82 research outputs found
Zariski density and computing with -integral groups
We generalize our methodology for computing with Zariski dense subgroups of
and , to accommodate
input dense subgroups of and . A key task, backgrounded by the Strong Approximation theorem, is
computing a minimal congruence overgroup of . Once we have this overgroup,
we may describe all congruence quotients of . The case receives
particular attention
The strong approximation theorem and computing with linear groups
We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group for . More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of SL(n,Q) for n > 2
Schmidt balls around the identity
Robustness measures as introduced by Vidal and Tarrach [PRA, 59, 141-155]
quantify the extent to which entangled states remain entangled under mixing.
Analogously, we introduce here the Schmidt robustness and the random Schmidt
robustness. The latter notion is closely related to the construction of Schmidt
balls around the identity. We analyse the situation for pure states and provide
non-trivial upper and lower bounds. Upper bounds to the random Schmidt-2
robustness allow us to construct a particularly simple distillability
criterion. We present two conjectures, the first one is related to the radius
of inner balls around the identity in the convex set of Schmidt number
n-states. We also conjecture a class of optimal Schmidt witnesses for pure
states.Comment: 7 pages, 1 figur
Freeness and -arithmeticity of rational M\"{o}bius groups
We initiate a new, computational approach to a classical problem: certifying
non-freeness of (-generator, parabolic) M\"{o}bius subgroups of
. The main tools used are algorithms for Zariski
dense groups and algorithms to compute a presentation of
for a localization of . We prove that
a M\"{o}bius subgroup is not free by showing that it has finite index in
the relevant . Further information about the structure of
is obtained; for example, we compute the minimal subgroup of finite index
in that contains
An ellipsoidal mirror for focusing neutral atomic and molecular beams
Manipulation of atomic and molecular beams is essential to atom optics applications including atom lasers, atom lithography, atom interferometry and neutral atom microscopy. The manipulation of charge-neutral beams of limited polarizability, spin or excitation states remains problematic, but may be overcome by the development of novel diffractive or reflective optical elements. In this paper, we present the first experimental demonstration of atom focusing using an ellipsoidal mirror. The ellipsoidal mirror enables stigmatic off-axis focusing for the first time and we demonstrate focusing of a beam of neutral, ground-state helium atoms down to an approximately circular spot, (26.8±0.5) ÎŒmĂ(31.4±0.8) ÎŒm in size. The spot area is two orders of magnitude smaller than previous reflective focusing of atomic beams and is a critical milestone towards the construction of a high-intensity scanning helium microscope
Unitarity as preservation of entropy and entanglement in quantum systems
The logical structure of Quantum Mechanics (QM) and its relation to other
fundamental principles of Nature has been for decades a subject of intensive
research. In particular, the question whether the dynamical axiom of QM can be
derived from other principles has been often considered. In this contribution,
we show that unitary evolutions arise as a consequences of demanding
preservation of entropy in the evolution of a single pure quantum system, and
preservation of entanglement in the evolution of composite quantum systems.Comment: To be submitted to the special issue of Foundations of Physics on the
occassion of the seventieth birthday of Emilio Santos. v2: 10 pages, no
figures, RevTeX4; Corrected and extended version, containing new results on
consequences of entanglement preservatio
Reflections upon separability and distillability
We present an abstract formulation of the so-called Innsbruck-Hannover
programme that investigates quantum correlations and entanglement in terms of
convex sets. We present a unified description of optimal decompositions of
quantum states and the optimization of witness operators that detect whether a
given state belongs to a given convex set. We illustrate the abstract
formulation with several examples, and discuss relations between optimal
entanglement witnesses and n-copy non-distillable states with non-positive
partial transpose.Comment: 12 pages, 7 figures, proceedings of the ESF QIT Conference Gdansk,
July 2001, submitted to special issue of J. Mod. Op
An ellipsoidal mirror for focusing neutral atomic and molecular beams
Manipulation of atomic and molecular beams is essential to atom optics applications including atom lasers, atom lithography, atom interferometry and neutral atom microscopy. The manipulation of charge-neutral beams of limited polarizability, spin or excitation states remains problematic, but may be overcome by the development of novel diffractive or reflective optical elements. In this paper, we present the first experimental demonstration of atom focusing using an ellipsoidal mirror. The ellipsoidal mirror enables stigmatic off-axis focusing for the first time and we demonstrate focusing of a beam of neutral, ground-state helium atoms down to an approximately circular spot, (26.8±0.5) ÎŒmĂ(31.4±0.8) ÎŒm in size. The spot area is two orders of magnitude smaller than previous reflective focusing of atomic beams and is a critical milestone towards the construction of a high-intensity scanning helium microscope
Characterizing Operations Preserving Separability Measures via Linear Preserver Problems
We use classical results from the theory of linear preserver problems to
characterize operators that send the set of pure states with Schmidt rank no
greater than k back into itself, extending known results characterizing
operators that send separable pure states to separable pure states. We also
provide a new proof of an analogous statement in the multipartite setting. We
use these results to develop a bipartite version of a classical result about
the structure of maps that preserve rank-1 operators and then characterize the
isometries for two families of norms that have recently been studied in quantum
information theory. We see in particular that for k at least 2 the operator
norms induced by states with Schmidt rank k are invariant only under local
unitaries, the swap operator and the transpose map. However, in the k = 1 case
there is an additional isometry: the partial transpose map.Comment: 16 pages, typos corrected, references added, proof of Theorem 4.3
simplified and clarifie
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