= Solution
In the $A$ <basis>, the transformations whose columns are the new <basis> <vectors> are
$$
U_B=\begin{pmatrix}1/2&\sqrt3/2\\\sqrt3/2&-1/2\end{pmatrix},
\qquad
U_C=\begin{pmatrix}1/2&\sqrt3/2\\-\sqrt3/2&1/2\end{pmatrix}.
$$
Thus $U_C$ is a rotation through $-\pi/3$, while $U_B$ is a real orthogonal reflection, equivalently a rotation with one basis-vector phase reversed.
For any real orthogonal <matrix> $U$ with columns $|u_j\rangle$,
$$
\sum_j|u_ju_j\rangle
=\sum_{j,k,l}U_{kj}U_{lj}|a_ka_l\rangle
=\sum_{k,l}(UU^T)_{kl}|a_ka_l\rangle
=\sum_k|a_ka_k\rangle.
$$
Applying this to $U_B$ and $U_C$ and normalizing proves
$$
\boxed{
\frac{|a_0a_0\rangle+|a_1a_1\rangle}{\sqrt2}
=\frac{|b_0b_0\rangle+|b_1b_1\rangle}{\sqrt2}
=\frac{|c_0c_0\rangle+|c_1c_1\rangle}{\sqrt2}.}
$$
Back to article page
