Universal points in the asymptotic spectrum of tensors
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Universal points in the asymptotic spectrum of tensors. / Christandl, Matthias; Vrana, Péter; Zuiddam, Jeroen.
I: Journal of the American Mathematical Society, Bind 36, Nr. 1, 2023, s. 31-79.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Universal points in the asymptotic spectrum of tensors
AU - Christandl, Matthias
AU - Vrana, Péter
AU - Zuiddam, Jeroen
N1 - Publisher Copyright: © 2021 American Mathematical Society.
PY - 2023
Y1 - 2023
N2 - 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 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 in terms of the asymptotic spectrum of , which is defined as the collection of semiring homomorphisms from 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.
AB - 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 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 in terms of the asymptotic spectrum of , which is defined as the collection of semiring homomorphisms from 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.
UR - http://www.scopus.com/inward/record.url?scp=85137212034&partnerID=8YFLogxK
U2 - 10.1090/jams/996
DO - 10.1090/jams/996
M3 - Journal article
AN - SCOPUS:85137212034
VL - 36
SP - 31
EP - 79
JO - Journal of the American Mathematical Society
JF - Journal of the American Mathematical Society
SN - 0894-0347
IS - 1
ER -
ID: 326729069