Quantum state tomography via sequential uses of the same informationally incomplete measuring apparatus

Abstract: 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 L\"{u}ders measurements with 2 outcomes are useless in the considered scenario.