Sensing quantities precisely is central to science and technology, from medical diagnostics to driverless vehicles. The fundamental limit to how precise any sensor can be is set by the so-called Holevo Cramér-Rao bound. Although the bound was first introduced in the 1970s by Alexander Holevo, one of the pioneers of quantum information theory, there is no explicit formula to compute it for generic quantum sensors. This is because its computation requires solving a non-linear optimisation problem. Yet, without this computation, we can never know how much any sensor can be improved.

In a new publication in Physical Review Letters (DOI:10.1103/PhysRevLett.123.200503) Francesco, Jamie, and Animesh present an efficient method to evaluate the Holevo Cramér-Rao bound for any quantum sensor. This method relies upon establishing that the optimisation problem involved is convex. Convexity is the property that the problem is shaped like a bowl such that its minimum lies at its bottom. This enables them to use established algorithms, called semi-definite programs, to solve the optimisation problem efficiently and compute the Holevo Cramér-Rao bound to the precision of sophisticated quantum sensors for the first time. In particular, they compute the bound for cases where weaker bounds are insufficient to reveal how precise quantum imaging and magnetometry can be in the real world.