In this article, a star-product scheme for a full set of mutually unbiased bases is established.
Abstract. Mutually unbiased bases (MUBs) are considered within the framework of a generic star-product scheme. We rederive that a full set of MUBs is adequate for a spin tomography, i.e. knowledge of all probabilities to find a system in each MUB-state is enough for a state reconstruction. Extending the ideas of the tomographic-probability representation and the star-product scheme to MUB-tomography, dequantizer and quantizer operators for MUB-symbols of spin states and operators are introduced, ordinary and dual star-product kernels are found. Since MUB-projectors are to obey specific rules of the star-product scheme, we reveal the Lie algebraic structure of MUB-projectors and derive new relations on triple- and four-products of MUB-projectors. Example of qubits is considered in detail. MUB-tomography by means of Stern-Gerlach apparatus is discussed.
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Article details: S. N. Filippov, V. I. Man'ko, Mutually unbiased bases: tomography of spin states and star-product scheme, arXiv:1008.2675v1 [quant-ph]
Article is available online at http://arxiv.org/abs/1008.2675