On the partially symmetric rank of tensor products of -states and other symmetric tensors
Edoardo Ballico
Università di Trento, ItalyAlessandra Bernardi
Università di Trento, ItalyMatthias Christandl
University of Copenhagen, DenmarkFulvio Gesmundo
University of Copenhagen, Denmark
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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