Citation: Sohal, Ramanjit, and Abhinav Prem. "A noisy approach to intrinsically mixed-state topological order." arXiv preprint arXiv:2403.13879 (2024).
Web: https://arxiv.org/abs/2403.13879
Tags: Statistical-mechanics, Monoidal-categories
This paper gives a nice interpretation of mixed state topological order on the doubled Hilbert space. Namely, the argue that if you try to gauge out non-bosons on one layer, you can think of that as condensing the doubled version of those non-bosons on the doubled Hilbert space. Here's how this works. First, you choose the anyons you want to condense. Then you pass to the doubled Hilbert space, and gauge the doubled anyons. Then, you trace out the top layer. The two layers are highly entangled after condensation, so after tracing out the top layer the bottom layer should be fundamentally mixed.
The authors also claim a connection mixed-state TQFTs, as introduced in
> Zini, Modjtaba Shokrian, and Zhenghan Wang. "Mixed-state TQFTs." arXiv preprint arXiv:2110.13946 (2021).
Their connection is a bit speculative, though. In fact, they say this: "To that end, it would be prudent to understand more fully the connections with the mixed TQFTs proposed by Zini and Wang". There are other mixed-state type things one could do (namely, things Dan, Alexei, and I are doing) which also look like what Zini and Wang did. It's unclear which physical thing mixed state TQFTs correspond to.
Another reference to keep in mind when it comes up to mixed-state topological order and noise is
> Fan, Ruihua, et al. "Diagnostics of mixed-state topological order and breakdown of quantum memory." PRX Quantum 5.2 (2024)
In Fan et al.'s paper they explicitly demonstrate the decoherence induced phase transition in the toric code, and use that fact to argue that the current decoders known for the toric code are essentially optimal. There's also another related work (which is similar to both these papers):
> Bao, Yimu, et al. "Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions." arXiv preprint arXiv:2301.05687 (2023).