TY - JOUR

T1 - Weighted slice rank and a minimax correspondence to Strassen's spectra

AU - Christandl, Matthias

AU - Lysikov, Vladimir

AU - Zuiddam, Jeroen

N1 - Publisher Copyright: © 2023 The Author(s)

PY - 2023

Y1 - 2023

N2 - 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. 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.

AB - 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. 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.

KW - Asymptotic spectrum

KW - Moment polytopes

KW - Slice rank

KW - Tensors

UR - http://www.scopus.com/inward/record.url?scp=85149769520&partnerID=8YFLogxK

U2 - 10.1016/j.matpur.2023.02.006

DO - 10.1016/j.matpur.2023.02.006

M3 - Journal article

AN - SCOPUS:85149769520

VL - 172

SP - 299

EP - 329

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

SN - 0021-7824

ER -