On the partially symmetric rank of tensor products of -states and other symmetric tensors

  • Edoardo Ballico

    Università di Trento, Italy
  • Alessandra Bernardi

    Università di Trento, Italy
  • Matthias Christandl

    University of Copenhagen, Denmark
  • Fulvio Gesmundo

    University of Copenhagen, Denmark
On the partially symmetric rank of tensor products of $W$-states and other symmetric tensors cover

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Abstract

Given tensors and of order and respectively, the tensor product is a tensor of order . It was recently shown that the tensor rank can be strictly submultiplicative under this operation ([Christandl–Jensen–Zuiddam]). We study this phenomenon for symmetric tensors where additional techniques from algebraic geometry are available. The tensor product of symmetric tensors results in a partially symmetric tensor and our results amount to bounds on the partially symmetric rank. Following motivations from algebraic complexity theory and quantum information theory, we focus on the so-called W-states, namely monomials of the form , and on products of such. In particular, we prove that the partially symmetric rank of is at most .

Cite this article

Edoardo Ballico, Alessandra Bernardi, Matthias Christandl, Fulvio Gesmundo, On the partially symmetric rank of tensor products of -states and other symmetric tensors. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 30 (2019), no. 1, pp. 93–124

DOI 10.4171/RLM/837