Factor-Group-Generated Polar Spaces and (Multi-)Qudits

Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce gradually necessary and sufficient conditions to be met in order to carry out the following programme: Given a group G, we first construct vector spaces over GF(p), p a prime, by factorising G over appropriate normal subgroups. Then, by expressing GF(p) in terms of the commutator subgroup of G, we construct alternating bilinear forms, which reflect whether or not two elements of G commute. Restricting to p = 2, we search for ''refinements'' in terms of quadratic forms, which capture the fact whether or not the order of an element of G is ≤ 2. Such factor-group-generated vector spaces admit a natural reinterpretation in the language of symplectic and orthogonal polar spaces, where each point becomes a ''condensation'' of several distinct elements of G. Finally, several well-known physical examples (single- and two-qubit Pauli groups, both the real and complex case) are worked out in detail to illustrate the fine traits of the formalism

The Veldkamp space of multiple qubits

We introduce a point-line incidence geometry in which the commutation relations of the real Pauli group of multiple qubits are fully encoded. Its points are pairs of Pauli operators differing in sign and each line contains three pairwise commuting operators any of which is the product of the other two (up to sign). We study the properties of its Veldkamp space enabling us to identify subsets of operators which are distinguished from the geometric point of view. These are geometric hyperplanes and pairwise intersections thereof. Among the geometric hyperplanes one can find the set of self-dual operators with respect to the Wootters spin-flip operation well-known from studies concerning multiqubit entanglement measures. In the two- and three-qubit cases a class of hyperplanes gives rise to Mermin squares and other generalized quadrangles. In the three-qubit case the hyperplane with points corresponding to the 27 Wootters self-dual operators is just the underlying geometry of the E6(6) symmetric entropy formula describing black holes and strings in five dimensions.Comment: 15 pages, 1 figure; added references, corrected typos; minor change

Relaxation et polarisation dynamique des protons dans un champ de 50 kilogauss

The results are presented of : 1) measurements of the spin-lattice relaxation of protons in high magnetic field (H ∼ 48 kG, 36 kG, 25 kG) and at T = 1 °K in ethylene glycol (E. G.) doped with a Crv complex ; 2) dynamic polarisation of protons in E. G. at H ∼ 48 kG, T = 4,2 °K and 1 °K, and 3) dynamic polarisation of protons in lanthanium magnesium nitrate (LMN) doped with Nd3+ at H = 35 kG and T = 1 °K. The mechanisms are discussed of the nuclear spin-lattice relaxation of the protons and of the electronic spin-lattice relaxation of the Crv complex in E. G. The experimental study has shown that the dynamic polarization of protons in E. G. (and probably also in similar substances) with an EPR linewidth comparable with the separation of the forbidden transitions can still increase considerably with increasing magnetic field between H = 25 kG and 48 kG at T = 1 °K . The maximum polarization PI obtained in samples of 0.06-0.1 cm3 was P I ≃ 80 %. Several models are discussed to explain the increase of PI with the ratio (H/T).On présente : 1) des résultats de mesure de la relaxation spin-réseau des protons dans des champs magnétiques élevés (H ~ 48 kG, 36 kG, 25 kG) et à T = 1 °K, pour l'éthylène-glycol (E. G.) dopé avec le complexe du Crv ; 2) des résultats de la polarisation dynamique des protons dans l'E. G. à H ~ 48 kG, T = 4,2 °K, 1 °K ; et 3) des résultats de la polarisation dynamique dans le double nitrate de lanthane (LMN) dopé au Nd 3+ à H ∼ 36 kG et T = 1 °K. On discute des mécanismes de la relaxation spin-réseau des protons et de la relaxation électronique du complexe de Crv dans l'E. G. L'étude expérimentale a montré que la polarisation dynamique des protons dans l'E. G. (et probablement aussi dans des substances similaires) avec une largeur de raie électronique comparable à la distance des transitions interdites, peut encore augmenter considérablement avec l'accroissement du champ magnétique entre H = 25 kG et 48 kG à T = 1 °K. Dans des échantillons de 0,06-0,1 cm 3 nous avons obtenu des valeurs maximales de cette polarisation PI ≃ 80 %. Nous présentons une discussion détaillée de quelques modèles pour expliquer l'augmentation de P I avec le rapport (H/T)

Relaxation et polarisation des cristaux de La2Mg3 (NO3)12, 24H2O dopé au Dy3+

This paper reports details of measurements of proton relaxation time and dynamic polarization in LMN doped with Dy3+. We have measured the electronic relaxation time for the parallel and perpendicular orientations in the static magnetic field. They confirm that there are two doublets X and Y whose admixture by the static magnetic field is orientation dependant. We also measured the nuclear relaxation time for different fields and temperatures, for the two orientations. The comparison with the results of various nuclear relaxation theories shows undoubtly the importance of the (1 — P20) term. High nuclear polarization in low magnetic fields has not been achieved due to the phononbottleneck and the fact that the forbidden lines are not sufficiently separated from the electronic line. At high fields and with 126 kMHz irradiation, we obtained polarization as high as in LMN doped with Nd3+.Nous analysons ici les mesures de temps de relaxation et la polarisation dynamique des protons dans LMN dopé au Dy3+. Nous avons mesuré le temps de relaxation électronique pour les orientations parallèle et perpendiculaire du cristal par rapport au champ magnétique statique. Les résultats confirment l'existence de deux doublets X et Y dont le mélange par le champ magnétique statique varie avec l'orientation. Nous avons mesuré les temps de relaxation nucléaire pour différents champs et températures aux différentes orientations. La comparaison de ces résultats avec les théories de la relaxation nucléaire confirme la prédominance du terme (1 — P2 0). Des polarisations nucléaires très élevées en champs bas n'ont pas été réalisées, à cause de l'engorgement des phonons, et de ce que les raies interdites ne sont pas séparées suffisamment de la raie électronique. Pour des champs élevés, on atteint une polarisation égale à celle obtenue avec le Nd3+ dans LMN, mais il est nécessaire d'irradier à une fréquence de 126 kMHz

On quadratic two-parameter families of spheres and their envelopes

In the present paper we investigate rational two-parameter families of spheres and their envelope surfaces in Euclidean R3. The four dimensional cyclographic model of the set of spheres in R3 is an appropriate framework to show that a quadratic triangular Bézier patch in R4 corresponds to a two-parameter family of spheres with rational envelope surface. The construction shows also that the envelope has rational offsets. Further we outline how to generalize the construction to obtain a much larger class of surfaces with similar properties

Spectromètre pour la polarisation dynamique des protons dans un champ de 50 kilogauss

A description is given of an apparatus for the dynamic polarization of protons in a field of 50 kG. The frequencies of the electron and nuclear paramagnetic resonance spectrometers are ∼ 132 GHz and 202 MHz respectively. The magnetic field of 50 kG is created by a superconducting coil. The temperature of the sample is around 1 °K. The results of the polarization of protons in ethyleneglycol suggest the construction of a similar apparatus for an operating polarized proton target which can give the same performances as the polarized targets now working at 25 kG and ∼ 0,5 °K.Nous donnons la description d'un appareillage pour la polarisation dynamique des protons dans un champ de 50 kG. Les fréquences des spectromètres de résonance paramagnétique électronique et nucléaire sont respectivement ∼ 132 GHz et 202 MHz. Le champ magnétique de 50 kG est produit par une bobine supraconductrice. La température de l'échantillon est environ 1 °K. Les résultats de la polarisation des protons dans l'éthylène glycol suggèrent la construction d'un appareil similaire pour une cible opérationnelle de protons polarisés qui peut donner les mêmes performances que les cibles polarisées fonctionnant actuellement à 25 kG et T ∼ 0,50 °K

3D shape recognition and reconstruction based on line element geometry

This paper presents a new method for the recognition and reconstruction of surfaces from 3D data. Line element geometry, which generalizes both line geometry and the Laguerre geometry of oriented planes, enables us to recognize a wide class of surfaces (spiral surfaces, cones, helical surfaces, rotational surfaces, cylinders, etc.) by fitting linear subspaces in an appropriate seven-dimensional image space. In combination with standard techniques such as PCA and RANSAC, line element geometry is employed to effectively perform the segmentation of complex objects according to surface type. Examples show applications in reverse engineering of CAD models and testing mathematical hypotheses concerning the exponential growth of sea shells. 1