The Best Rank-1 Approximation of a Symmetric Tensor and Related Spherical Optimization Problems

Zhang, Xinzhen; Ling, Chen; Qi, Liqun

doi:10.1137/110835335

Cited by 88 publications

(74 citation statements)

References 31 publications

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“…We have to show that every coefficient of each f i in the Bombieri-Weyl basis { d α X α } is N (0, 1)-distributed. Fix 1 ≤ j ≤ n, and suppose that (25) f j (X) =…”

Section: Gaussian Tensors and Gaussian Polynomial Systems Recall Thamentioning

confidence: 99%

The Expected Number of Eigenvalues of a Real Gaussian Tensor

Breiding¹

2017

SIAM J. Appl. Algebra Geometry

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“…We have to show that every coefficient of each f i in the Bombieri-Weyl basis { d α X α } is N (0, 1)-distributed. Fix 1 ≤ j ≤ n, and suppose that (25) f j (X) =…”

Section: Gaussian Tensors and Gaussian Polynomial Systems Recall Thamentioning

confidence: 99%

The Expected Number of Eigenvalues of a Real Gaussian Tensor

Breiding¹

2017

SIAM J. Appl. Algebra Geometry

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“…Mathematicians have always studied the best rank-one tensor approximation problems over the real field [19,47,46,42,48]. Assume that a d-order…”

Section: Best Real Rank-one Approximation and Z-eigenvaluesmentioning

confidence: 99%

Geometric Measure of Entanglement and U-Eigenvalues of Tensors

Bai

2014

SIAM J. Matrix Anal. & Appl.

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We study tensor analysis problems motivated by the geometric measure of quantum entanglement. We define the concept of the unitary eigenvalue (U-eigenvalue) of a complex tensor, the unitary symmetric eigenvalue (US-eigenvalue) of a symmetric complex tensor, and the best complex rank-one approximation. We obtain an upper bound on the number of distinct US-eigenvalues of symmetric tensors and count all US-eigenpairs with nonzero eigenvalues of symmetric tensors. We convert the geometric measure of the entanglement problem to an algebraic equation system problem. A numerical example shows that a symmetric real tensor may have a best complex rankone approximation that is better than its best real rank-one approximation, which implies that the absolute-value largest Z-eigenvalue is not always the geometric measure of entanglement.Key words. unitary eigenvalue (U-eigenvalue), Z-eigenvalue, symmetric real tensor, geometric measure of entanglement, the best rank-one approximation 1. Introduction. Entanglement of compound systems is a key resource in quantum information processing. In many practical applications it is of fundamental importance to know whether a state is entangled or not. However, this information is often not sufficient, and it is also required to know how much a state is entangled. A useful tool for quantifying the amount of entanglement of a state is given by the so-called entanglement measures [1, 2].A widely used entanglement measure is provided by the geometric measure of entanglement [3,4,5] that is defined for a pure state. The geometric measure of entanglement was first proposed by Shimony [6] and extended to multipartite systems by Wei and Goldbart [7]. It has applications in various different topics, including many body physics [4], local discrimination, quantum computation, condensed matter systems, entanglement witnesses, and the study of quantum channel capacities. The geometric measure of entanglement is nothing but the injective tensor norm itself [8], which appears in the theory of operator algebra [9] and has now become increasingly important in theoretical physics-particularly in quantum channel capacities [10,11,12,13,14,15,16,17,18].A tensor is a multidimensional array [19]. Tensor decompositions originated with Hitchcock in 1927 [20], and the idea of a multiway model is attributed to Cattell in 1944 [21]. Recently, interest in tensor decompositions has expanded to other fields. Examples include signal processing [22,23], numerical linear algebra [24,25], computer

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“…In addition, it is much easier to compute than a full CP decomposition [43] [28]. This problem may be seen to be related to tensor eigenvalues [17] [53] [59] [35] [88]. It has been proved recently that the best rank-1 approximation of a symmetric tensor is symmetric [34]; a shorter proof can be found in [35], as well as uniqueness issues.…”

Section: The Case Of Rank-one Approximatementioning

confidence: 99%