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Formula for unbiased bases. (English) Zbl 1209.81049

Summary: 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.

MSC:

81P45 Quantum information, communication, networks (quantum-theoretic aspects)
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
81R15 Operator algebra methods applied to problems in quantum theory
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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References:

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