Citation: Liu, Shang. "Anyon quantum dimensions from an arbitrary ground state wave function." Nature Communications 15.1 (2024): 5134.
Web: https://arxiv.org/abs/2304.13235
Tags: MTC-reconstruction, Information-theory
This paper gives a new protocol for obtaining anyon quantum dimensions from a ground state wavefunction. The protocol goes as follows. If you have access to a wavefunction of a topological order on the torus with vacuum charge along one axis, then computing its Renyi entropies across two disjoint sectors of the torus gives a value which depends on the index of the Renyi entropy. Computing this Renyi entropy for many values gives enough information to recover the quantum dimensions.
Of course, the most naive protocol to test quantum dimensions is to excite the system. It will generate some mixture of anyons, and the probability that it creates an anyon of type a is exactly d_a/D^2. This gives a way of measuring d_a. Alternatively, given a way of creating specific anyons then creating two pairs and recoding the probability of annihilation gives you the quantum dimension as well.
This protocol is thus useful in a regime where you cannot more anyons or measure anyon type even, but you CAN create a trial wavefunction on the torus with a specific charge along its meridian, perform tomography to recover its associated wavefunction amplitudes, and then numerically compute its entropy between two different regions. The way Shang Liu proposes to do this is to create a state on a large disk, couple it to a second large disk with the complex conjugate topological order, and then to perform Bell-type measurements to mimic a gapped boundary on the interface between the disks. This new state is like the state on a torus.
All in all, this paper does not seem very practically relevant in the arena of MTC reconstruction. In particular, it is able to reconstruct anyon dimensions but it does so assuming a-priori a field theory description and using that to make calculations.