I have the following question coming from quantum information theory. Consider linear space over $C$ of dimension 3, and a tensor product of three such spaces. So, there are 27 basis vectors of the following form:
$(i,j,k)$, where $i,j,k=0,1,2. $
Let’s take a local unitary matrix, i.e. a tensor product of three unitary matrices $U= U_1 \otimes U_2 \otimes U_3 $ which preserves the subspace generated by three following vectors:
$s_1 = (0,0,0) + (1,1,1) + (2,2,2)$,
$s_2 = (0,1,2) + (1,2,0) + (2,0,1)$,
$s_1 = (0,2,1) + (1,0,2) + (2,1,0)$.
Show, that $U_1 , U_2 , U_3 $ are permutation matrices.
In fact, it is enough to show, that $U= U_1 \otimes U_2 \otimes U_3 $ has to permute vectors $s_1, s_2, s_3$. The rest will follow from this fact.