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The present work is devoted to the development of a method for high-precisiontomography of ion qubit registers under conditions of limiteddistinguishability of the states of a logical value 0 and a logical value 1. Inthe considered ion qubits, the identification of the quantum state is achievedby measuring the fluorescence of the ion by repeated excitation of the cyclictransition, which includes only the lower energy state that sets a logicalvalue 0 and becomes bright, but does not include the upper metastable statethat remains dark and sets logical value 1. It is important to note that it isnot always possible to achieve low levels of registration errors due to thefinite lifetime of excited levels, photon scattering, dark noise, low numericalaperture values, etc. However, even under such conditions, with use of themodel of fuzzy quantum measurements, it is possible to provide precise controlof quantum states. We show that a model that is characterized by relativelyhigh levels of errors under conditions, where we have a reliable statisticalmodel of their occurrence, is more accurate than the case when the considerederrors are small, but we do not have an adequate statistical model for theoccurrence of these errors. In the given illustrative examples, we show thatthe factor of reducing the loss of accuracy with the use of the model of fuzzymeasurements can reach values of the order of 1000 or more in comparison withstandard measurements. The obtained results are essential for the developmentof high-precision methods for controlling the technology of quantum computingon the ion platform.

High-fidelity tomography of fluorescent ion qubits under conditions of limited discrimination between bright and dark levels