The Projective Line Over the Finite Quotient Ring GF(2)[$x$]/$< x^{3} - x>$ and Quantum Entanglement I. Theoretical Background

Details The Projective Line Over the Finite Quotient Ring GF(2)[$x$]/$< x^{3} - x>$ and Quantum Entanglement I. Theoretical Background The Projective Line Over the Finite Quotient Ring GF(2)[$x$]/$< x^{3} - x>$ and Quantum Entanglement I. Theoretical Background

2006-03-07

arxiv.org/abs/quant-ph/0603051v2

Theoretical and Mathematical Physics 151 (2007) 475-482

Physics

Quantum Physics

8 pages, 2 figures

Scientific paper

The paper deals with the projective line over the finite factor ring $R\_{\clubsuit} \equiv$ GF(2)[$x$]/$$. The line is endowed with 18 points, spanning the neighbourhoods of three pairwise distant points. As $R\_{\clubsuit}$ is not a local ring, the neighbour (or parallel) relation is not an equivalence relation so that the sets of neighbour points to two distant points overlap. There are nine neighbour points to any point of the line, forming three disjoint families under the reduction modulo either of two maximal ideals of the ring. Two of the families contain four points each and they swap their roles when switching from one ideal to the other; the points of the one family merge with (the image of) the point in question, while the points of the other family go in pairs into the remaining two points of the associated ordinary projective line of order two. The single point of the remaining family is sent to the reference point under both the mappings and its existence stems from a non-trivial character of the Jacobson radical, ${\cal J}\_{\clubsuit}$, of the ring. The factor ring $\widetilde{R}\_{\clubsuit} \equiv R\_{\clubsuit}/ {\cal J}\_{\clubsuit}$ is isomorphic to GF(2) $\otimes$ GF(2). The projective line over $\widetilde{R}\_{\clubsuit}$ features nine points, each of them being surrounded by four neighbour and the same number of distant points, and any two distant points share two neighbours. These remarkable ring geometries are surmised to be of relevance for modelling entangled qubit states, to be discussed in detail in Part II of the paper.

Affiliated with

Also associated with

No associations

Rating

The Projective Line Over the Finite Quotient Ring GF(2)[$x$]/$< x^{3} - x>$ and Quantum Entanglement I. Theoretical Background does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The Projective Line Over the Finite Quotient Ring GF(2)[$x$]/$< x^{3} - x>$ and Quantum Entanglement I. Theoretical Background, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Projective Line Over the Finite Quotient Ring GF(2)[$x$]/$< x^{3} - x>$ and Quantum Entanglement I. Theoretical Background will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-366449