Optimal state tomography by measuring the qubit of a qubit-qutrit system

Measuring only one qubit in a composite quantum system is described by projectors on half-dimensional subspaces of the full Hilbert space. Despite the limitation these measurements can be combined with unitary operations in order to perform full quantum state tomography (QST). Moreover, for a system of qubits, measurement operators can be arranged in a fashion optimal for QST, i.e. such that the corresponding subspaces are mutually unbiased [1].
Here, we consider a qubit-qutrit system, e.g. formed by the N-14 nuclear spin-1 and two states of the electron spin in a nitrogen-vacancy center in diamond where the electronic state could be measured using resonance fluorescence. We formulate the search for the most efficient QST scheme as a high-dimensional optimization problem and approach this problem numerically as we did previously for QST schemes with rank-1 measurement operators [2]. For the rank-3 operators considered here, we find that our numerical solution approximates the result of mutually unbiased subspaces.

[1] Bodmann, Haas, Proc. Amer. Math. Soc. 146, 2601 (2018)
[2] Ivanova-Rohling, Rohling, Phys. Rev. A 100, 032332 (2019)