Zariski density and computing with SS-integral groups

We generalize our methodology for computing with Zariski dense subgroups of $\mathrm{SL}(n, \mathbb{Z})$ and $\mathrm{Sp}(n, \mathbb{Z})$, to accommodate input dense subgroups $H$ of $\mathrm{SL}(n, \mathbb{Q})$ and $\mathrm{Sp}(n, \mathbb{Q})$. A key task, backgrounded by the Strong Approximation theorem, is computing a minimal congruence overgroup of $H$. Once we have this overgroup, we may describe all congruence quotients of $H$. The case $n=2$ 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 SS-arithmeticity of rational M\"{o}bius groups

We initiate a new, computational approach to a classical problem: certifying non-freeness of ($2$-generator, parabolic) M\"{o}bius subgroups of $\mathrm{SL}(2,\mathbb{Q})$. The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of $\mathrm{SL}(2, R)$ for a localization $R= \mathbb{Z}[\frac{1}{b}]$ of $\mathbb{Z}$. We prove that a M\"{o}bius subgroup $G$ is not free by showing that it has finite index in the relevant $\mathrm{SL}(2, R)$. Further information about the structure of $G$ is obtained; for example, we compute the minimal subgroup of finite index in $\mathrm{SL}(2,R)$ that contains $G$

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) μm×(31.4±0.8) μm 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) μm×(31.4±0.8) μm 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