  We develop a new method for high-precision tomography of ion qubit registers
under conditions of limited distinguishability of its logical states. It is not
always possible to achieve low error rates during the readout of the quantum
states of ion qubits due to the finite lifetime of excited levels, photon
scattering, dark noise, low numerical aperture, etc. However, the model of
fuzzy quantum measurements makes it possible to ensure precise tomography of
quantum states. To do this, we developed a fuzzy measurement model based on
counting the number of fluorescent photons. A statistically adequate algorithm
for the reconstruction of quantum states of ion qubit registers based on fuzzy
measurement operators is proposed. The algorithm uses the complete information
available in the experiment and makes it possible to account for systematic
measurement errors associated with the limited distinguishability of the
logical states of ion qubits. We show that the developed model, although
computationally more complex, contains significantly more information about the
state of the qubit and provides a higher accuracy of state reconstruction
compared to the model based on the threshold algorithm.
