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The present paper deals with mutually unbiased bases for systems of qudits in dimensions. Such bases are of considerable interest in quantum information. A formula for deriving a complete set of mutually unbiased bases is given for where is a prime integer. The formula follows from a nonstandard approach to the representation theory of the group . A particular case of the formula is derived from the introduction of a phase operator associated with a generalized oscillator algebra. The case...
We introduce dynamical analogues of the free orthogonal and free unitary quantum groups, which are no longer Hopf algebras but Hopf algebroids or quantum groupoids. These objects are constructed on the purely algebraic level and on the level of universal C*-algebras. As an example, we recover the dynamical studied by Koelink and Rosengren, and construct a refinement that includes several interesting limit cases.
It is shown, using geometric methods, that there is a C*-algebraic quantum group related to any double Lie group (also known as a matched pair of Lie groups or a bicrossproduct Lie group). An algebra underlying this quantum group is the algebra of a differential groupoid naturally associated with the double Lie group.
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