Multicomplementary operators via finite Fourier transform

Details Multicomplementary operators via finite Fourier transform Multicomplementary operators via finite Fourier transform

2004-10-20

arxiv.org/abs/quant-ph/0410155v1

Journal of Physics A 38, 2747--2760 (2005)

Physics

Quantum Physics

15 pages, no figures

Scientific paper

10.1088/0305-4470/38/12/015

A complete set of d+1 mutually unbiased bases exists in a Hilbert spaces of dimension d, whenever d is a power of a prime. We discuss a simple construction of d+1 disjoint classes (each one having d-1 commuting operators) such that the corresponding eigenstates form sets of unbiased bases. Such a construction works properly for prime dimension. We investigate an alternative construction in which the real numbers that label the classes are replaced by a finite field having d elements. One of these classes is diagonal, and can be mapped to cyclic operators by means of the finite Fourier transform, which allows one to understand complementarity in a similar way as for the position-momentum pair in standard quantum mechanics. The relevant examples of two and three qubits and two qutrits are discussed in detail.

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