744 research outputs found
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 -dimensional qudit is made, with 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 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)[]/ and Quantum Entanglement I. Theoretical Background
The paper deals with the projective line over the finite factor ring
GF(2)[]/. The line is endowed with 18
points, spanning the neighbourhoods of three pairwise distant points. As
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, , of the ring. The factor ring is isomorphic to GF(2)
GF(2). The projective line over 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 -dimensional qudit, 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 () and clock () 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 () the absolute efficiency of the harmonics
declines for the steepest plasma density scale-length , 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
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