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Spectral norm of a symmetric tensor and its computation

Authors: Shmuel Friedland and Li Wang
Journal: Math. Comp. 89 (2020), 2175-2215
MSC (2010): Primary 13P15, 15A69, 65H04, 81P40
Published electronically: May 15, 2020
MathSciNet review: 4109564
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Abstract: We show that the spectral norm of a $ d$-mode real or complex symmetric tensor in $ n$ variables can be computed by finding the fixed points of the corresponding polynomial map. For a generic complex symmetric tensor the number of fixed points is finite, and we give upper and lower bounds for the number of fixed points. For $ n=2$ we show that these fixed points are the roots of a corresponding univariate polynomial of degree at most $ (d-1)^2+1$, except certain cases, which are completely analyzed. In particular, for $ n=2$ the spectral norm of $ d$-symmetric tensor is polynomially computable in $ d$ with a given relative precision. For a fixed $ n>2$ we show that the spectral norm of a $ d$-mode symmetric tensor is polynomially computable in $ d$ with a given relative precision with respect to the Hilbert-Schmidt norm of the tensor. These results show that the geometric measure of entanglement of $ d$-mode symmetric qunits on $ \mathbb{C}^n$ are polynomially computable for a fixed $ n$.

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Additional Information

Shmuel Friedland
Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois, 851 South Morgan Street, Chicago, Illinois 60607-7045

Li Wang
Affiliation: Department of Mathematics, University of Texas at Arlington, 411 S. Nedderman Drive, 478 Pickard Hall, Arlington, Texas 76019-0408

Keywords: Symmetric tensors, homogeneous polynomials, spectral norm, anti-fixed and fixed points, computation of spectral norm, $d$-mode symmetric qubits, $d$-mode symmetric qunits on $\mathbb{C}^n$, geometric measure of entanglement.
Received by editor(s): August 15, 2018
Received by editor(s) in revised form: August 17, 2018, February 6, 2019, August 1, 2019, and January 12, 2020
Published electronically: May 15, 2020
Article copyright: © Copyright 2020 American Mathematical Society