Quantum State Tomography Via Sequential Uses of the Same Informationally Incomplete Measuring Apparatus
V. A. Zhuravlev, Sergey N. Filippov · Lobachevskii Journal of Mathematics · 2020
State of a $$d$$ -dimensional quantum system can only be inferred by performing an informationally complete measurement with $$m\geqslant d^{2}$$ outcomes. However, an experimentally accessible measurement can be informationally incomplete. Here we show that a single informationally incomplete measuring apparatus is still able to provide all the information about the quantum system if applied several times in a row. We derive a necessary and sufficient condition for such a measuring apparatus and give illustrative examples for qubits, qutrits, general $$d$$ -level systems, and composite systems of $$n$$ qubits, where such a measuring apparatus exists. We show that projective measurements and Lüders measurements with 2 outcomes are useless in the considered scenario.