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

Метод Санига, Michel R. P. Planat

The paper deals with the projective line over the finite factor ring $R_{\clubsuit} \equiv$ GF(2)[ $x$ ]/ $\langle x^{3} - x \rangle$ . 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.The paper deals with the projective line over the finite factor ring $R_{\clubsuit} \equiv$ GF(2)[$x$]/$\langle x^{3} - x \rangle$. 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.

Jan 1, 20070 citations

Four-Qubit Geometry

Zhang Chi Liu

Sep 1, 20260 citations

On the geometry of two qubits

Т.Э. Кренкель

Journal of Mathematical SciencesAug 8, 20130 citations

Quantum Entanglement and Projective Ring Geometry

Michel R. P. Planat

The paper explores the basic geometrical properties of the observables characterizing two-qubit systems by employing a novel projective ring geometric approach.After introducing the basic facts about quantum complementarity and maximal quantum entanglement in such systems, we demonstrate that the 15 × 15 multiplication table of the associated four-dimensional matrices exhibits a so-far-unnoticed geometrical structure that can be regarded as three pencils of lines in the projective plane of order two.In one of the pencils, which we call the kernel, the observables on two lines share a base of Bell states.In the complement of the kernel, the eight vertices/observables are joined by twelve lines which form the edges of a cube.A substantial part of the paper is devoted to showing that the nature of this geometry has much to do with the structure of the projective lines defined over the rings that are the direct product of n copies of the Galois field GF (2), with n = 2, 3 and 4.The paper explores the basic geometrical properties of the observables characterizing two-qubit systems by employing a novel projective ring geometric approach.After introducing the basic facts about quantum complementarity and maximal quantum entanglement in such systems, we demonstrate that the 15 × 15 multiplication table of the associated four-dimensional matrices exhibits a so-far-unnoticed geometrical structure that can be regarded as three pencils of lines in the projective plane of order two.In one of the pencils, which we call the kernel, the observables on two lines share a base of Bell states.In the complement of the kernel, the eight vertices/observables are joined by twelve lines which form the edges of a cube.A substantial part of the paper is devoted to showing that the nature of this geometry has much to do with the structure of the projective lines defined over the rings that are the direct product of n copies of the Galois field GF (2), with n = 2, 3 and 4.

arXiv:quant-ph/0605239Symmetry Integrability and Geometry Methods and ApplicationsAug 17, 200625 citations

GENERAL OPEN QUANTUM SYSTEMS ON A RING

WORLD SCIENTIFIC eBooksJan 1, 20220 citations

A Fine Structure of Finite Projective Ring Lines

Метод Санига

The lecture introduced and discussed novel, and rather unexpected, properties of the fine structure of the projective lines defined over finite rings, which emerged as a by-product of our recent applications of these remarkable geometries in quantum physics. The corner-stone concept of the talk was a free cyclic submodule (fcs) over a finite ring and the particular focus was on those rings which give rise to fcs's not containing any admissible pairs. It was first shown that one can fine-tune the neighbour relation by taking into account the cardinality of shared pairs of the fcs's representing given points. This was illustrated by examples of projective lines defined over local rings; here, in the case of rings of type 8/4 we find two different kinds of a projective line, and as many as four kinds for the 16/8 type. The difference between the individual kinds of lines was shown to be intimately related with the number of pairs not lying on any fcs generated by an admissible pair – so-called “outliers”. It was subsequently demonstrated that there exist finite rings (some of) whose outliers even generate fcs's. The smallest case when this occurs is the non-commutative ring of type 8/6, but also some other examples – several non-commutative rings of order sixteen – were given to illustrate the phenomenon. The talk was finished by musing about possible physical implications of these intriguing features in quantum (information) theory.The lecture introduced and discussed novel, and rather unexpected, properties of the fine structure of the projective lines defined over finite rings, which emerged as a by-product of our recent applications of these remarkable geometries in quantum physics. The corner-stone concept of the talk was a free cyclic submodule (fcs) over a finite ring and the particular focus was on those rings which give rise to fcs's not containing any admissible pairs. It was first shown that one can fine-tune the neighbour relation by taking into account the cardinality of shared pairs of the fcs's representing given points. This was illustrated by examples of projective lines defined over local rings; here, in the case of rings of type 8/4 we find two different kinds of a projective line, and as many as four kinds for the 16/8 type. The difference between the individual kinds of lines was shown to be intimately related with the number of pairs not lying on any fcs generated by an admissible pair – so-called “outliers”. It was subsequently demonstrated that there exist finite rings (some of) whose outliers even generate fcs's. The smallest case when this occurs is the non-commutative ring of type 8/6, but also some other examples – several non-commutative rings of order sixteen – were given to illustrate the phenomenon. The talk was finished by musing about possible physical implications of these intriguing features in quantum (information) theory.

Feb 25, 20081 citation

The Projective Line Over the Finite Quotient Ring GF(2)[

x

\langle x^{3} - x \rangle

and Quantum Entanglement I. Theoretical Background

The paper deals with the projective line over the finite factor ring

R_{\clubsuit} \equiv

GF(2)[

x

\langle x^{3} - x \rangle

. 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.