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The next gap in the subrank of 3-tensors

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arxiv 2307.06115 v2 pith:2O67GRCB submitted 2023-07-12 math.AG cs.CCmath.COquant-ph

Fulvio Gesmundo , Jeroen Zuiddam This is my paper · ORCID
classification math.AGcs.CCmath.COquant-ph
keywords asymptoticequalsubranktensorleastnextnonzeropossible
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Recent works of Costa-Dalai, Christandl-Gesmundo-Zuiddam, Blatter-Draisma-Rupniewski, and Bri\"et-Christandl-Leigh-Shpilka-Zuiddam have investigated notions of discreteness and gaps in the possible values that asymptotic tensor ranks can take. In particular, it was shown that the asymptotic subrank and asymptotic slice rank of any nonzero 3-tensor is equal to 1, equal to 1.88, or at least 2 (over any field), and that the set of possible values of these parameters is discrete (in several regimes). We determine exactly the next gap, showing that the asymptotic subrank and asymptotic slice rank of any nonzero 3-tensor is equal to 1, equal to 1.88, equal to 2, or at least 2.68.

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  1. Asymptotic tensor rank is characterized by polynomials

    cs.CC 2024-11 conditional novelty 8.0 of 10

    Sublevel sets of asymptotic tensor rank are Zariski-closed, making the parameter well-ordered in value, complete over the complex numbers, and computable from above.

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