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"Mixed-state TQFTs", Modjtaba Shokrian Zini, Zhenghan Wang, 2021

Reviewed August 31, 2023

Citation: Zini, Modjtaba Shokrian, and Zhenghan Wang. "Mixed-state TQFTs." arXiv preprint arXiv:2110.13946 (2021).

Web: https://arxiv.org/abs/2110.13946

Tags: Mathematical, TQFT

This paper introduces a generalization of TQFTs, known as mixed-state TQFTs. The point is very simple. A TQFT assigns to each topological space a Hilbert space. In other words, a space whose states are all pure. In practice, every system is noisy and hence the image should contain mixed states. These functors which allow for mixed states are known as mixed state TQFTs.

The most interesting point of this paper is that it helps give a philosophical understanding of the Drinfeld center. There is a natural connection V_TV = V_RT \otimes V_RT^* where V_TV is the Hilbert space of the Turaev-Viro TQFT and V_RT is the vector space of the Reshetikhin-Turaev TQFT. In particular, this means that tracing out V_RT^* we naturally get a mixed state TQFT. The elementary quantum double procedure sending mixed states on V to pure states on V\otimes V^* naturally inverses this process. Hence, the relationship between TV and RT is exactly that of the elementary quantum double.

However, we also know from

> Balsam, Benjamin. "Turaev-Viro invariants as an extended TQFT III." arXiv preprint arXiv:1012.0560 (2010).

That the TV and RT TQFTs are related by a Drinfeld center. Hence, the Drinfeld center is the categorification of the elementary quantum double procedure.