A tiny, arbitrary window is all that is needed to see and understand the state of an entire quantum system.
Quantum systems are usually hard to track because you cannot see every part of them at once. This proof shows that the Schrödinger equation is observable from any nonempty open subset of a torus at any time. This means that looking through one small keyhole allows for the total determination of the system across its entire space. It resolves a long-standing conjecture about how information is distributed in quantum mechanics. This finding has huge implications for how we monitor and control quantum computers through limited sensors.
On the observability of the Schrödinger equation in the torus from open sets
arXiv · 2605.02480
We study the observability of the Schrödinger equation on the $d$-dimensional torus $\mathbb T^d$, $d \geq 1$, from an open subset $\omega \subset \mathbb T^d$. Our first main result establishes a quantitative observability estimate for the free Schrödinger equation in the regime of small times $T$ and for small observation sets of the form $\omega = \prod_{j=1}^{d}(a_j,b_j)$. Our second main result shows that observability holds for the Schrödinger equation with a merely bounded potential $V \i