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Tight informationally complete quantum measurements

A J Scott

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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

03.65.Ta Foundations of quantum mechanics; measurement theory

02.30.Tb Operator theory

03.67.-a Quantum information

03.65.Wj State reconstruction, quantum tomography

02.10.Ud Linear algebra

MSC

81P15 Quantum measurement theory

81R15 Operator algebra methods (See also 46Lxx, 81T05)

Subjects

Mathematical physics

Computational physics

Quantum information and quantum mechanics

Dates

Issue 43 (27 October 2006)

Received 18 July 2006, in final form 5 September 2006

Published 11 October 2006



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