Sequential generalized measurements: Asymptotics, typicality, and emergent projective 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 prove it from the perspective of quantum channels. We show that projective measurements naturally arise from sequential generalized measurements in the asymptotic limit, when the measurement operators are normal and commuting with each other. Specifically, a selective projective measurement arises from a set of typical sequences of selective generalized measurements. We also provide an explicit scheme to construct projective measurements of a quantum system with sequential generalized measurements. Remarkably, a single ancilla qubit is sufficient to mediate sequential generalized measurements for constructing arbitrary projective measurements of a generic system, which can have applications in readout, initialization, and feedback control of a quantum system. As an example, we present a protocol to measure the modular excitation numbers of a bosonic mode with an ancilla qubit.

Figure 1

(a) Schematic of sequential nonselective GMs and sequences of selective GMs in the asymptotic limit. (b) Emergent PMs arising from summation over the sets of typical sequences of selective GMs. (c) The emergent projections in the operator space of the quantum system.

Figure 2

(a) Quantum circuit diagram to realize sequential GMs on the target system with PMs of an ancilla qubit. (b) Distributions of eigenvalues of $U_{\pm }\left(t\right)=e^{\pm i\chi a^{†}at/2}={\sum }_{j=0}^{\infty }{\sum }_{l=0}^{2N-1}{e}^{\pm i{\omega }_{2jN+l}}|2jN+l\rangle \langle 2jN+l|$ in the complex unit circle to detect the $k$ mod $N$ excitation numbers of a bosonic mode with $t=2\pi /\left(N\chi \right)$ and $N=2$, 3.