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11
On the Pauli graphs of Nqudits
, 2007
"... A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of Nqudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per twoqubits, all basic properties and ..."
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Cited by 17 (15 self)
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A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of Nqudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per twoqubits, all basic properties and partitionings of the corresponding Pauli graph are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin’s array and a bipartitepart and (c) an independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin’s square, and set of five mutually noncommuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) Nqubit Pauli graph is shown to be pseudogeometric, i. e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a twoqutrit system is introduced, leaving open its possible link to more abstract and exotic finite geometries.
Projective ring line encompassing twoqubits
 Theoret. and Math. Phys
, 2008
"... The projective line over the (noncommutative) ring of twobytwo matrices with coefficients in GF(2) is found to fully accommodate the algebra of 15 operators — generalized Pauli matrices — characterizing twoqubit systems. The relevant subconfiguration consists of 15 points each of which is eithe ..."
Abstract

Cited by 15 (14 self)
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The projective line over the (noncommutative) ring of twobytwo matrices with coefficients in GF(2) is found to fully accommodate the algebra of 15 operators — generalized Pauli matrices — characterizing twoqubit systems. The relevant subconfiguration consists of 15 points each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified with the points in such a onetoone manner that their commutation relations are exactly reproduced by the underlying geometry of the points, with the ring geometrical notions of neighbor/distant answering, respectively, to the operational ones of commuting/noncommuting. This remarkable configuration can be viewed in two principally different ways accounting, respectively, for the basic 9+6 and 10+5 factorizations of the algebra of the observables. First, as a disjoint union of the projective line over GF(2) × GF(2) (the “Mermin” part) and two lines over GF(4) passing through the two selected points, the latter omitted. Second, as the generalized quadrangle of order two, with its ovoids and/or spreads standing for (maximum) sets of five mutually noncommuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open up rather unexpected vistas for an algebraic geometrical modelling of finitedimensional quantum systems and give their numerous applications a wholly new perspective.
Projective ring line of an arbitrary single qudit
 J. Phys. A
, 2008
"... As a continuation of our previous work (arXiv:0708.4333) an algebraic geometrical study of a single ddimensional 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 struc ..."
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Cited by 15 (9 self)
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As a continuation of our previous work (arXiv:0708.4333) an algebraic geometrical study of a single ddimensional 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 Zd. 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 Z 2 d. We find, remarkably, that a perpset is not a settheoretic 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 multiplequdit cases is made.
Pauli graph and finite projective lines/geometries
, 2007
"... The commutation relations between the generalized Pauli operators of Nqudits (i. e., N plevel quantum systems), and the structure of their maximal sets of commuting bases, follow a nice graph theoretical/geometrical pattern. One may identify vertices/points with the operators so that edges/lines j ..."
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Cited by 6 (4 self)
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The commutation relations between the generalized Pauli operators of Nqudits (i. e., N plevel quantum systems), and the structure of their maximal sets of commuting bases, follow a nice graph theoretical/geometrical pattern. One may identify vertices/points with the operators so that edges/lines join commuting pairs of them to form the socalled Pauli graph P p N. As per twoqubits (p = 2, N = 2) all basic properties and partitionings of this graph are embodied in the geometry of the symplectic generalized quadrangle of order two, W(2). The structure of the twoqutrit (p = 3, N = 2) graph is more involved; here it turns out more convenient to deal with its dual in order to see all the parallels with the twoqubit case and its surmised relation with the geometry of generalized quadrangle Q(4, 3), the dual of W(3). Finally, the generalized adjacency graph for multiple (N> 3) qubits/qutrits is shown to follow from symplectic polar spaces of order two/three. The relevance of these mathematical concepts to mutually unbiased bases and to quantum entanglement is also highlighted in some detail.
Vectors, Cyclic Submodules, and Projective Spaces Linked with Ternions
, 2008
"... Given a ring of ternions R, i. e., a ring isomorphic to that of upper triangular 2 ×2 matrices with entries from an arbitrary commutative field F, a complete classification is performed of the vectors from the free left Rmodule R n+1, n ≥ 1, and of the cyclic submodules generated by these vectors. ..."
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Cited by 3 (3 self)
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Given a ring of ternions R, i. e., a ring isomorphic to that of upper triangular 2 ×2 matrices with entries from an arbitrary commutative field F, a complete classification is performed of the vectors from the free left Rmodule R n+1, n ≥ 1, and of the cyclic submodules generated by these vectors. The vectors fall into 5 + F  and the submodules into 6 distinct orbits under the action of the general linear group GLn+1(R). Particular attention is paid to free cyclic submodules generated by nonunimodular vectors, as these are linked with the lines of PG(n, F), the ndimensional projective space over F. In the finite case, F = GF(q), explicit formulas are derived for both the total number of nonunimodular free cyclic submodules and the number of such submodules passing through a given vector. These formulas yield a combinatorial approach to the lines and points of PG(n, q), n ≥ 2, in terms of vectors and nonunimodular free cyclic submodules of R n+1.
Jury: M. H. de Guise
, 2009
"... Thèse de l’Université de Lyon (École doctorale de Physique et Astrophysique de Lyon) présentée devant ..."
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Thèse de l’Université de Lyon (École doctorale de Physique et Astrophysique de Lyon) présentée devant
Theoretical and Mathematical Physics, 155(3): 905–913 (2008) PROJECTIVE RING LINE ENCOMPASSING TWOQUBITS
"... We find that the projective line over the (noncommutative) ring of 2×2 matrices with coefficients in GF (2) fully accommodates the algebra of 15 operators (generalized Pauli matrices) characterizing twoqubit systems. The relevant subconfiguration consists of 15 points, each of which is either simul ..."
Abstract
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We find that the projective line over the (noncommutative) ring of 2×2 matrices with coefficients in GF (2) fully accommodates the algebra of 15 operators (generalized Pauli matrices) characterizing twoqubit systems. The relevant subconfiguration consists of 15 points, each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified onetoone with the points such that their commutation relations are exactly reproduced by the underlying geometry of the points with the ring geometric notions of neighbor and distant corresponding to the respective operational notions of commuting and noncommuting. This remarkable configuration can be viewed in two principally different ways accounting for the basic corresponding 9+6 and 10+5 factorizations of the algebra of observables: first, as a disjoint union of the projective line over GF (2) × GF (2) (the “Mermin ” part) and two lines over GF (4) passing through the two selected points that are omitted; second, as the generalized quadrangle of order two with its ovoids and/or spreads corresponding to (maximum) sets of five mutually noncommuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open unexpected possibilities for an algebrogeometric modeling of finitedimensional quantum systems and completely new prospects for their numerous applications.