Projective Ring Line Encompassing Two-Qubits - Archive ouverte HAL Access content directly
Journal Articles Theoretical and Mathematical Physics Year : 2008

Projective Ring Line Encompassing Two-Qubits


The projective line over the (non-commutative) ring of two-by-two matrices with coefficients in GF(2) is found to fully accommodate the algebra of 15 operators - generalized Pauli matrices - characterizing two-qubit systems. The relevant sub-configuration 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 with the points in such a one-to-one manner that their commutation relations are exactly reproduced by the underlying geometry of the points, with the ring geometrical notions of neighbor/distant answering, respectively, to the operational ones of commuting/non-commuting. This remarkable configuration can be viewed in two principally different ways accounting, respectively, for the basic 9+6 and 10+5 factorizations of the algebra of the observables. First, as a disjoint union of the projective line over GF(2) x GF(2) (the "Mermin" part) and two lines over GF(4) passing through the two selected points, the latter omitted. Second, as the generalized quadrangle of order two, with its ovoids and/or spreads standing for (maximum) sets of five mutually non-commuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open up rather unexpected vistas for an algebraic geometrical modelling of finite-dimensional quantum systems and give their numerous applications a wholly new perspective.
Fichier principal
Vignette du fichier
prletq_final.pdf (131.08 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-00111733 , version 1 (06-11-2006)
hal-00111733 , version 2 (13-11-2006)
hal-00111733 , version 3 (05-12-2006)
hal-00111733 , version 4 (11-12-2006)
hal-00111733 , version 5 (28-12-2006)



Metod Saniga, Michel R. P. Planat, Petr Pracna. Projective Ring Line Encompassing Two-Qubits. Theoretical and Mathematical Physics, 2008, 155 (3), pp.905-913. ⟨10.1007/s11232-008-0076-x⟩. ⟨hal-00111733v5⟩
277 View
122 Download



Gmail Facebook X LinkedIn More