# Jones index, data hiding and total quantum dimension

The paper *Jones index, data hiding and total quantum dimension*, together with Leander Fiedler and Tobias Osborne is available on the arXiv. We started this project already a few years ago, so I am happy that after changing the scope a few times and finding better ways to explain things, the paper is finaly ready. Below is a high level summary of the paper, explaining why we think it is interesting.

In the paper we study topologically ordered states in the thermodynamic limit. This has significant advantages over the more prevalent finite dimensional approach, and opens up the possibility of using deep results only available in infinite dimensions, as we demonstrate. We illustrate this by a data hiding scheme and connect these concepts to more familiar quantities such as topological entanglement entropy. This way we are able to give a new interpretation of the quantum dimension. We expect our approach to open up new possibilities in the study of topological phases.

One of the key points of our work is that it forms a bridge between different fields, from quantum information and topologically ordered condensed matter systems in physics, to operator algebra and index theory in pure mathematics. These are quite distinct fields, so a major goal of our paper is to communicate our results in a way that is accessible to researchers working in any of these fields. To achieve this, we have included brief discussions of known results and techniques, along with illustrating how they are related to our new results. We first illustrate the main intuition behind our results in a finite dimensional setting, and explain which complications arise there, and how they can be resolved by going to the thermodynamic limit.

The starting point of our paper is the observation made by one of us earlier that the total quantum dimension, an invariant related to the anyonic excitations of the system, can be obtained by considering the Jones index (or rather, its generalisation) of an inclusion of operator algebras. What is new in the present paper is that we reinterpret the quantum dimension (again, in the thermodynamic limit setting) in terms of a secret sharing scheme. This interpretation can then be made precise using properties of the index. In particular, it is possible to exactly quantify the amount of information that can he hidden in this way, and to identify a quantum channel. This uses deep results on the Jones index, that at the same time can be given a natural physical interpretation in our context.

To summarise, our work sheds new light on the total quantum dimension. Using completely different methods than considered before in the literature, we in the end obtain similar interpretations. We expect that these new methods will be useful in the further study of topologically ordered systems, for one since it allows us to tap into a vast wealth of mathematical results. At the same time, we hope it will be a useful and accessible introduction to the relatively unknown (but in our opinion extremely useful) operator algebraic methods.