On decompositions and approximations of conjugate partial-symmetric complex tensors

Fu, Taoran; Jiang, Bo; Li, Zhening

doi:10.48550/arxiv.1802.09013

Cited by 4 publications

(3 citation statements)

References 27 publications

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“…Lots of study have been conducted regarding properties of tensors such as tensor eigenvalues [4,5,6], the best rank-one approximation [7,8,9], tensor rank [3,10], tensor and symmetric tensor decomposition [11,12,13], symmetric tensor [14,15], nonnegative tensor [16], copositive tensors [17], and completely positive tensor [18,19,20]. Many tensor computation methods are also proposed including tensor eigenvalue computation [21,22,23,24,25,26], tensor system solution [27,28,29,30,31,32,33], and tensor decomposition [34].…”

Section: Introductionmentioning

confidence: 99%

Hermitian tensor and quantum mixed state

Ni¹

2019

Preprint

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An order 2m complex tensor H is said to be Hermitian ifIt can be regarded as an extension of Hermitian matrix to higher order. A Hermitian tensor is also seen as a representation of a quantum mixed state. Motivated by the separability discrimination of quantum states, we investigate properties of Hermitian tensors including: unitary similarity relation, partial traces, nonnegative Hermitian tensors, Hermitian eigenvalues, rank-one Hermitian decomposition and positive Hermitian decomposition, and their applications to quantum states.

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Section: Introductionmentioning

confidence: 99%

Hermitian tensor and quantum mixed state

Ni¹

2019

Preprint

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“…An adaptive gradient method for computing generalized tensor eigenpairs has been developed in [14]. Fu et al [15] derived new algorithms to compute best rank-one approximation of conjugate partial-symmetric (CPS) tensors by unfolding CPS tensors to Hermitian matrices. Che et al [16] proposed a neural networks method for computing the generalized eigenvalues of complex tensors, and Che et al [17] also derived an iterative algorithm for computing US-eigenpairs of complex symmetric tensors and U-eigenpairs of complex tensors based on the Takagi factorization of complex symmetric matrices.…”

Section: Introductionmentioning

confidence: 99%

Iterative methods for computing U-eigenvalues of non-symmetric complex tensors with application in quantum entanglement

Zhang

2019

Comput Optim Appl

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The purpose of this paper is to study the problem of computing unitary eigenvalues (U-eigenvalues) of non-symmetric complex tensors. By means of symmetric embedding of complex tensors, the relationship between U-eigenpairs of a non-symmetric complex tensor and unitary symmetric eigenpairs (US-eigenpairs) of its symmetric embedding tensor is established. An algorithm (Algorithm 3.1) is given to compute the U-eigenvalues of nonsymmetric complex tensors by means of symmetric embedding. Another algorithm, Algorithm 3.2, is proposed to directly compute the U-eigenvalues of non-symmetric complex tensors, without the aid of symmetric embedding. Finally, a tensor version of the well-known Gauss-Seidel method is developed. Efficiency of these three algorithms are compared by means of various numerical examples. These algorithms are applied to compute the geometric measure of entanglement of quantum multipartite non-symmetric pure states.Keywords complex tensor · unitary eigenvalue · iterative method · geometric measure of entanglement Mathematics Subject Classification (2000) 15A18 · 15A69 · 81P40

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“…We refer to [9,12,32,36] for the work on symmetric tensor decompositions. Symmetric tensors can be generalized to partial symmetric tensors [24] and conjugate partial symmetric tensors [19]. A class of interesting symmetric tensors are Hankel tensors [35].…”

Section: Introductionmentioning

confidence: 99%

Hankel Tensor Decompositions and Ranks

Nie¹,

Ye²

2019

SIAM J. Matrix Anal. Appl.

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Hermitian tensors are generalizations of Hermitian matrices, but they have very different properties. Every complex Hermitian tensor is a sum of complex Hermitian rank-1 tensors. However, this is not true for the real case. We study basic properties for Hermitian tensors such as Hermitian decompositions and Hermitian ranks. For canonical basis tensors, we determine their Hermitian ranks and decompositions. For real Hermitian tensors, we give a full characterization for them to have Hermitian decompositions over the real field. In addition to traditional flattening, Hermitian tensors specially have Hermitian and Kronecker flattenings, which may give different lower bounds for Hermitian ranks. We also study other topics such as eigenvalues, positive semidefiniteness, sum of squares representations, and separability.2010 Mathematics Subject Classification. 15A69,15B48,65F99.

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On decompositions and approximations of conjugate partial-symmetric complex tensors

Cited by 4 publications

References 27 publications

Hermitian tensor and quantum mixed state

Hermitian tensor and quantum mixed state

Iterative methods for computing U-eigenvalues of non-symmetric complex tensors with application in quantum entanglement

Hankel Tensor Decompositions and Ranks

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