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Full-Text Articles in Algebra
Information-Preserving Structures: A General Framework For Quantum Zero-Error Information, Robin Blume-Kohout, Hui Khoon Ng, David Poulin, Lorenza Viola
Information-Preserving Structures: A General Framework For Quantum Zero-Error Information, Robin Blume-Kohout, Hui Khoon Ng, David Poulin, Lorenza Viola
Dartmouth Scholarship
Quantum systems carry information. Quantum theory supports at least two distinct kinds of information (classical and quantum), and a variety of different ways to encode and preserve information in physical systems. A system’s ability to carry information is constrained and defined by the noise in its dynamics. This paper introduces an operational framework, using information-preserving structures, to classify all the kinds of information that can be perfectly (i.e., with zero error) preserved by quantum dynamics. We prove that every perfectly preserved code has the same structure as a matrix algebra, and that preserved information can always be corrected. We …
The Life Of Evariste Galois And His Theory Of Field Extension, Felicia N. Adams
The Life Of Evariste Galois And His Theory Of Field Extension, Felicia N. Adams
Senior Honors Theses
Evariste Galois made many important mathematical discoveries in his short lifetime, yet perhaps the most important are his studies in the realm of field extensions. Through his discoveries in field extensions, Galois determined the solvability of polynomials. Namely, given a polynomial P with coefficients is in the field F and such that the equation P(x) = 0 has no solution, one can extend F into a field L with α in L, such that P(α) = 0. Whereas Galois Theory has numerous practical applications, this thesis will conclude with the examination and proof of the fact that it is impossible …