Abstract. We introduce a class of informationally complete positive-operator-valued measures which are, in analogy with a tight frame, 'as close as possible' to orthonormal bases for the space of quantum states. These measures are distinguished by an exceptionally simple state-reconstruction formula which allows 'painless' quantum state tomography. Complete sets of mutually unbiased bases and symmetric informationally complete positive-operator-valued measures are both members of this class, the latter being the unique minimal rank-one members. Recast as ensembles of pure quantum states, the rank-one members are in fact equivalent to weighted 2-designs in complex projective space. These measures are shown to be optimal for quantum cloning and linear quantum state tomography.

PACS numbers: 03.65.Wj, 03.67.−a, 02.10.Ud

Print publication: Issue 43 (27 October 2006)
Received 18 July 2006, in final form 5 September 2006
Published 11 October 2006