*Citation:* Bonderson, Parsa. "Measuring topological order." Physical Review Research 3.3 (2021): 033110.

*Web:* https://arxiv.org/abs/2102.05677

*Tags:* Expository, Hardware, Modular-tensor-categories, Pedagogical

In this paper, Parsa Bonderson gives a lovely overview of the ability of experimenters to observe the data of a unitary modular tensor category. This paper brings to light a lot of folklore reasoning about the results of doing physical experiments on MTCs and is extremely clearly written. This is perhaps the best source out there for mathematically-minded people trying to get physical insight about what MTCs mean.

A good point to remember, that this article really stresses, is that topological order is not just determined by a UMTC. It is determined by a UMTC along with a choice of chiral central charge. It is known that this chiral central charge quantity is measured by the Hall conductance:

> Kane, C. L., and Matthew PA Fisher. "Quantized thermal transport in the fractional quantum Hall effect." Physical Review B 55.23 (1997): 15832.

To get detailed information about which UMTC describes your topological order, you need to get more creative. One of the main ways is to make a topological computer and see what you get. That is, braid particles around and observe the results of fusion. This allows you to get most of the data of your MTC. However, it doesn't get you everything. In particular, you can get the absolute values of every F-symbol but you can't get the phases.

To get phases you need to resort to experiments which can detect phases, like interferometry. This interferometer poses some difficulty. For one, you need to allow particles to travel in a superposition of different possible trajectories. If all of your particles are being carefully controlled, this is impossible. Also, to make an interferometry experiment work your quasiparticles need to be brought very close to each other, but at such levels of proximity non-topological effects might start to dominate so your data won't measure what you want it to. These are all things to keep in mind. Interferometry isn't impossible though, and it has been performed in various experiments to mixed success.

This paper is also a very good reference to the fact that the direct sum in MTCs can be thought of as a classical probability distribution. That is, if you pair create a/a^* and b/b^* at separate locations and then bring a and b together, the probability that a and b will fuse into c is N^{a,b}_c d_c/(d_a d_b). This computation follows directly from the definition of projective charge measurement, and is a good first exercise for somebody which wants to go from MTC data to observable probabilities.

A great insight from this paper is that performing variations of the above experiment can let you know things about how robust your topological order. Namely, repeating the charge measurement you can see whether or the result will change, and if it does change that means that the energy splitting between the fusion channels is showing up. This allows you to measure the splitting between fusion channel energy levels. Performing this experiment at various spacial separations can let you know the correlation distance of your sample.

Another good insight highlighted in this paper is that the fractionalized electric charge of a FQHE anyon can be related to MTC invariants. In particular, if an anyon X has charge Q, then there exists an anyon q such that braiding q around X results in the phase exp(2*pi *i *Q) up to proper normalization. This is a fantastic way to connect the braiding phases of MTCs to charge fractionalization in the FQHE.

This paper also demonstrates a well-known adage that measurement-only braiding is not as good as true transport braiding. This is because the map isn't from X \otimes Y \to Y\otimes X, but instead from X\otimes Y\to Y \otimes X' where X' is an object isomorphic to X. The non-canonical-ness of the isomorphism X\to X' loses crucial phase information.

Overall, this is a fantastic paper which I highly recommend.