Asymptotics and typicality of sequential generalized measurements

The relation between projective measurements and generalized quantum measurements is a fundamental problem in quantum physics, and clarifying this issue is also important to quantum technologies. While it has been intuitively known that projective measurements can be constructed from sequential generalized or weak measurements, there is still lack of a proof of this hypothesis in general cases. Here we rigorously prove it from the perspective of quantum channels. We show that projective measurements naturally arise from sequential generalized measurements in the asymptotic limit. Specifically, a selective projective measurement arises from a set of typical sequences of sequential generalized measurements. We provide an explicit scheme to construct projective measurements of a quantum system with sequential generalized quantum measurements. Remarkably, a single ancilla qubit is sufficient to mediate a sequential weak measurement for constructing arbitrary projective measurements of a generic system. As an example, we present a protocol to measure the modular excitation number of a bosonic mode with an ancilla qubit.