Measurement Incompatibility under Loss

We investigate the measurement incompatibility of continuous-variable systems with infinite-dimensional Hilbert spaces under the influence of pure losses, a fundamental noise source in quantum optics, and a significant challenge for long-distance quantum communication. We show that loss channels with transmissivities less than 1/n make any set of n measurements compatible. Additionally, we design a set of measurements that remains incompatible even under extreme losses, where the number of measurements in the set increases with the amount of loss. These measurements rely on on-off photodetectors and linear optics, making them feasible for implementation under realistic laboratory conditions. Furthermore, we demonstrate that no loss channel can break the incompatibility of all measurements. As a result, quantum steering remains achievable in the presence of pure loss.