Kybernetika 46 no. 6, 1122-1137, 2010

Formula for unbiased bases

Maurice R. Kibler


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 $SU(2)$. A particular case of the formula is derived from the introduction of a phase operator associated with a generalized oscillator algebra. The case when $d = p^e$ ($e \geq 2$), corresponding to the power of a prime integer, is briefly examined. Finally, complete sets of mutually unbiased bases are analysed through a Lie algebraic approach.


mutually unbiased bases, Weyl pairs, phase states, Lie algebras


81R05, 81R10, 81R15, 81R50


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