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arXiv:2108.04191 · Journal of Physics A Mathematical and Theoretical

MUB-like structures and tomographic reconstruction for N -ququart systems

Juan Jesús Jesús Díaz Guevara, Isabel Sainz, A. B. Klimov

Published Jun 3, 2021• 2 citations
arXivHTMLPDF

Abstract

Abstract We construct an informationally complete set of mutually unbiased—like bases for any number of ququarts. These bases are used in an explicit tomographic protocol which performance is analysed by estimating quadratic errors and compared to other reconstruction schemes.
DOI
10.1088/1751-8121/ac07ea

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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. An example of qubits is considered in detail. MUB tomography by means of the Stern–Gerlach apparatus is discussed.
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The standard construction of complete sets of mutually unbiased bases (MUBs) in prime power dimensions is based on the quadratic Gauss sums. We introduce complete MUB sets for three, four, and five qubits that are unitarily inequivalent to all existing MUB sets. These sets are constructed by using certain exponential sums, where the degree of the polynomial appearing in the exponent can be higher than 2. Every basis of these MUBs (except the computational) consists of two disjoint blocks of vectors with different factorization structures and associated with a unique hypergraph (or graph) that represents an interaction between the qubits.