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Discreteness of Asymptotic Tensor Ranks (Extended Abstract)

Authors Jop Briët , Matthias Christandl , Itai Leigh , Amir Shpilka , Jeroen Zuiddam

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Author Details

Jop Briët
  • Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
Matthias Christandl
  • University of Copenhagen, Denmark
Itai Leigh
  • Tel Aviv University, Israel
Amir Shpilka
  • Tel Aviv University, Israel
Jeroen Zuiddam
  • University of Amsterdam, The Netherlands

Funding

  • Briët, Jop: Dutch Research Council (NWO) as part of the NETWORKS programme (Grant No. 024.002.003).
  • Christandl, Matthias: European Research Council (ERC Grant Agreement No. 81876), VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059) and the Novo Nordisk Foundation (Grant NNF20OC0059939 "Quantum for Life").
  • Leigh, Itai: Erasmus+ International Credit Mobility grant, Israel Science Foundation (Grant No. 514/20), the Len Blavatnik and the Blavatnik Family foundation, and the Deutsch foundation.
  • Shpilka, Amir: Israel Science Foundation (Grant No. 514/20) and the Len Blavatnik and the Blavatnik Family foundation.
  • Zuiddam, Jeroen: Dutch Research Council (NWO) Veni grant VI.Veni.212.284.

Acknowledgements

We thank the Centre de Recherches Mathématiques Montréal, where this work was initiated. We also thank Jurij Volcic and Vladimir Lysikov for helpful and stimulating discussions. MC Thanks the National Center for Competence in Research SwissMAP of the Swiss National Science Foundation and the Section of Mathematics at the University of Geneva for their hospitality. IL thanks the QMATH Center at the University of Copenhagen for the hospitality during his stay.

Cite AsGet BibTex

Jop Briët, Matthias Christandl, Itai Leigh, Amir Shpilka, and Jeroen Zuiddam. Discreteness of Asymptotic Tensor Ranks (Extended Abstract). In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.20

Abstract

Tensor parameters that are amortized or regularized over large tensor powers, often called "asymptotic" tensor parameters, play a central role in several areas including algebraic complexity theory (constructing fast matrix multiplication algorithms), quantum information (entanglement cost and distillable entanglement), and additive combinatorics (bounds on cap sets, sunflower-free sets, etc.). Examples are the asymptotic tensor rank, asymptotic slice rank and asymptotic subrank. Recent works (Costa-Dalai, Blatter-Draisma-Rupniewski, Christandl-Gesmundo-Zuiddam) have investigated notions of discreteness (no accumulation points) or "gaps" in the values of such tensor parameters. We prove a general discreteness theorem for asymptotic tensor parameters of order-three tensors and use this to prove that (1) over any finite field (and in fact any finite set of coefficients in any field), the asymptotic subrank and the asymptotic slice rank have no accumulation points, and (2) over the complex numbers, the asymptotic slice rank has no accumulation points. Central to our approach are two new general lower bounds on the asymptotic subrank of tensors, which measures how much a tensor can be diagonalized. The first lower bound says that the asymptotic subrank of any concise three-tensor is at least the cube-root of the smallest dimension. The second lower bound says that any concise three-tensor that is "narrow enough" (has one dimension much smaller than the other two) has maximal asymptotic subrank. Our proofs rely on new lower bounds on the maximum rank in matrix subspaces that are obtained by slicing a three-tensor in the three different directions. We prove that for any concise tensor, the product of any two such maximum ranks must be large, and as a consequence there are always two distinct directions with large max-rank.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Quantum information theory
  • Mathematics of computing → Combinatoric problems
Keywords
  • Tensors
  • Asymptotic rank
  • Subrank
  • Slice rank
  • Restriction
  • Degeneration
  • Diagonalization
  • SLOCC

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