Mutually unbiased bases: tomography of spin states and star-product scheme

TL;DR

This paper explores the use of mutually unbiased bases (MUBs) in spin state tomography, introducing a star-product scheme, and demonstrates how MUBs enable complete state reconstruction with concrete algebraic relations, exemplified by qubits.

Contribution

It develops a star-product framework for MUB-based spin tomography, including new operators, kernels, and algebraic relations, extending the tomographic-probability representation.

Findings

MUBs are sufficient for spin state reconstruction.

New relations for MUB-projectors are derived.

Star-product kernels for MUB-symbols are formulated.

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.