Browsing by Subject "quantum"
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Item On simple modules for certain pointed Hopf algebras(Texas A&M University, 2007-04-25) Pereira Lopez, MarianaIn 2003, Radford introduced a new method to construct simple modules for the Drinfel??????d double of a graded Hopf algebra. Until then, simple modules for such algebras were usually constructed by taking quotients of Verma modules by maximal submodules. This new method gives a more explicit construction, in the sense that the simple modules are given as subspaces of the Hopf algebra and one can easily find spanning sets for them. I use this method to study the representations of two types of pointed Hopf algebras: restricted two-parameter quantum groups, and the Drinfel??????d double of rank one pointed Hopf algebras of nilpotent type. The groups of group-like elements of these Hopf algebras are abelian; hence, they fall among those Hopf algebras classified by Andruskiewitsch and Schneider. I study, in particular, under what conditions a simple module can be factored as the tensor product of a one dimensional module with a module that is naturally a module for a special quotient. For restricted two-parameter quantum groups, given ???? a primitive ??????th root of unity, the factorization of simple u????y,????z (sln)-modules is possible, if and only if gcd((y ?????? z)n, ??????) = 1. I construct simple modules using the computer algebra system Singular::Plural and present computational results and conjectures about bases and dimensions. For rank one pointed Hopf algebras, given the data D = (G, ????, a), the factorization of simple D(HD)-modules is possible if and only if |????(a)| is odd and |????(a)| = |a| = |????|. Under this condition, the tensor product of two simple D(HD)-modules is completely reducible, if and only if the sum of their dimensions is less or equal than |????(a)| + 1.Item Tubulin in vitro, in vivo and in silico(Texas A&M University, 2005-02-17) Mershin, AndreasTubulin, microtubules and associated proteins were studied theoretically, computationally and experimentally in vitro and in vivo in order to elucidate the possible role these play in cellular information processing and storage. Use of the electric dipole moment of tubulin as the basis for binary switches (biobits) in nanofabricated circuits was explored with surface plasmon resonance, refractometry and dielectric spectroscopy. The effects of burdening the microtubular cytoskeleton of olfactory associative memory neurons with excess microtubule associated protein TAU in Drosophila fruitflies were determined. To investigate whether tubulin may be used as the substrate for quantum computation as a bioqubit, suggestions for experimental detection of quantum coherence and entanglement among tubulin electric dipole moment states were developed.Item Upper bounds on minimum distance of nonbinary quantum stabilizer codes(Texas A&M University, 2005-11-01) Kumar, SantoshThe most popular class of quantum error correcting codes is stabilizer codes. Binary quantum stabilizer codes have been well studied, and Calderbank, Rains, Shor and Sloane (July 1998) have constructed a table of upper bounds on the minimum distance of these codes using linear programming methods. However, not much is known in the case of nonbinary stabilizer codes. In this thesis, we establish a bridge between selforthogonal classical codes over the finite field containing q2 elements and quantum codes, extending and unifying previous work by Matsumoto and Uyematsu (2000), Ashikhmin and Knill (November 2001), Kim and Walker (2004). We construct a table of upper bounds on the minimum distance of the stabilizer codes using linear programming methods that are tighter than currently known bounds. Finally, we derive code construction techniques that will help us find new codes from existing ones. All these results help us to gain a better understanding of the theory of nonbinary stabilizer codes.