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

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 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

U2 - 10.4171/RLM/837

DO - 10.4171/RLM/837

M3 - Journal article

SN - 1120-6330

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

IS - 1

ER -