The Projective Line Over the Finite Quotient Ring GF(2)[x]/⟨x3−x⟩ and Quantum Entanglement I. Theoretical Background
The paper deals with the projective line over the finite factor ring R♣≡ GF(2)[x]/⟨x3−x⟩. The line is endowed with 18 points, spanning the neighbourhoods of three pairwise distant points. As R♣ 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, J♣, of the ring. The factor ring R♣≡R♣/J♣ is isomorphic to GF(2) ⊗ GF(2). The projective line over R♣ 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 Projective Line Over the Finite Quotient Ring GF(2)[x]/⟨x3−x⟩ and Quantum Entanglement\\ II. The Mermin ``Magic" Square/Pentagram
In 1993, Mermin (Rev. Mod. Phys. 65, 803--815) gave lucid and strikingly simple proofs of the Bell-Kochen-Specker (BKS) theorem in Hilbert spaces of dimensions four and eight by making use of what has since been referred to as the Mermin(-Peres) ``magic square" and the Mermin pentagram, respectively. The former is a 3×3 array of nine observables commuting pairwise in each row and column and arranged so that their product properties contradict those of the assigned eigenvalues. The latter is a set of ten observables arranged in five groups of four lying along five edges of the pentagram and characterized by similar contradiction. An interesting one-to-one correspondence between the operators of the Mermin-Peres square and the points of the projective line over the product ring GF(2)⊗GF(2) is established. Under this mapping, the concept ``mutually commuting" translates into ``mutually distant" and the distinguishing character of the third column's observables has its counterpart in the distinguished properties of the coordinates of the corresponding points, whose entries are both either zero-divisors, or units. The ten operators of the Mermin pentagram answer to a specific subset of points of the line over GF(2)[x]/⟨x3−x⟩. The situation here is, however, more intricate as there are two different configurations that seem to serve equally well our purpose. The first one comprises the three distinguished points of the (sub)line over GF(2), their three ``Jacobson" counterparts and the four points whose both coordinates are zero-divisors; the other features the neighbourhood of the point (1,0) (or, equivalently, that of (0,1)). Some other ring lines that might be relevant for BKS proofs in higher dimensions are also mentioned.