You can keep tabs on quantum particles inside a 'donut' of space just by watching a path that technically doesn't even exist.
arXiv · March 17, 2026 · 2603.15193
The Takeaway
Standard physics intuition suggests you need to sample a substantial portion of a region to know what is happening inside it. This mathematical proof shows that for certain quantum systems, an observer moving along an infinitesimally thin, 'zero-measure' line can gain 100% of the information about the entire system.
From the abstract
We prove that solutions of the toroidal Schr{รถ}dinger equation can be observed from suitably curved space-time trajectories, thus of zero Lebesgue measure. To do so, we establish new upper and lower bounds for certain trigonometric sums along curves, in the spirit of the celebrated Ingham inequality. In a second part, we establish observability properties over arbitrarily short curves of the low-and high-frequency components separately. For the low-frequency component, we establish strong restri