'schrodinger' picture in measurement based topological quantum computation

I am looking at the measurement processes in topological quantum computation (TQC) as mentioned here http://arxiv.org/abs/1210.7929 and in other measurement based TQC papers. Let's say I start with pairs of Majorana fermions 1+2 and 3+4 and both pairs have zero topological charge to begin with such that I can write the state $\left|0\right\rangle _{12}\left|0\right\rangle _{34}$. Suppose now I want to write this in a different basis where 1 and 3 form one pair and 2 and 4 one pair. I think I could write this as $\alpha \left|0\right\rangle _{13}\left|0\right\rangle _{24} +\beta \left|1\right\rangle _{13}\left|1\right\rangle _{24}$ but how do I determine $\alpha$ and $\beta$ ? I want to work in this picture because it looks simpler instead of following anyonic rules.

Answer

For four Majorana zero modes, if the total topological charge is $1$ there are two states $|0\rangle_{12}0\rangle_{34}$ and $|1\rangle_{12}|1\rangle_{34}$ ($i\gamma_1\gamma_2\cdot i\gamma_3\gamma_4=1$. So this system can be mapped to a qubit, with $i\gamma_1\gamma_2=\sigma_z, i\gamma_1\gamma_3=\sigma_x, i\gamma_2\gamma_3=\sigma_y$ (I did not check the signs carefully). Now what you want is just to do a basis transformation and rewrite the state in a basis which diagonalizes $i\gamma_1\gamma_3=\sigma_x$, which should be rather straightforward.

More generally, this kind of basis transformation is encoded in the $F$ symbols of the anyon model.