# Computing with qutrits

I'm doing some calculations with qutrits and I need a unitary matrix $$U$$ that does the following:

$$U|00\rangle = |12 \rangle - | 21\rangle$$

$$U|11\rangle = |20 \rangle - | 02\rangle$$

$$U|22\rangle = |01 \rangle - | 10\rangle$$

where $$|0\rangle$$,$$|1\rangle$$, $$|2\rangle$$ are the basis states. I don't care how it acts on the other states as long as it acts in the way prescribed earlier. Until now I've been using ad-hoc methods for finding such unitaries. In summary, I require two things: the unitary that fulfills the conditions mentioned above and if possible some general method to find unitaries that perform certain transformations over qutrits or qudits.

## 1 Answer

Simply find any set of normalised vectors that are mutually orthogonal and orthogonal to all your target vectors. Then define these to be the outputs for all the basis states whose outputs are not already defined.

In practice, the easiest thing is probably to put the output vectors as rows on a non-square matrix and compute the null space. That’ll give you a suitable set of vectors. Not all code necessarily orthogonalises the vectors in the null space (depends what you’re using) but, if not Gram-Schmidt will do the job for you.

Of course this method makes no attempt for find a separable decomposition if one exists, but you suggested that you don’t care about that.