Canonical forms of 2 × 2 × 2 and 2 × 2 × 2 × 2 arrays over 𝔽2 and 𝔽3

Linear and Multilinear Algebra

Volumn 61, Issue 7, 2013, Pages 986-997

  Bremner, Murray R ^a   Stavrou, Stavros G ^a  

^a UNIVERSITY OF SASKATCHEWAN   (Canada)

Author keywords

canonical forms; computer algebra; finite fields; group actions; multidimensional arrays

Indexed keywords

EID: 84879657985     PISSN: 03081087     EISSN: 15635139     Source Type: Journal    
DOI: 10.1080/03081087.2012.721362     Document Type: Article

References (33)

1. Borsten, L, Dahanayake, D, Duff, MJ, Marrani, A and Rubens, W. 2010. Four-qubit entanglement classification from string theory. Phys. Rev. Lett., 105: 4
2. Bremner, MR. 2012. On the hyperdeterminant for 2 × 2 × 3 arrays. Linear Multilinear Algebra, 60: 921 - 932.
3. Bremner, MR, Bickis, MG and Soltanifar, M. 2012. Cayley's hyperdeterminant: a combinatorial approach via representation theory. Linear Algebra Appl., 437: 94 - 112.
4. 3(ℂ) on 3 × 3 × 3 arrays. Math. Comput., (to appear)
5. Briand, E, Luque, J-G and Thibon, J-Y. 2003. A complete set of covariants of the four qubit system. J. Phys. A, 36: 9915 - 9927.
6. Brylinski, J-L. 2002. " Algebraic Measures of Entanglement ". In Mathematics of Quantum Computation, Boca Raton, FL: Computational Mathematics Series, Chapman & Hall/CRC.
7. Cayley, A. 1845. On the theory of linear transformations. Cambridge Math. J., 4: 193 - 209.
8. Chterental, O and Djoković, DZ. Normal forms and tensor ranks of pure states of four qubits., arXiv: quant-ph/0612184v2 (submitted on 21 December 2006)
9. Cichocki, A, Zdunek, R, Phan, AH and Amari, S. 2009. Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation, Chichester, West Sussex: Wiley.
10. de Silva, V and Lim, L-H. 2008. Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl., 30: 1084 - 1127.
11. Djoković, DZ and Osterloh, A. 2009. On polynomial invariants of several qubits. J. Math. Phys., 50: 23
12. Ehrenborg, R. 1999. Canonical forms of two by two by two matrices. J. Algebra, 213: 195 - 224.
13. Gelfand, IM, Kapranov, MM and Zelevinsky, AV. 1992. Hyperdeterminants. Adv. Math., 96: 226 - 263.
14. Gelfand, IM, Kapranov, MM and Zelevinsky, AV. 1994. Discriminants, Resultants, and Multidimensional Determinants, Boston, Basel: Birkhäuser.
15. Gour, G and Wallach, NR. 2010. All maximally entangled four-qubit states. J. Math. Phys., 51: 24
16. Huggins, P, Sturmfels, B, Yu, J and Yuster, DS. 2008. The hyperdeterminant and triangulations of the 4-cube. Math. Comput., 77: 1653 - 1679.
17. Klyachko, A. Coherent states, entanglement, and geometric invariant theory., arXiv: quant-ph/0206012v1 (submitted on 3 June 2002)
18. Kolda, TG and Bader, BW. 2009. Tensor decompositions and applications. SIAM Rev., 51: 455 - 500.
19. Kroonenberg, PM. 2008. Applied Multiway Data Analysis, Hoboken, NJ: Wiley.
20. Lamata, L, León, J, Salgado, D and Solano, E. 2007. Inductive entanglement classification of four qubits under stochastic local operations and classical communication. Phys. Rev. A, 75: 9
21. Landsberg, JM. 2012. Tensors: Geometry and Applications. American Mathematical Society, Providence, RI,: 446
22. Le Paige, C. 1881. Sur les formes trilinéaires. Comp. Acad. Sci., 92: 1103 - 1105.
23. Lévay, P. 2006. On the geometry of four-qubit invariants. J. Phys. A, 39: 9533 - 9545.
24. Luque, J-G and Thibon, J-Y. 2003. Polynomial invariants of four qubits. Phys. Rev. A, 67: 5
25. Mandilara, A and Viola, L. 2007. Nilpotent polynomial approach to four-qubit entanglement. J. Phys. B, 40: S167 - S180.
26. Oldenburger, R. 1932. On canonical binary trilinear forms. Bull. Am. Math. Soci., 38: 385 - 387.
27. Oldenburger, R. 1937. Real canonical binary trilinear forms. Am. J. Math., 59: 427 - 435.
28. Schwartz, E. 1922. Über binäre trilineare Formen. Math. Zeitsch., 12: 18 - 35.
29. Smilde, A, Bro, R and Geladi, P. 2004. Multi-way Analysis: Applications in the Chemical Sciences, Chichester, West Sussex: Wiley.
30. Smith, L and Stong, RE. 2010. Invariants of binary bilinear forms modulo two. Proc. Am. Math. Soc., 138: 17 - 26.
31. ten Berge, JMF. 1991. Kruskal's polynomial for 2 × 2 × 2 arrays, and a generalization to 2 × n × n arrays. Psychometrika, 56: 631 - 636.
32. Verstraete, F, Dehaene, J, De Moor, B and Verschelde, H. 2002. Four qubits can be entangled in nine different ways. Phys. Rev. A, 65: 5
33. Wallach, NR. 2005. The Hilbert series of measures of entanglement for 4 qubits. Acta Appl. Math., 86: 203 - 220.