Polynomial invariants of four qubits

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 67, Issue 4, 2003, Pages 423031-423035

  Luque, Jean Gabriel ^a   Thibon, Jean Yves ^a  

^a Université de Marne la Vallée   (France)

Author keywords

[No Author keywords available]

Indexed keywords

MATRIX ALGEBRA; POLYNOMIALS; PROBLEM SOLVING;

FOUR-QUBIT STATES;

QUANTUM THEORY;

EID: 4243462237     PISSN: 10502947     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (27)

1. W. Dür, G. Vidai, and J. Cirac, Phys. Rev. A 62, 062314 (2000).
2. J.-L. Brylinski, e-print quant-ph/0008031.
3. J.-L. Brylinski and R. Brylinski, e-print quant-ph/0010101.
4. F. Verstraete, J. Dehaene, B.D. Moor, and H. Verschelde, Phys. Rev. A 65, 052112 (2002).
5. A. Miyaké, Phys. Rev. A 67, 012108 (2003).
6. A.A. Klyachko, e-print quant-ph/0206012.
7. C. Bennett, S. Popoescu, D. Rohrlich, J. Smolin, and A. Thap- liyal, Phys. Rev. A 63, 012307 (2001).
8. In the physics literature, the solution appears in Ref. [1], but an equivalent problem had been solved in classical invariant theory since at least 1881 [21,22], Note also that a problem equivalent to the classification of four-qubit states had been studied by Segre in 1922 [17], but he only obtained a partial classification.
9. We note that a combinatorial method for computing invariants of fourth-rank tensors is proposed in Ref. [10]. The point in the present paper is that we can prove that our system of invariants is complete.
10. V. Tapia, e-print math-ph/0208010.
11. J.-Y. Thibon, Int. J. Alg. Comp. 2, 207 (1991).
12. T. Scharf, J.-Y. Thibon, and B. Wybourne, J. Phys. A 24, 7461 (1993).
13. This is a tedious method. With the help of computer algebra system, the Hubert series is more straightforwardly obtained by a residue calculation. Similar computations [for SU(2) and U(2) invariants] appear in Ref. [14].
14. M. Grassl, T. Both, M. Rötteler, and Y. Makhlin, Talk at MSRI, University of California at Berkeley 2002 (unpublished).
15. Cayley actually considered several different notions under the same generic name, see Ref. [16].
16. J.-G. Luque and J.-Y. Thibon, e-print math-ph/02011044.
17. C. Segre, Ann. Mat., Ser. III 29, 105 (1922).
18. P. Olver, Classical Invariant Theory (Cambridge University Press, Cambridge, Cambridge, 1999).
19. I. Gelfand, M. Kapranov, and A. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser, Boston, 1994).
20. L. Schläfli, Denkschr. Kaiserl. Akad. Wiss., Math-naturwiss. Klasse, 4(1852).
21. C. Le Paige, Bull. Acad. R. Sci. Belgique (3) 2, 40 (1881).
22. E. Schwartz, Math. Z. 12, 18 (1922).
23. 2 will not depend on these choices.
24. For five qubits, the generic orbits depend on 16 free parameters, and for eight qubits, the moduli space would be of dimension 231.
25. R. Thrall and J. Chanler, Duke Math. J. 4, 678 (1938).
26. J. Chanler, Duke Math. J. 5, 552 (1939).
27. 4 on the four-dimensional vector space spanned by a,b,c,d.