version