**Waves carry information about their surroundings. An exact theory has now been developed at TU Wien - with astonishing results that can be used for technical purposes.**

No matter whether ultrasound is used to study the body, radar systems to study airspace or seismic waves to study the interior of our planet: You are always dealing with waves that are deflected, scattered or reflected by their surroundings. As a result, these waves carry a certain amount of information about their surroundings, and this information must then be extracted as comprehensively and precisely as possible.

The best way to do this has been the subject of research around the world for many years. TU Wien has now succeeded in describing the information that a wave carries about its surroundings with mathematical precision. This has made it possible to show how waves pick up information about an object and then transport it to a measuring device. This can now be used to generate tailor-made waves that can be used to elicit maximum information from the environment - for more precise imaging processes, for example. Microwave experiments have confirmed this theory. The results were published in the journal "Nature Physics".

## Where exactly is the information located?

"The basic idea is quite commonplace: you send a wave at an object and the part of the wave that is scattered back from the object is measured at a detector," says Stefan Rotter from the Institute of Theoretical Physics at TU Wien. "You can then learn something about the object from the measurement data - for example, its precise position, speed or size." This information about the environment that this wave carries with it is known as "Fisher information".

However, it is often not possible to capture the entire wave. Usually only part of the wave reaches the detector. This raises the question: Where exactly is this information actually located in the wave? Are there parts of the wave that can be safely ignored? Would a different waveform perhaps provide more information to the detector?

"To get to the bottom of these questions, we took a closer look at the mathematical properties of this Fisher information and came up with some astonishing results," says Stefan Rotter. "The information fulfills a so-called conservation equation - the information in the wave is preserved as it moves through space, according to very similar laws to those known for the conservation of energy, for example."

## A comprehensible path of information

Using the newly developed formalism, the research team has now been able to calculate exactly at which point in space the wave actually carries how much information about the object. It turns out that the information about different properties of the object (such as position, speed and size) can be hidden in completely different parts of the wave.

As the theoretical calculations show, the information content of the wave depends precisely on how strongly the wave is influenced by the object properties we are looking for. "For example, if we want to measure whether an object is a little further to the left or a little further to the right, then the Fisher information is carried precisely by the area of the wave that comes into contact with the right and left edges of the object," says Jakob Hüpfl, the doctoral student who played a key role in the study. This information then spreads out, and the more of this information reaches the detector, the more precisely the position of the object can be read from it."

## Microwave experiments confirm the theory

In Ulrich Kuhl’s group at the University of Cote d’Azur in Nice, experiments were carried out by Felix Russo as part of his master’s thesis: A disordered environment was created in a microwave chamber using randomly positioned Teflon objects. Between these objects was a metallic rectangle whose position was to be determined. Microwaves were sent through the system and then picked up by a detector. The question now was: How well can the position of the metal rectangle be deduced from the waves caught in the detector in such a complicated physical situation and how does the information flow from the rectangle to the detector?

By precisely measuring the microwave field, it was possible to show exactly how the information spreads across the horizontal and vertical position of the rectangle: it emanates from the respective edges of the rectangle and then moves along with the wave - without any information being lost - exactly as predicted by the newly developed theory.

## Possible applications in many areas

"This new mathematical description of Fisher information has the potential to improve the quality of very different imaging methods," says Stefan Rotter. If you can quantify where the desired information is located and how it propagates, you can, for example, position the detector appropriately or calculate customized waves that unload this information in the detector to the greatest possible extent.

"We tested our theory with microwaves, but it is equally valid for a wide variety of waves with arbitrary wavelengths," emphasizes Rotter. "We provide simple formulas that can be used to improve microscopy methods as well as quantum physical sensors."

## Original publication

Hüpfl, Russo, Rachbauer, Bouchet, Lu, Kuhl & Rotter, Nature Physics (2024), DOI: 10.1038/s41567’024 -02519-8

Freely available version: https://arxiv.org/abs/2309.00010