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

Vitamin A supplements for preventing mortality, illness, and blindness in children aged under 5: systematic review and meta-analysis

Objective To determine if vitamin A supplementation is associated with reductions in mortality and morbidity in children aged 6 months to 5 years

Effect of the hard magnetic inclusion on the macroscopic anisotropy of nanocrystalline magnetic-materials

It is shown that the presence of highly anisotropic magnetic precipitates in a soft multiphased matrix can produce a remarkable hardening, even when the volume fraction of the precipitates is small. The exchange coupling between the matrix and the precipitates is the relevant parameter. In particular, the simplified analysis we develop in this paper accounts for the magnetic hardening observed in very soft Fe-rich nanocrystals after annealing at higher temperatures

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

Size effects in the magnetic behaviour of TbAl_2 milled alloys

The study of the magnetic properties depending upon mechanical milling of the ferromagnetic polycrystalline TbAl_2 material is reported. The Rietveld analysis of the X-ray diffraction data reveals a decrease of the grain size down to 14 nm and -0.15 % of variation of the lattice parameter, after 300 hours of milling time. Irreversibility in the zero field cooled - field cooled (ZFC-FC) DC-susceptibility and clear peaks in the AC susceptibility between 5 and 300 K show that the long-range ferromagnetic structure is inhibited in favour of a disordered spin arrangement below 45 K. This glassy behaviour is also deduced from the variation of the irreversibility transition with the field (H^{2/3}) and frequency. The magnetization process of the bulk TbAl_2 is governed by domain wall thermal activation processes. By contrast, in the milled samples, cluster-glass properties arise as a result of cooperative interactions due to the substitutional disorder. The interactions are also influenced by the nanograin structure of the milled alloys, showing a variation of coercivity with the grain size, below the crossover between the multi- and single-domain behaviours.Comment: 23 pages, 11 figures, to appear in J. Phys.: Condens. Ma

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