Projective Ring Line Encompassing Two-Qubits
Résumé
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 finding opens up rather unexpected vistas for an algebraic geometrical modelling of finite-dimensional quantum systems and gives their numerous applications a wholly new perspective.