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.
2 Answers
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.
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$\begingroup$ Thanks. There is an overall ambiguous phase when we do this kind of transformation. Does that phase have some restrictions by imposing the pentagon condition from anyonic rules ? $\endgroup$– user56199Commented Jun 9, 2015 at 16:21
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1$\begingroup$ The basis transformation related different basis states. So one can always redefine the basis states by phases and mathematically this is the gauge transformation on $F$ symbols. There is an overall $\pm 1$ on the particular $F$, which is a gauge-invariant quantity that depends on the topological order. $\endgroup$ Commented Jun 9, 2015 at 16:36
The state you are writing is annihilated by $c_1+ic_2$ and $c_3+ic_4$. So you need to find a state in the new basis which has the same property.
If you do this, you should indeed find that this state is of the form $\vert0\rangle\vert1\rangle-i\vert1\rangle\vert0\rangle$ (or similar, depending on your conventions).
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$\begingroup$ Norbert, thanks. Total topological charge (which can be thought here as the parity) is conserved; so I think it can't go to the state you wrote, let's say, if we do measurements. $\endgroup$– user56199Commented May 27, 2015 at 21:13
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1$\begingroup$ It depends how you define your states w.r.t. to the old Majorana modes. Changing e.g. $c_1\pm ic_3$ will change $\vert0\rangle$ to $\vert1\rangle$. So I don't see why there should be a conserved quantity independent of your basis choice. -- In either case, that's the way should try, and see what you get with your convention. There should be a single state which satisfies both constraints. $\endgroup$ Commented May 27, 2015 at 21:17