The Projective Line Over the Finite Quotient Ring GF(2)[$x$]/$\langle x^{3} - x \rangle$ and Quantum Entanglement\\ II. The Mermin ``Magic" Square/Pentagram
Метод Санига, Michel R. P. Planat, Milan Minarovjech
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 $3 \times 3$ 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 ${\rm GF}(2) \otimes \rm{GF}(2)$ 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)[$x$]/$\langle x^{3} - x \rangle$. 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.
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 $3 \times 3$ 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 ${\rm GF}(2) \otimes \rm{GF}(2)$ 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)[$x$]/$\langle x^{3} - x \rangle$. 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.
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.
A ``magic rectangle" of eleven observables of four qubits, employed by Harvey and Chryssanthacopoulos (2008) to prove the Bell-Kochen-Specker theorem in a 16-dimensional Hilbert space, is given a neat finite-geometrical reinterpretation in terms of the structure of the symplectic polar space W(7,2) of the real four-qubit Pauli group. Each of the four sets of observables of cardinality five represents an elliptic quadric in the three-dimensional projective space of order two (PG(3,2)) it spans, whereas the remaining set of cardinality four corresponds to an affine plane of order two. The four ambient PG(3,2)s of the quadrics intersect pairwise in a line, the resulting six lines meeting in a point. Projecting the whole configuration from this distinguished point (observable) one gets another, complementary ``magic rectangle" of the same qualitative structure.
Mermin's pentagram, a specific set of ten three-qubit observables arranged in quadruples of pairwise commuting ones into five edges of a pentagram and used to provide a very simple proof of the Kochen-Specker theorem, is shown to be isomorphic to an ovoid (elliptic quadric) of the three-dimensional projective space of order two, PG(3, 2). This demonstration employs properties of the real three-qubit Pauli group embodied in the geometry of the symplectic polar space W (5, 2) and rests on the facts that: 1) the four observables/operators on any of the five edges of the pentagram can be viewed as points of an affine plane of order two, 2) all the ten observables lie on a hyperbolic quadric of the five-dimensional projective space of order two, PG(5, 2), and 3) that the points of this quadric are in a well-known bijective correspondence with the lines of PG(3, 2).
We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension q, containing a square, into its factors. Illustrative low dimensional examples are the quartit (q=4) and two-qubit (q=22) systems, the octit (q=8), qubit/quartit (q=2×4) and three-qubit (q=23) systems, and so on. In the single qudit case, e.g. q=4,8,12,..., one defines a bijection between the σ(q) maximal commuting sets [with σ[q) the sum of divisors of q] of Pauli observables and the maximal submodules of the modular ring Zq2, that arrange into the projective line P1(Zq) and a independent set of size σ(q)−ψ(q) [with ψ(q) the Dedekind psi function]. In the multiple qudit case, e.g. q=22,23,32,..., the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if q=22) and GQ(3,3) (if q=32). More precisely, in dimension pn (p a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the 2n-dimensional vector space over the field Fp. In this space, one makes use of the commutator to define a symplectic polar space W2n−1(p) of cardinality σ(p2n−1), that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of W2n−1(p) are punctured polar spaces (i.e. a observable and all maximum cliques passing to it are removed) of size given by the Dedekind psi function ψ(p2n−1). For multiple qudit mixtures (e.g. qubit/quartit, qubit/octit and so on), one finds multiple copies of polar spaces, ponctured polar spaces, hypercube geometries and other intricate structures. Such structures play a role in the science of quantum information.