3,101 research outputs found
Two-letter words and a fundamental homomorphism ruling geometric contextuality
It has recently been recognized by the author that the quantum contextuality
paradigm may be formulated in terms of the properties of some subgroups of the
two-letter free group and their corresponding point-line incidence geometry
. I introduce a fundamental homomorphism mapping the
(infinitely many) words of G to the permutations ruling the symmetries of
. The substructure of is revealing the essence of geometric
contextuality in a straightforward way.Comment: 18 pages, 11 figures, 2 tables to appear in "Symmetry: Culture and
Science
Zoology of Atlas-groups: dessins d'enfants, finite geometries and quantum commutation
Every finite simple group P can be generated by two of its elements. Pairs of
generators for P are available in the Atlas of finite group representations as
(not neccessarily minimal) permutation representations P. It is unusual but
significant to recognize that a P is a Grothendieck's dessin d'enfant D and
that most standard graphs and finite geometries G-such as near polygons and
their generalizations-are stabilized by a D. In our paper, tripods P -- D -- G
of rank larger than two, corresponding to simple groups, are organized into
classes, e.g. symplectic, unitary, sporadic, etc (as in the Atlas). An
exhaustive search and characterization of non-trivial point-line configurations
defined from small index representations of simple groups is performed, with
the goal to recognize their quantum physical significance. All the defined
geometries G' s have a contextuality parameter close to its maximal value 1.Comment: 19 page
On finite groups acting on homology 4-spheres and finite subgroups of SO(5)
We show that a finite group which admits a faithful, smooth,
orientation-preserving action on a homology 4-sphere, and in particular on the
4-sphere, is isomorphic to a subgroup of the orthogonal group SO(5), by
explicitly determining the various groups which can occur (up to an
indetermination of index two in the case of solvable groups). As a consequence
we obtain also a characterization of the finite groups which are isomorphic to
subgroups of the orthogonal groups SO(5) and O(5).Comment: 13 page
The tame-wild principle for discriminant relations for number fields
Consider tuples of separable algebras over a common local or global number
field, related to each other by specified resolvent constructions. Under the
assumption that all ramification is tame, simple group-theoretic calculations
give best possible divisibility relations among the discriminants. We show that
for many resolvent constructions, these divisibility relations continue to hold
even in the presence of wild ramification.Comment: 31 pages, 11 figures. Version 2 fixes a normalization error: |G| is
corrected to n in Section 7.5. Version 3 fixes an off-by-one error in Section
6.
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 and (for
single qubits), and (for two qubits), and (for three
qubits), the complex reflection groups 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 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
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