QuantumTopologicalOrder 2009-04-11 11:09:08 2009-04-11 11:45:06 Topic quantum topological order high temperature superconductors quantum algebroid quantum double groupoid quantum double groupoid representations Anderson localization delocalized quantum states edge quantum states QTO QES topological order spontaneous symmetry breaking Mott-Anderson transition Mott-Hubbard transition glass-transition Landau symmetry-breaking long-range coupling quantum extended symmetry topology order topolgical order symmetry quantum topological order \section{Quantum Topological Order and Extended Quantum Symmetries} In noncrystalline systems and certain quantum (Hall) liquids with long-range coupling symmetry-breaking descriptions of phase transitions were suggested to be insufficient and alternative theories in terms of topological order were proposed to replace previous Landau symmetry-breaking models. Such glassy systems with short-range structural order, but long-range correlations in magnetic and/or electrical properties may thus exhibit several topological orders both in hower and higher dimensions. Quantum states with different topological orders can be interchanged only through a phase transition-- a result that should be provable by means of quantum algebraic topology means in terms of quantum operator algebras and locally compact quantum groupoid representations \cite{QNAT2k9}. A related concept is that of ``quantum glassiness'' \cite{CC2k5} which incorporates many concepts from topological order theories. A basic concept in topological order theories is that of an ordered, entangled ground state for a many body system with long-range coupling(s) (as for example magnetic dipole-dipole coupled ferromagnets, high or low temperature superconductors, and so on). Therefore, \emph{quantum topological order (QTO)} can be described as a pattern of {\em long-range quantum entanglement in quantum states}, and it can be classified as an {\em extended quantum symmetry} in terms of categorical representations, categorical groups, locally compact quantum groupoid representations, braided tensor categories/monoidal categories, quantum algebroids or quantum double groupoid representations. Topological order theories and topological quantum computation were also recently reported to be of interest for the design of quantum computers \cite{QNAT2k9}, and thus such fundamental topological order theories might conceivably lead to practical applications in developing ultra-fast quantum supercomputers. \begin{thebibliography}{99} \bibitem{LMWX2k5} Levin M. and Wen X-G., Colloquium: Photons and electrons as emergent phenomena, Rev. Mod. 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