RESEARCH ARTICLE

Symmetric Hermitian decomposability criterion, decomposition, and its applications

  • Guyan NI ,
  • Bo YANG
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  • Department of Mathematics, National University of Defense Technology, Changsha 410073, China

Received date: 16 Jan 2021

Accepted date: 24 Mar 2021

Published date: 15 Oct 2022

Copyright

2022 Higher Education Press

Abstract

The Hermitian tensor is an extension of Hermitian matrices and plays an important role in quantum information research. It is known that every symmetric tensor has a symmetric CP-decomposition. However, symmetric Hermitian tensor is not the case. In this paper, we obtain a necessary and sufficient condition for symmetric Hermitian decomposability of symmetric Hermitian tensors. When a symmetric Hermitian decomposable tensor space is regarded as a linear space over the real number field, we also obtain its dimension formula and basis. Moreover, if the tensor is symmetric Hermitian decomposable, then the symmetric Hermitian decomposition can be obtained by using the symmetric Hermitian basis. In the application of quantum information, the symmetric Hermitian decomposability condition can be used to determine the symmetry separability of symmetric quantum mixed states.

Cite this article

Guyan NI , Bo YANG . Symmetric Hermitian decomposability criterion, decomposition, and its applications[J]. Frontiers of Mathematics in China, 2022 , 17(5) : 961 -986 . DOI: 10.1007/s11464-021-0927-4

1
Bohnet-Waldraff F, Braun D, Giraud O. Tensor eigenvalues and entanglement of symmetric states. Phys Rev A, 2016, 94: 042324

DOI

2
Bohnet-Waldraff F, Braun D, Giraud O. Entanglement and the truncated moment problem. Phys Rev A, 2017, 96: 032312

DOI

3
Brachat J, Comon P, Mourrain B, Tsigaridas E. Symmetric tensor decomposition. Linear Algebra Appl, 2010, 433: 1851–1872

DOI

4
Breiding P, Vannieuwenhoven N. A Riemannian trust region method for the canonical tensor rank approximation problem. SIAM J Optim, 2018, 28: 2435–2465

DOI

5
Chang K-C, Pearson K, Zhang T. On eigenvalue problems of real symmetric tensors. J Math Anal Appl, 2009, 350: 416–422

DOI

6
Chiantini L, Ottaviani G, Vannieuwenhoven N. Effective criteria for specific identifiability of tensors and forms. SIAM J Matrix Anal Appl, 2017, 38: 656–681

DOI

7
Comon P, Golub G, Lim L -H, Mourrain B. Symmetric tensors and symmetric tensor rank. SIAM J Matrix Anal Appl, 2008, 30: 1254–1279

DOI

8
Comon P, Lim L-H, Qi Y, Ye K. Topology of tensor ranks. Adv Math, 2020, 367: 107–128

DOI

9
Derksen H, Friedland S, Lim L-H, Wang L. Theoretical and computational aspects of entanglement. arXiv: 1705.07160

10
Domanov I, Lathauwer L De.. Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL. SIAM J Matrix Anal Appl, 2015, 36: 1567–1589

DOI

11
Dressler M, Nie J, Yang Z. Separability of Hermitian tensors and PSD decompositions. Linear Algebra Appl, doi.org/10.1080/03081087.2021.1965078

12
Fu T, Jiang B, Li Z. On decompositions and approximations of conjugate partial-symmetric complex tensors. arXiv: 1802.09013

13
Galuppi F, Mella M. Identifiability of homogeneous polynomials and Cremona transformations. J Reine Angew Math, 2019, 757: 279–308

DOI

14
Giraud O, Braun D, Baguette D, Bastin T, Martin J. Tensor representation of spin states. Phys Rev Lett, 2015, 114: 080401

DOI

15
Horodecki M, Horodecki P. Reduction criterion of separability and limits for a class of distillation protocols. Phys Rev A, 1999, 59: 4206–4216

DOI

16
Jiang B, Li Z, Zhang S. Characterizing real-valued multivariate complex polynomials and their symmetric tensor representations. SIAM J Matrix Anal Appl, 2016, 37: 381–408

DOI

17
Kolda T G, Bader B W. Tensor decompositions and applications. SIAM Rev, 2009, 51: 455–500

DOI

18
Kruskal J. Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Algebra Appl, 1977, 18: 95–138

DOI

19
Landsberg J M. Tensors: Geometry and Applications. Grad Stud Math, Vol 128. Providence: Amer Math Soc, 2012

20
Landsberg J M, Teitler Z. On the ranks and border ranks of symmetric tensors. Found Comput Math, 2010, 10: 339–366

DOI

21
Lathauwer L De.. A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization. SIAM J Matrix Anal Appl, 2006, 28: 642–666

DOI

22
Lathauwer L De, Moor B De, Vandewalle J. Computation of the canonical decomposition by means of a simultaneous generalized Schur decomposition. SIAM J Matrix Anal Appl, 2004, 26: 295–327

DOI

23
Li Y, Ni G. Separability discrimination and decomposition of m-partite quantum mixed states. Phys Rev A, 2020, 102: 012402

DOI

24
Li Z, Nakatsukasa Y, Soma T, Uschmajew A. On orthogonal tensors and best rank-one approximation ratio. SIAM J Matrix Anal Appl, 2018, 39: 400–425

DOI

25
Lim L-H. Tensors and hypermatrices. In: Hogben L, ed. Handbook of Linear Algebra. 2nd ed. Discrete Math Appl (Boca Raton). Boca Raton: CRC Press, 2014, 15-1–15-30

26
Milazzo N, Braun D, Giraud O. Truncated moment sequences and a solution to the channel separability problem. arXiv: 2006.15003

27
Ni G. Hermitian tensor and quantum mixed state. arXiv: 1902.02640

28
Ni G, Qi L, Bai M. Geometric measure of entanglement and U-eigenvalues of tensors. SIAM J Matrix Anal Appl, 2014, 35: 73–87

DOI

29
Nie J. Generating polynomials and symmetric tensor decompositions. Found Comput Math, 2017, 17: 423–465

DOI

30
Nie J. Low rank symmetric tensor approximations. SIAM J Matrix Anal Appl, 2017, 38: 1517–1540

DOI

31
Nie J, Yang Z. Hermitian tensor decompositions. SIAM J Matrix Anal Appl, 2020, 41(3): 1115–1144

DOI

32
Nie J, Zhang X. Positive maps and separable matrices. SIAM J Optim, 2018, 26(2): 1236-1256

DOI

33
Qi L, Xu C, Xu Y. Nonnegative tensor factorization, completely positive tensors and a hierarchical elimination algorithm. SIAM J Matrix Anal Appl, 2014, 35: 1227–1241

DOI

34
Qi L, Zhang G, Braun D, Bohnet-Waldraff F, Giraud O. Regularly decomposable tensors and classical spin states. Commun Math Sci, 2017, 15: 1651–1665

DOI

35
Qi L, Zhang G, Ni G. How entangled can a multi-party system possibly be? Phys Lett A, 2018, 382: 1465–1471

DOI

36
Sidiropoulos N, Bro R. On the uniqueness of multilinear decomposition of N-way arrays. J Chemometrics, 2000, 14: 229–239

DOI

37
Sorber L, Barel M Van, Lathauwer L De.. Optimization-based algorithms for tensor decompositions: canonical polyadic decomposition, decomposition in rank-(Lr, Lr, 1) terms, and a new generalization. SIAM J Optim, 2013, 23: 695–720

DOI

38
Wei T-C, Goldbart P M. Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys Rev A, 2003, 68: 042307

DOI

39
Zhou A, Fan J. Completely positive tensor recovery with minimal nuclear value. Comput Optim Appl, 2018, 70: 419–441

DOI

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