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

Edoardo Ballico; Alessandra Bernardi; Matthias Christandl; Fulvio Gesmundo

doi:10.4171/RLM/837

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

Edoardo Ballico, Alessandra Bernardi, Matthias Christandl, Fulvio Gesmundo

Institut for Matematiske Fag

10 Citationer (Scopus)

Abstract

— Given tensors T and T ⁰ of order k and k⁰ respectively, the tensor product T n T ⁰ is a tensor of order k þ k⁰. 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 x^d1y, and on products of such. In particular, we prove that the partially symmetric rank of x^d11y n n x^dk1y is at most 2^k1ðd₁ þ þ d_kÞ.

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	https://doi.org/10.4171/RLM/837
Status	Udgivet - 2019

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10.4171/RLM/837

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On the partially symmetric rank of tensor products of W-states and other symmetric tensors. / Ballico, Edoardo; Bernardi, Alessandra; Christandl, Matthias et al.
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 › peer review

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title = "On the partially symmetric rank of tensor products of W-states and other symmetric tensors",

abstract = "— Given tensors T and T 0 of order k and k0 respectively, the tensor product T n T 0 is a tensor of order k {\th} k0. 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 xd1y, and on products of such. In particular, we prove that the partially symmetric rank of xd11y n n xdk1y is at most 2k1{\dh}d1 {\th} {\th} dk{\TH}.",

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",

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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 0 of order k and k0 respectively, the tensor product T n T 0 is a tensor of order k þ k0. 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 xd1y, and on products of such. In particular, we prove that the partially symmetric rank of xd11y n n xdk1y is at most 2k1ðd1 þ þ dkÞ.

AB - — Given tensors T and T 0 of order k and k0 respectively, the tensor product T n T 0 is a tensor of order k þ k0. 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 xd1y, and on products of such. In particular, we prove that the partially symmetric rank of xd11y n n xdk1y is at most 2k1ðd1 þ þ dkÞ.

KW - Partially symmetric rank

KW - cactus rank

KW - tensor rank

KW - W-state

KW - entanglement

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DO - 10.4171/RLM/837

M3 - Journal article

SN - 1120-6330

VL - 30

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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

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

Abstract

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