@@ -267,7 +267,7 @@ eigenvalues of $\hat\sigma_z$, namely $V_+=(0,1)^\dagger$ and $V_-=(1,0)^\dagger
The matrix $\opkappa$ has the eigenvalues $\evp_\pm$,
and the same eigenvectors as $\eB\Pauli$.
We introduce the eigenvector matrix
\begin{equation}
\begin{equation}\label{DQofB}
\hat{Q}\coloneqq\left(V_+, V_-\right).
\end{equation}
Then $\opkappa$ has the eigenvalue decomposition
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@@ -455,7 +455,7 @@ This expression is evaluated via the eigenvalue decomposition
\UU{Q^\dagger_{a}}
\,,
\end{align}
where we have defined $\UU{\sigma'_a}=\left(\pi/2\right)^{3/2}\UU{\sigma_a}$ and the transformation matrix $\UU Q$ is the same as in equation \cref{eq:transformation_matrix_q}.
where we have defined $\UU{\sigma'_a}=\left(\pi/2\right)^{3/2}\UU{\sigma_a}$ and the transformation matrix $\UU Q$ is the same as in equation \cref{DQofB}.
The case of zero magnetic field $\vec e_b =0$ needs to be treated separately again, in that case we have
