221 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
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
Modelle und Algorithmen fĂŒr mobile Datenobjekte und Umgebungen
Im ersten Teil dieser Arbeit werden Anforderungen an die interne ReprĂ€sentation der Bewegungen mobiler Objekte beschrieben und eine Modellierung eines Rahmenwerks vorgestellt, das diverse ReprĂ€sentationskonzepte fĂŒr Bewegungen integriert, um auch verschieden charakterisierte Bewegungen realistisch darstellen zu können; hierfĂŒr wurde eine umfassende, effiziente und verlĂ€ssliche Anfragebearbeitung, die auf verschiedensten BewegungsreprĂ€sentationen operieren kann, konzipiert. Im zweiten Teil dieser Arbeit wird die speichereffiziente Verarbeitung von geometrischen Daten betrachtet, wie sie in mobilen Umgebungen, in denen Datenverarbeitung von ressourcenbeschrĂ€nkten Kleinstrechnern durchzufĂŒhren ist, von Nutzen sein kann. Konkret werden speicher- und laufzeiteffiziente Algorithmen zum Berechnen von Ausgleichsgeraden, die fĂŒr die Aggregation von geometrischen und zeitvarianten Daten relevant sind, sowie fĂŒr die Berechnung der Maximamengen und -schichten von Punktmengen, die fĂŒr die prioritĂ€tsgesteuerte Datenselektion und -gruppierung eingesetzt werden können, prĂ€sentiert
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
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Comparison of the Accuracy and Applicability of Forebody Wake Effect Models for Parachute System Design
The forebody wake effect (FWE) is important to consider when designing parachute systems because it can affect parachute performance. Parachutes work by altering the aerodynamic properties of an attached forebody to control a descent. The FWE can reduce parachute drag, causing the system to descend faster than desired. This drag reduction coupled with wind or other factors can change the descent trajectory and landing impact speed. Modeling the FWE is important for ensuring that the parachute system descends and lands safely at the desired location. In the available literature there are three prominent FWE models for parachute system design. These models fall under two general modelling methods. The first method is to generate a statistical or empirical model based on a high number of full-scale experimental flights or tests. The second method is to create a case specific model with computational fluid dynamics(CFD). The goal of this paper is to explore the limitations and appropriate uses of the existing FWE models. Their applications and limitations were evaluated and compared in terms of modelling method, accuracy in determining drag reduction and breadth of situational applicability. The investigation showed that the models are applicable in specific design cases and vary in accuracy. The three models presented have different strengths and limitations, as expected. This review lays the foundation for developing a more comprehensive FWE model for parachute system design
Qudits of composite dimension, mutually unbiased bases and projective ring geometry
The Pauli operators attached to a composite qudit in dimension may
be mapped to the vectors of the symplectic module
( the modular ring). As a result, perpendicular vectors
correspond to commuting operators, a free cyclic submodule to a maximal
commuting set, and disjoint such sets to mutually unbiased bases. For
dimensions , and 18, the fine structure and the incidence
between maximal commuting sets is found to reproduce the projective line over
the rings , , ,
and ,
respectively.Comment: 10 pages (Fast Track communication). Journal of Physics A
Mathematical and Theoretical (2008) accepte
Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators
The commutation relations of the generalized Pauli operators of a
qubit-qutrit system are discussed in the newly established graph-theoretic and
finite-geometrical settings. The dual of the Pauli graph of this system is
found to be isomorphic to the projective line over the product ring Z2xZ3. A
"peculiar" feature in comparison with two-qubits is that two distinct
points/operators can be joined by more than one line. The multi-line property
is shown to be also present in the graphs/geometries characterizing two-qutrit
and three-qubit Pauli operators' space and surmised to be exhibited by any
other higher-level quantum system.Comment: 8 pages, 6 figures. International Journal of Theoretical Physics
(2007) accept\'
On Invariant Notions of Segre Varieties in Binary Projective Spaces
Invariant notions of a class of Segre varieties \Segrem(2) of PG(2^m - 1,
2) that are direct products of copies of PG(1, 2), being any positive
integer, are established and studied. We first demonstrate that there exists a
hyperbolic quadric that contains \Segrem(2) and is invariant under its
projective stabiliser group \Stab{m}{2}. By embedding PG(2^m - 1, 2) into
\PG(2^m - 1, 4), a basis of the latter space is constructed that is invariant
under \Stab{m}{2} as well. Such a basis can be split into two subsets whose
spans are either real or complex-conjugate subspaces according as is even
or odd. In the latter case, these spans can, in addition, be viewed as
indicator sets of a \Stab{m}{2}-invariant geometric spread of lines of PG(2^m
- 1, 2). This spread is also related with a \Stab{m}{2}-invariant
non-singular Hermitian variety. The case is examined in detail to
illustrate the theory. Here, the lines of the invariant spread are found to
fall into four distinct orbits under \Stab{3}{2}, while the points of PG(7,
2) form five orbits.Comment: 18 pages, 1 figure; v2 - version accepted in Designs, Codes and
Cryptograph
Performance of prototype BTeV silicon pixel detectors in a high energy pion beam
The silicon pixel vertex detector is a key element of the BTeV spectrometer.
Sensors bump-bonded to prototype front-end devices were tested in a high energy
pion beam at Fermilab. The spatial resolution and occupancies as a function of
the pion incident angle were measured for various sensor-readout combinations.
The data are compared with predictions from our Monte Carlo simulation and very
good agreement is found.Comment: 24 pages, 20 figure
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