  The present work is devoted to the development of a method for high-precision
tomography of ion qubit registers under conditions of limited
distinguishability of the states of a logical value 0 and a logical value 1. In
the considered ion qubits, the identification of the quantum state is achieved
by measuring the fluorescence of the ion by repeated excitation of the cyclic
transition, which includes only the lower energy state that sets a logical
value 0 and becomes bright, but does not include the upper metastable state
that remains dark and sets logical value 1. It is important to note that it is
not always possible to achieve low levels of registration errors due to the
finite lifetime of excited levels, photon scattering, dark noise, low numerical
aperture values, etc. However, even under such conditions, with use of the
model of fuzzy quantum measurements, it is possible to provide precise control
of quantum states. We show that a model that is characterized by relatively
high levels of errors under conditions, where we have a reliable statistical
model of their occurrence, is more accurate than the case when the considered
errors are small, but we do not have an adequate statistical model for the
occurrence of these errors. In the given illustrative examples, we show that
the factor of reducing the loss of accuracy with the use of the model of fuzzy
measurements can reach values of the order of 1000 or more in comparison with
standard measurements. The obtained results are essential for the development
of high-precision methods for controlling the technology of quantum computing
on the ion platform.
