### A recurrence relation approach to higher order quantum superintegrability.

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The present paper deals with mutually unbiased bases for systems of qudits in $d$ dimensions. Such bases are of considerable interest in quantum information. A formula for deriving a complete set of $1+p$ mutually unbiased bases is given for $d=p$ where $p$ is a prime integer. The formula follows from a nonstandard approach to the representation theory of the group $SU\left(2\right)$. A particular case of the formula is derived from the introduction of a phase operator associated with a generalized oscillator algebra. The case...

We present a new proof of Janson’s strong hypercontractivity inequality for the Ornstein-Uhlenbeck semigroup in holomorphic algebras associated with CAR (canonical anticommutation relations) algebras. In the one generator case we calculate optimal bounds for t such that ${U}_{t}$ is a contraction as a map $L\u2082\left(\right)\to {L}_{p}\left(\right)$ for arbitrary p ≥ 2. We also prove a logarithmic Sobolev inequality.