Decomposing positive operator valued measurements on qubits into destructive weak measurements

Many quantum measurements are destructive. In an example of photodetection, when the detector "clicks" a photon is absorbed. In monitoring the emission of photons from a source, one decomposes the strong measurement into a series of weak measurements, which describes the evolution of the state during the process. Motivated by this destructive photodetection, we study a model of destructive weak measurements using qubits. A system qubit is coupled to a series of ancilla qubits, each prepared in the |0> state. The coupling performs a weak SWAP between the system and the ancilla, which is followed by a strong measurement of the ancilla. The weak measurement is destructive in the sense that the |0> state “leaks" into the system. Surprisingly, this destructive model can achieve any POVM on the system. Showing this requires a chain of arguments. First, any POVM with linearly dependent POVM elements is equivalent to choosing randomly among a set of POVMs with linearly independent elements. Each such POVM, in turn, is equivalent to a POVM whose elements are all proportional to projectors, followed by a probabilistic output step. Finally, any POVM whose elements are proportional to a linearly independent set of projectors can be achieved by a sequence of destructive weak measurements.