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

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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 tidsskriftTidsskriftartikelfagfællebedømt

Harvard

Ballico, E, Bernardi, A, Christandl, M & Gesmundo, F 2019, 'On the partially symmetric rank of tensor products of W-states and other symmetric tensors', Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni, bind 30, nr. 1, s. 93-124. https://doi.org/10.4171/RLM/837

APA

Ballico, E., Bernardi, A., Christandl, M., & Gesmundo, F. (2019). On the partially symmetric rank of tensor products of W-states and other symmetric tensors. Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni, 30(1), 93-124. https://doi.org/10.4171/RLM/837

Vancouver

Ballico E, Bernardi A, Christandl M, Gesmundo F. On the partially symmetric rank of tensor products of W-states and other symmetric tensors. Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni. 2019;30(1):93-124. https://doi.org/10.4171/RLM/837

Author

Ballico, Edoardo ; Bernardi, Alessandra ; Christandl, Matthias ; Gesmundo, Fulvio. / On the partially symmetric rank of tensor products of W-states and other symmetric tensors. I: Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni. 2019 ; Bind 30, Nr. 1. s. 93-124.

Bibtex

@article{65f7e6a3458442928612df099eb65d89,
title = "On the partially symmetric rank of tensor products of W-states and other symmetric tensors",
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).",
keywords = "Partially symmetric rank, cactus rank, tensor rank, W-state, entanglement",
author = "Edoardo Ballico and Alessandra Bernardi and Matthias Christandl and Fulvio Gesmundo",
year = "2019",
doi = "10.4171/RLM/837",
language = "English",
volume = "30",
pages = "93--124",
journal = "Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni",
issn = "1120-6330",
publisher = "European Mathematical Society Publishing House",
number = "1",

}

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