Dip your finger into water once, and from that single sensation, try to guess its temperature. In the quantum world, such one-shot measurements happen all the time, except you have to guess invisible parameters: say, the squeezing strength of a laser beam. A mistake can be costly, and there is barely any data to be confident about.

For a long time, the mathematics of such problems felt like assembling a puzzle in the dark. Integrals sprawled out of control, and even powerful computers stalled. The authors of the new work found a workaround. Instead of dealing with any conceivable measurement, they proposed to work only with those described by polynomials in the “coordinates” and “momenta.” This is like giving a chef a limited set of knives, but honed to perfection. In this restricted world, the problem becomes linear and is solved by a single simple formula.

The scientists derived a clear condition under which the shadow coincides with the ideal. In practice, the method suggests using standard Gaussian operations, those used in optical chips, and measuring “quadratures,” that is, simultaneously measuring the amplitude and phase of the wave. And if, instead of the estimate dictated by the formula, you take a simple average over the remaining uncertainty, the result gets even closer to the theoretical limit. Even with a single detector click, you can squeeze out almost all the available information.