Exploring the Accuracy of Interferometric Quantum Measurements under Conservation Laws

A (target) quantum system is often measured through observations performed on a second (meter) system to which the target is coupled. In the presence of global conservation laws holding on the joint meter-target system, the Wigner-Araki-Yanase theorem and its generalizations predict a lower bound on the measurement's error (Ozawa's bound). While practically negligible for macroscopic meters, it becomes relevant for microscopic ones. Here, we propose a simple interferometric setup, arguably within reach of present technology, in which a flying particle (a microscopic quantum meter) is used to measure a qubit by interacting with it in one arm of the interferometer. In this scenario, the globally conserved quantity is the total energy of particle and qubit. We show how the measurement error µ is linked to the nonstationary nature of the measured observable and the finite duration of the target-meter interaction while Ozawa's bound µB only depends on the momentum uncertainty of the meter's wave packet. When considering short wave packets with respect to the evolution time of the qubit, we show that µ/ µB is strictly tied to the position-momentum uncertainty of the meter's wave packet and µ/ µB→1 only when employing Gaussian wave packets. On the contrary, long wave packets of any shape lead to µ/ µB→2. In addition to their fundamental relevance, our findings have important practical consequences for optimal resource management in quantum technologies.