Abstract
We find that the projective line over the (noncommutative) ring of 2×2 matrices with coefficients in GF(2) fully accommodates the algebra of 15 operators (generalized Pauli matrices) characterizing two-qubit systems. The relevant subconfiguration consists of 15 points, each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified one-to-one with the points such that their commutation relations are exactly reproduced by the underlying geometry of the points with the ring geometric notions of neighbor and distant corresponding to the respective operational notions of commuting and noncommuting. This remarkable configuration can be viewed in two principally different ways accounting for the basic corresponding 9+6 and 10+5 factorizations of the algebra of observables: first, as a disjoint union of the projective line over GF(2) × GF(2) (the “Mermin” part) and two lines over GF(4) passing through the two selected points that are omitted; second, as the generalized quadrangle of order two with its ovoids and/or spreads corresponding to (maximum) sets of five mutually noncommuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open unexpected possibilities for an algebro-geometric modeling of finite-dimensional quantum systems and completely new prospects for their numerous applications.
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References
A. Blunck and H. Havlicek, Abh. Math. Sem. Univ. Hamburg, 70, 287–299 (2000).
A. Blunck and H. Havlicek, Math. Pannon., 14, 113–127 (2003).
A. Blunck and H. Havlicek, Aequationes Math., 69, 146–163 (2005).
H. Havlicek,Quad. Sem. Matem. di Brescia, 11, 1–63 (2006); http://www.geometrie.tuwien.ac.at/havlicek/pdf/dd-laguerre.pdf (2004).
A. Herzer, “Chain geometries,” in: Handbook of Incidence Geometry (F. Buekenhout, ed.), Elsevier, Amsterdam (1995), p. 781–842.
M. Saniga, M. Planat, M. R. Kibler, and P. Pracna, Chaos Solitons Fractals, 33, 1095–1102 (2007); arXiv:math/0605301v4 [math.AG] (2006).
M. Saniga and M. Planat, Chaos Solitons Fractals, 37, 337–345 (2008); arXiv:math/0604307v2 [math.AG] (2006).
M. Saniga and M. Planat, Theor. Math. Phys., 151, 474–481 (2007); arXiv:quant-ph/0603051v2 (2006).
M. Saniga, M. Planat, and M. Minarovjech, Theor. Math. Phys., 151, 625–631 (2007); arXiv:quant-ph/0603206v1 (2006).
M. Planat, M. Saniga, and M. R. Kibler, SIGMA, 0602, 066 (2006); arXiv:quant-ph/0605239v4 (2006).
M. Saniga, M. Planat, and P. Pracna, “A classification of the projective lines over small rings II: Non-commutative case,” arXiv:math.AG/0606500v1 (2006).
J. B. Fraleigh, A First Course in Abstract Algebra, Addison-Wesley, Reading, Mass. (1967).
B. R. McDonald, Finite Rings with Identity, Dekker, New York (1974).
R. Raghavendran, Compos. Math., 21, 195–229 (1969).
C. Nöbauer, “The book of the rings: Part I,” http://www.algebra.uni-linz.ac.at/~noebsi/pub/rings.ps (2000).
H. Van Maldeghem, Generalized Polygons, Birkhäuser, Basel (1998).
B. Polster and H. Van Maldeghem, J. Combin. Theory Ser. A, 96, 162–179 (2001).
B. Polster, A Geometrical Picture Book, Springer, New York (1998).
B. Polster, Bull. Belg. Math. Soc. Simon Stevin, 5, 417–425 (1998).
B. Polster, A. E. Schroth, and H. Van Maldeghem, Math. Intelligencer, 23, 33–47 (2001).
J. Lawrence, Č. Brukner, and A. Zeilinger, Phys. Rev. A, 65, 032320 (2002).
S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, Pitman, Boston (1984).
H. Havlicek, Mitt. Math. Ges. Hamburg, 26, 75–94 (2007); http://www.geometrie.tuwien.ac.at/havlicek/pub/pentazyklisch.pdf.
M. Saniga and M. Planat, Adv. Stud. Theor. Phys., 1, 1–4 (2007); arXiv:quant-ph/0612179v1 (2006).
M. Planat and M. Saniga, Quantum Inf. Comput., 8, 127–146 (2008); arXiv: quant-ph/0701211v3 (2007).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 3, pp. 463–473, June, 2008.
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Saniga, M., Planat, M. & Pracna, P. Projective ring line encompassing two-qubits. Theor Math Phys 155, 905–913 (2008). https://doi.org/10.1007/s11232-008-0076-x
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DOI: https://doi.org/10.1007/s11232-008-0076-x