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arXiv:quant-ph/0508003 · International Journal of Modern Physics B

SIMPLE MINIMAL INFORMATIONALLY COMPLETE MEASUREMENTS FOR QUDITS

Stefan Weigert

Published May 20, 2006• 24 citations
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Abstract

Simple minimal but informationally complete positive operator-valued measures are constructed out of the expectation-value representation for qudits. Upon suitable modification, the procedure transforms any set of d2 linearly independent hermitean operators into such an observable. Minor changes in the construction lead to closed-form expressions for informationally complete positive measures in the spaces ℂd.

DOI
10.1142/s021797920603442x

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Simple minimal but informationally complete positive operator-valued measures are constructed out of the expectation-value representation for qudits. Upon suitable modification, the procedure transforms any set of d2 linearly independent hermitean operators into such an observable. Minor changes in the construction lead to closed-form expressions for informationally complete positive measures in the spaces ℂd.
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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 general quantum states. 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 members. Such measures share a particularly simple state reconstruction formula, allowing painless quantum state tomography, and are shown to be optimal for quantum cloning.
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.
We consider the existence in arbitrary finite dimensions d of a positive operator valued measure (POVM) comprised of d2 rank-one operators all of whose operator inner products are equal. Such a set is called a “symmetric, informationally complete” POVM (SIC–POVM) and is equivalent to a set of d2 equiangular lines in Cd. SIC–POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC–POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.