Abstract
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).
Original language | English |
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Journal | Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni |
Volume | 30 |
Issue number | 1 |
Pages (from-to) | 93-124 |
ISSN | 1120-6330 |
DOIs | |
Publication status | Published - 2019 |
Keywords
- Partially symmetric rank
- cactus rank
- tensor rank
- W-state
- entanglement