Projective Ring Line of an Arbitrary Single Qudit

As a continuation of our previous work (arXiv:0708.4333) an algebraic geometrical study of a single $d$-dimensional qudit is made, with $d$ being {\it 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 \bZ_{d}. 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 \bZ^{2}_{d}. 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)[xx]/<x3−x>< x^{3} - x> and Quantum Entanglement I. Theoretical Background

The paper deals with the projective line over the finite factor ring $R\_{\clubsuit} \equiv$ GF(2)[$x$]/$$. 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.Comment: 8 pages, 2 figure

Projective Ring Line of a Specific Qudit

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 \bZ_{d} is established, where the integer exponents of the generating shift ($X$) and clock ($Z$) operators are associated with submodules of \bZ^{2}_{d}. Under this correspondence, the set of operators commuting with a given one -- a perp-set -- represents a \bZ_{d}-submodule of \bZ^{2}_{d}. A crucial novel feature here is that the operators are also represented by {\it non}-admissible pairs of \bZ^{2}_{d}. This additional degree of freedom makes it possible to view any perp-set as a {\it set-theoretic} union of the corresponding points of the associated projective line

Projective Ring Line Encompassing Two-Qubits

The projective line over the (non-commutative) ring of two-by-two matrices with coefficients in GF(2) is found to fully accommodate the algebra of 15 operators - generalized Pauli matrices - characterizing two-qubit systems. The relevant sub-configuration 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 one-to-one 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/non-commuting. 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) x 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 non-commuting 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 finite-dimensional quantum systems and give their numerous applications a wholly new perspective.Comment: 8 pages, three tables; Version 2 - a few typos and one discrepancy corrected; Version 3: substantial extension of the paper - two-qubits are generalized quadrangles of order two; Version 4: self-dual picture completed; Version 5: intriguing triality found -- three kinds of geometric hyperplanes within GQ and three distinguished subsets of Pauli operator

Harmonic Generation from Relativistic Plasma Surfaces in Ultra-Steep Plasma Density Gradients

Harmonic generation in the limit of ultra-steep density gradients is studied experimentally. Observations demonstrate that while the efficient generation of high order harmonics from relativistic surfaces requires steep plasma density scale-lengths ($L_p/\lambda < 1$) the absolute efficiency of the harmonics declines for the steepest plasma density scale-length $L_p \to 0$, thus demonstrating that near-steplike density gradients can be achieved for interactions using high-contrast high-intensity laser pulses. Absolute photon yields are obtained using a calibrated detection system. The efficiency of harmonics reflected from the laser driven plasma surface via the Relativistic Oscillating Mirror (ROM) was estimated to be in the range of 10^{-4} - 10^{-6} of the laser pulse energy for photon energies ranging from 20-40 eV, with the best results being obtained for an intermediate density scale-length

Structure peculiarities of cementite and their influence on the magnetic characteristics

The iron carbide $Fe_3C$ is studied by the first-principle density functional theory. It is shown that the crystal structure with the carbon disposition in a prismatic environment has the lowest total energy and the highest energy of magnetic anisotropy as compared to the structure with carbon in an octahedron environment. This fact explains the behavior of the coercive force upon annealing of the plastically deformed samples. The appearance of carbon atoms in the octahedron environment can be revealed by Mossbauer experiment.Comment: 10 pages, 3 figures, 3 tables. submitted to Phys.Rev.

Notas para el estudio de la política de regulación dominial de la provincia de Buenos Aires, 1990-2004

La irregularidad desde el punto de vista del dominio refiere a posesiones de inmuebles a los que se accede por vías no formales y/o en los que no se inicia o queda inconcluso el debido proceso registral, con la confección de la escritura a nombre del o los titulares del dominio. Podemos agrupar las situaciones de irregularidad dominial según tres diferentes formas de acceso al suelo (que a su vez encierran diferentes casos): la adquisición regular sin culminar el proceso registral, la ocupación directa de tierras, la compra en el mercado irregular (Clichevsky 2003). Cuando la irregularidad está referida a las condiciones urbanoambientales se trata de casos que no cumplen con las normas urbanísticas que fijan estándares mínimos de habitabilidad: son áreas habitadas emplazadas en zonas no aptas para residencia (planicies de inundación de los cursos de agua, bañados, áreas contaminadas o de pendiente pronunciada, entre otras) o de parcelas y/o construcciones que no cumplen con la normativa vigente al respecto.Facultad de Humanidades y Ciencias de la Educació

De la faille alpine à la fosse de Puysegur (Nouvelle-Zélande) : résultats de la campagne de cartographie multifaisceaux GEODYNZ-SUD, Leg 2

Le Leg 2 de la campagne GEODYNZ-SUD, menée au SW de la Nouvelle-Zélande, a permis de reconnaître les structures qui accompagnent du Nord au Sud le passage de la faille alpine à la subduction oblique sous la marge du Fiodland, puis à celle naissante, intra-océanique sous la ride de Macquarie. Au Nord et au-dessus de la plaque australienne subductée vers l'Est, un faisceau longitudinal de décrochements converge vers le système transpressif de la faille alpine en découpant la marge continentale. Au Sud, la déformation décrochante est strictement localisée au sommet de la ride de Macquarie. (Résumé d'auteur