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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Universal points in the asymptotic spectrum of tensors
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by Matthias Christandl, Péter Vrana and Jeroen Zuiddam HTML | PDF
J. Amer. Math. Soc. 36 (2023), 31-79 Request permission

Abstract:

Motivated by the problem of constructing fast matrix multiplication algorithms, Strassen (FOCS 1986, Crelle 1987–1991) introduced and developed the theory of asymptotic spectra of tensors. For any sub-semiring $\mathcal {X}$ of tensors (under direct sum and tensor product), the duality theorem that is at the core of this theory characterizes basic asymptotic properties of the elements of $\mathcal {X}$ in terms of the asymptotic spectrum of $\mathcal {X}$, which is defined as the collection of semiring homomorphisms from $\mathcal {X}$ to the non-negative reals with a natural monotonicity property. The asymptotic properties characterized by this duality encompass fundamental problems in complexity theory, combinatorics and quantum information.

Universal spectral points are elements in the asymptotic spectrum of the semiring of all tensors. Finding all universal spectral points suffices to find the asymptotic spectrum of any sub-semiring. The construction of non-trivial universal spectral points has been an open problem for more than thirty years. We construct, for the first time, a family of non-trivial universal spectral points over the complex numbers, called quantum functionals. We moreover prove that the quantum functionals precisely characterise the asymptotic slice rank of complex tensors. Our construction, which relies on techniques from quantum information theory and representation theory, connects the asymptotic spectrum of tensors to the quantum marginal problem and entanglement polytopes.

References
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Additional Information
  • Matthias Christandl
  • Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark
  • MR Author ID: 729711
  • Email: christandl@math.ku.dk
  • Péter Vrana
  • Affiliation: Department of Geometry, Budapest University of Technology and Economics, Egry József u. 1., 1111 Budapest, Hungary; and MTA-BME Lendület Quantum Information Theory Research Group, Müegyetem rkp 3., 1111 Budapest, Hungary
  • ORCID: 0000-0003-0770-0432
  • Email: vranap@math.bme.hu
  • Jeroen Zuiddam
  • Affiliation: Centrum Wiskunde & Informatica, Science Park 123, Amsterdam, Netherlands
  • Address at time of publication: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 105-107, 1098 XG Amsterdam, Netherlands
  • MR Author ID: 1206023
  • ORCID: 0000-0003-0651-6238
  • Email: j.zuiddam@uva.nl
  • Received by editor(s): October 14, 2018
  • Received by editor(s) in revised form: June 23, 2021, and August 28, 2021
  • Published electronically: November 23, 2021
  • Additional Notes: The authors were financially supported by the European Research Council (ERC Grant Agreements no. 337603 and 81876), the Danish Council for Independent Research (Sapere Aude), and VILLUM FONDEN via the QMATH Centre of Excellence (Grant no. 10059). The third author was supported by NWO (617.023.116), National Science Foundation (Grant no. CCF-1900460) and the Simons Society of Fellows
  • © Copyright 2021 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 36 (2023), 31-79
  • MSC (2020): Primary 15A69, 14L24, 68Q17
  • DOI: https://doi.org/10.1090/jams/996