Qubit positive-operator-valued measurements by destructive weak measurements

Many quantum measurements, such as photodetection, can be destructive. In photodetection, when the detector ``clicks'' a photon has been absorbed and destroyed. Yet the lack of a click also gives information about the presence or absence of a photon. In monitoring the emission of photons from a source, one decomposes the strong measurement into a series of weak measurements, which describe the evolution of the state during the measurement process. Motivated by this example of destructive photon detection, a simple model of destructive weak measurements using qubits was studied in earlier work by the authors. It has shown that the model can achieve any positive-operator-valued measurement (POVM) with commuting POVM elements including projective measurements. In this paper, we use a different approach for decomposing any POVM into a series of weak measurements. The process involves three steps: randomly choose, with certain probabilities, a set of linearly independent POVM elements; perform that POVM by a series of destructive weak measurements; and output the result of the series of weak measurements with certain probabilities. The probabilities of the outcomes from this process agree with those from the original POVM, and hence this model of destructive weak measurements can perform any qubit POVM.