# Measuring magnets: yes, you can do better with quantum

November 5, 2014

We have another new paper out on the arXiv today: https://arxiv.org/abs/1411.0716

This one is about measuring magnetic fields with very high precision by harnessing quantum effects. And more generally about showing that quantum effects can be useful for precision measurements even when there is some noise present.

Measuring magnetic fields precisely is useful for imaging brain activity as well as lots of other applications (see for example this list on Wikipedia https://en.wikipedia.org/wiki/Magnetometer#Uses). More broadly, precise estimation of parameters is fundamental in science. For example, the most precise clocks we have are atomic clocks which are based on measuring the frequency of an atomic transition. Another example is big experiments like LIGO and GEO600 which are looking for signs of gravitational waves. They split a laser beam in two, send the parts along different directions, and then look for a tiny phase difference between them when they are reflected back.

It has long been known that in the estimation of a phase or a frequency, in principle the precision can be improved a lot by harnessing quantum effects. Imagine that the parameter is estimated by using N probe particles. Classically, each of these probe particles sense the parameter independently. Taking the average from measurements on each particle, the uncertainty in your best estimate then goes down with the square root of N (this follows from a very general result about probabilities know as the Central Limit Theorem). One the other hand, if the N particles are prepared in a so-called entangled quantum state, then one can arrange it so that the estimate error goes down linearly with N. There is a quadratic improvement in precision. If there are N = 10^12 probe particles, as one might have for example in an atomic magnetometer, then this is an improvement in precision by a factor of one million!

The quadratic improvement, however, holds in an ideal case free of the noise and imperfections which are always present in real systems. More recently, researchers have shown that quite generally, as soon as you put a little bit of noise in the system, the quadratic improvement goes away. The estimate error you get with quantum probes goes down with the square root of N, like in the classical case. Quantum methods may still improve the precision by a constant factor, but it is not going to scale with N.

This is a bit disappointing. Fortunately though, there are some loopholes in the ‘quite generally’ of these no-go results. That is, there are cases which are not covered and where quantum may still give a scaling advantage. The question is, are these exceptions relevant in practice? And if so, how much of an advantage can you still get?

In our paper we show that one of the noise models which isn’t covered seems to apply well to an actual atomic magnetometry setup which was realised recently, and we show that with reasonable entangled states and measurements which can be done in the lab, one still gets a scaling quantum advantage. This is nice, because it shows that quantum techniques still have a large potential to improve precision measurements. Depending on experimental parameters, the improvement for the specific magnetometer could be anything from ten-fold to thousand-fold. It is also nice because even if there is a mismatch between our noise model and the noise in the real experiment, such that the no-go results do actually apply, if the mismatch is not too big, the constant (non-scaling) improvement that quantum techniques give can be large.

Published paper: https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031010