Unitary reflection groups for quantum fault tolerance

This paper explores the representation of quantum computing in terms of unitary reflections (unitary transformations that leave invariant a hyperplane of a vector space). The symmetries of qubit systems are found to be supported by Euclidean real reflections (i.e., Coxeter groups) or by specific imprimitive reflection groups, introduced (but not named) in a recent paper [Planat M and Jorrand Ph 2008, {\it J Phys A: Math Theor} {\bf 41}, 182001]. The automorphisms of multiple qubit systems are found to relate to some Clifford operations once the corresponding group of reflections is identified. For a short list, one may point out the Coxeter systems of type $B_3$ and $G_2$ (for single qubits), $D_5$ and $A_4$ (for two qubits), $E_7$ and $E_6$ (for three qubits), the complex reflection groups $G(2^l,2,5)$ and groups No 9 and 31 in the Shephard-Todd list. The relevant fault tolerant subsets of the Clifford groups (the Bell groups) are generated by the Hadamard gate, the $\pi/4$ phase gate and an entangling (braid) gate [Kauffman L H and Lomonaco S J 2004 {\it New J. of Phys.} {\bf 6}, 134]. Links to the topological view of quantum computing, the lattice approach and the geometry of smooth cubic surfaces are discussed.Comment: new version for the Journal of Computational and Theoretical Nanoscience, focused on "Technology Trends and Theory of Nanoscale Devices for Quantum Applications

International Migration, Transfers of Norms and Home Country Fertility

This paper examines the relationship between international migration and source country fertility. The impact of international migration on source country fertility may have a number of causes, including a transfer of destination countries’ fertility norms and an incentive to acquire more education. We provide a rigorous test of the diffusion of fertility norms using original and detailed data on migration. Our results provide evidence of a strong transfer of fertility norms from migrants to their country of origin.international migration, endogenous fertility, human capital, social norms

International migration, transfers of norms and home country fertility

This paper examines the relationship between international migration and source country fertility. The impact of international migration on source country fertility may have a number of causes, including a transfer of destination countries'fertility norms and an incentive to acquire more education. It provides provide a rigorous test of the diffusion on of fertility norms using original and detailed data on migration. The results provide evidence of a significant transfer of fertility norms from migrants to their country of origin: a one percent decrease in the fertility norm to which migrants are exposed reduces home country fertility by about 0.3 percent for origin countries.Population Policies,Gender and Social Development,Reproductive Health,Human Migrations&Resettlements,Anthropology

Brain Drain and its Determinants: A Major Issue for Small States

This paper examines the relationship between the brain drain and country size, as well as the extent of small states’ overall loss of human capital. We find that small states are the main losers because they i) lose a larger proportion of their skilled labor force and ii) exhibit stronger reactions to standard push factors. We also observe that the correlation between human capital indicators and country size is close to zero. This suggests that small states are more successful in producing skilled natives and less successful in retaining them.brain drain, small states, human capital, openness

International Migration, Transfers of Norms and Home Country Fertility

This paper examines the relationship between international migration and source country fertility. The impact of international migration on source country fertility may have a number of causes, including a transfer of destination countries' fertility norms and an incentive to acquire more education. We provide a rigorous test of the diffusion of fertility norms using original and detailed data on migration. Our results provide evidence of a strong transfer of fertility norms from migrants to their country of origin.international migration, endogenous fertility, human capital, social norms

A Classification of the Projective Lines over Small Rings

A compact classification of the projective lines defined over (commutative) rings (with unity) of all orders up to thirty-one is given. There are altogether sixty-five different types of them. For each type we introduce the total number of points on the line, the number of points represented by coordinates with at least one entry being a unit, the cardinality of the neighbourhood of a generic point of the line as well as those of the intersections between the neighbourhoods of two and three mutually distant points, the number of `Jacobson' points per a neighbourhood, the maximum number of pairwise distant points and, finally, a list of representative/base rings. The classification is presented in form of a table in order to see readily not only the fine traits of the hierarchy, but also the changes in the structure of the lines as one goes from one type to the other. We hope this study will serve as an impetus to a search for possible applications of these remarkable geometries in physics, chemistry, biology and other natural sciences as well.Comment: 7 pages, 1 figure; Version 2: classification extended up to order 20, references updated; Version 3: classification extended up to order 31, two more references added; Version 4: references updated, minor correctio

International Migration, Transfers of Norms and Home Country Fertility

This paper examines the relationship between international migration and source country fertility. The impact of international migration on source country fertility may have a number of causes, including a transfer of destination countries’ fertility norms and an incentive to acquire more education. We provide a rigorous test of the diffusion of fertility norms using original and detailed data on migration. Our results provide evidence of a strong transfer of fertility norms from migrants to their country of origin.International migration; endogenous fertility; human capital; social norms

Quantum Entanglement and Projective Ring Geometry

The paper explores the basic geometrical properties of the observables characterizing two-qubit systems by employing a novel projective ring geometric approach. After introducing the basic facts about quantum complementarity and maximal quantum entanglement in such systems, we demonstrate that the 15$\times$15 multiplication table of the associated four-dimensional matrices exhibits a so-far-unnoticed geometrical structure that can be regarded as three pencils of lines in the projective plane of order two. In one of the pencils, which we call the kernel, the observables on two lines share a base of Bell states. In the complement of the kernel, the eight vertices/observables are joined by twelve lines which form the edges of a cube. A substantial part of the paper is devoted to showing that the nature of this geometry has much to do with the structure of the projective lines defined over the rings that are the direct product of $n$ copies of the Galois field GF(2), with $n$ = 2, 3 and 4.Comment: 13 pages, 6 figures Fig. 3 improved, typos corrected; Version 4: Final Version Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA