On the partially symmetric rank of tensor products of W-states and other symmetric tensors
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- ON THE PARTIALLY SYMMETRIC RANK OF TENSOR PRODUCTS OF
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Given tensors T and T′ of order k and k′ respectively, the tensor product T⊗T′ is a tensor of order k+k′.
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 xd−1y, and on products of such. In particular, we prove that the partially symmetric rank of xd1−1y⊗⋯⊗xdk−1y is at most 2k−1(d1+⋯+dk).
Originalsprog | Engelsk |
---|---|
Tidsskrift | Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni |
Vol/bind | 30 |
Udgave nummer | 1 |
Sider (fra-til) | 93-124 |
ISSN | 1120-6330 |
DOI | |
Status | Udgivet - 2019 |
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