On the partially symmetric rank of tensor products of W-states and other symmetric tensors
Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
Standard
On the partially symmetric rank of tensor products of W-states and other symmetric tensors. / Ballico, Edoardo; Bernardi, Alessandra; Christandl, Matthias; Gesmundo, Fulvio.
I: Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni, Bind 30, Nr. 1, 2019, s. 93-124.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
Harvard
APA
Vancouver
Author
Bibtex
}
RIS
TY - JOUR
T1 - On the partially symmetric rank of tensor products of W-states and other symmetric tensors
AU - Ballico, Edoardo
AU - Bernardi, Alessandra
AU - Christandl, Matthias
AU - Gesmundo, Fulvio
PY - 2019
Y1 - 2019
N2 - 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).
AB - 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).
KW - Partially symmetric rank
KW - cactus rank
KW - tensor rank
KW - W-state
KW - entanglement
U2 - 10.4171/RLM/837
DO - 10.4171/RLM/837
M3 - Journal article
VL - 30
SP - 93
EP - 124
JO - Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni
JF - Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni
SN - 1120-6330
IS - 1
ER -
ID: 230842665