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A Fine Structure of Finite Projective Ring Lines

Метод Санига

Published Feb 25, 2008• 1 citation
OpenAlex

Abstract

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.

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A very particular connection between the commutation relations of the elements of the generalized Pauli group of a d -dimensional qudit, d being a product of distinct primes, and the structure of the projective line over the (modular) ring is established, where the integer exponents of the generating shift ( X ) and clock ( Z ) operators are associated with submodules of . Under this correspondence, the set of operators commuting with a given one—a perp-set—represents a submodule of . A crucial novel feature here is that the operators are also represented by non -admissible pairs of . This additional degree of freedom makes it possible to view any perp-set as a set-theoretic union of the corresponding points of the associated projective line.

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FINITE GEOMETRIES IN QUANTUM THEORY: FROM GALOIS (FIELDS) TO HJELMSLEV (RINGS)

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Geometries over Galois fields (and related finite combinatorial structures/algebras) have recently been recognized to play an ever-increasing role in quantum theory, especially when addressing properties of mutually unbiased bases (MUBs). The purpose of this contribution is to show that completely new vistas open up if we consider a generalized class of finite (projective) geometries, viz. those defined over Galois rings and/or other finite Hjelmslev rings. The case is illustrated by demonstrating that the basic combinatorial properties of a complete set of MUBs of a q-dimensional Hilbert space [Formula: see text], q = prwith p being a prime and r a positive integer, are qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p2and rank r. The q vectors of a basis of [Formula: see text] correspond to the q points of a (so-called) neighbour class and the q + 1 MUBs answer to the total number of (pairwise disjoint) neighbour classes on the conic. Although this remarkable analogy is still established at the level of cardinalities only, we currently work on constructing an explicit mapping by associating a MUB to each neighbour class of the points of the conic and a state vector of this MUB to a particular point of the class. Further research in this direction may prove to be of great relevance for many areas of quantum information theory, in particular for quantum information processing.

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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.

•Jan 1, 2007•0 citations

Qudits of composite dimension, mutually unbiased bases and projective ring geometry

Michel R. P. Planat, Anne-Céline Baboin

The $d^2$ Pauli operators attached to a composite qudit in dimension $d$ may be mapped to the vectors of the symplectic module $\mathcal{Z}_d^{2}$ ($\mathcal{Z}_d$ the modular ring). As a result, perpendicular vectors correspond to commuting operators, a free cyclic submodule to a maximal commuting set, and disjoint such sets to mutually unbiased bases. For dimensions $d=6,~10,~15,~12$, and 18, the fine structure and the incidence between maximal commuting sets is found to reproduce the projective line over the rings $\mathcal{Z}_{6}$, $\mathcal{Z}_{10}$, $\mathcal{Z}_{15}$, $\mathcal{Z}_6 \times \mathbf{F}_4$ and $\mathcal{Z}_6 \times \mathcal{Z}_3$, respectively.

arXiv:0709.2623•Journal of Physics A Mathematical and Theoretical•Oct 31, 2007•15 citations
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.
A very particular connection between the commutation relations of the elements of the generalized Pauli group of a d -dimensional qudit, d being a product of distinct primes, and the structure of the projective line over the (modular) ring is established, where the integer exponents of the generating shift ( X ) and clock ( Z ) operators are associated with submodules of . Under this correspondence, the set of operators commuting with a given one—a perp-set—represents a submodule of . A crucial novel feature here is that the operators are also represented by non -admissible pairs of . This additional degree of freedom makes it possible to view any perp-set as a set-theoretic union of the corresponding points of the associated projective line.
Geometries over Galois fields (and related finite combinatorial structures/algebras) have recently been recognized to play an ever-increasing role in quantum theory, especially when addressing properties of mutually unbiased bases (MUBs). The purpose of this contribution is to show that completely new vistas open up if we consider a generalized class of finite (projective) geometries, viz. those defined over Galois rings and/or other finite Hjelmslev rings. The case is illustrated by demonstrating that the basic combinatorial properties of a complete set of MUBs of a q-dimensional Hilbert space [Formula: see text], q = prwith p being a prime and r a positive integer, are qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p2and rank r. The q vectors of a basis of [Formula: see text] correspond to the q points of a (so-called) neighbour class and the q + 1 MUBs answer to the total number of (pairwise disjoint) neighbour classes on the conic. Although this remarkable analogy is still established at the level of cardinalities only, we currently work on constructing an explicit mapping by associating a MUB to each neighbour class of the points of the conic and a state vector of this MUB to a particular point of the class. Further research in this direction may prove to be of great relevance for many areas of quantum information theory, in particular for quantum information processing.
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.
As a continuation of our previous work ( Havlicek and Saniga 2007 J. Phys. A: Math. Gen. 40 F943–52 ( Preprint 0708.4333)) an algebraic geometrical study of a single d -dimensional qudit is made, with d being any positive integer. The study is based on an intricate relation between the symplectic module of the generalized Pauli group of the qudit and the fine structure of the projective line over the (modular) ring . Explicit formulae are given for both the number of generalized Pauli operators commuting with a given one and the number of points of the projective line containing the corresponding vector of . We find, remarkably, that a perp-set is not a set-theoretic union of the corresponding points of the associated projective line unless d is a product of distinct primes. The operators are also seen to be structured into disjoint ‘layers’ according to the degree of their representing vectors. A brief comparison with some multiple-qudit cases is made.
The Projective Line Over the Finite Quotient Ring GF(2)[xxx]/⟨x3−x⟩\langle x^{3} - x \rangle⟨x3−x⟩ and Quantum Entanglement I. Theoretical Background
The paper deals with the projective line over the finite factor ring R♣≡R_{\clubsuit} \equivR♣​≡ GF(2)[xxx]/⟨x3−x⟩\langle x^{3} - x \rangle⟨x3−x⟩. The line is endowed with 18 points, spanning the neighbourhoods of three pairwise distant points. As R♣R_{\clubsuit}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♣{\cal J}_{\clubsuit}J♣​, of the ring. The factor ring R~♣≡R♣/J♣\widetilde{R}_{\clubsuit} \equiv R_{\clubsuit}/ {\cal J}_{\clubsuit}R♣​≡R♣​/J♣​ is isomorphic to GF(2) ⊗\otimes⊗ GF(2). The projective line over R~♣\widetilde{R}_{\clubsuit}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 d2d^2d2 Pauli operators attached to a composite qudit in dimension ddd may be mapped to the vectors of the symplectic module Zd2\mathcal{Z}_d^{2}Zd2​ (Zd\mathcal{Z}_dZd​ the modular ring). As a result, perpendicular vectors correspond to commuting operators, a free cyclic submodule to a maximal commuting set, and disjoint such sets to mutually unbiased bases. For dimensions d=6, 10, 15, 12d=6,~10,~15,~12d=6, 10, 15, 12, and 18, the fine structure and the incidence between maximal commuting sets is found to reproduce the projective line over the rings Z6\mathcal{Z}_{6}Z6​, Z10\mathcal{Z}_{10}Z10​, Z15\mathcal{Z}_{15}Z15​, Z6×F4\mathcal{Z}_6 \times \mathbf{F}_4Z6​×F4​ and Z6×Z3\mathcal{Z}_6 \times \mathcal{Z}_3Z6​×Z3​, respectively.