Coherent numerical range

Definition

Coherent numerical range $W_{\mathrm{CS}}(A)$ of a matrix $A$ is a subset of the complex plane defined as $W_\mathrm{CS}(A)=\{z:z=\bra{\vartheta, \varphi}A\ket{\vartheta, \varphi},\braket{\vartheta, \varphi}{\vartheta, \varphi}=1\},$ where $\ket{\vartheta, \varphi}$ is an SU(2) coherent state.

Definition of $SU(2)$ coherent states

$SU(2)$ coherent states are related to the $SU(2)$ algebra of the components of the angular momentum operator $J = \{J_x , J_y , J_z \}$. Let us choose a reference state $\ket{\kappa}$, usually taken as the maximal eigenstate $\ket{j,j}$ of the component $J_z$ acting on $ \mathcal{H}_d $, $ d = 2j + 1 $, $ j = 1/2, 1/3, \ldots $. This state, pointing toward the ‘north pole’ of the sphere, enjoys the minimal uncertainty equal to $j$ . Then, the vector coherent state is defined by the Wigner rotation matrix $R_{\vartheta, \varphi}$

\[\ket{\vartheta,\varphi} = R_{\vartheta,\varphi} \ket{\kappa} = (1 + \|\gamma\|^2)^{−j} \ee^{\gamma J_-} \ket{j, j},\]

where $R_{\vartheta, \varphi} = \exp[(\ii \vartheta( \cos\varphi J_x − \sin\varphi J_y)]$, $J_- = J_x - \ii J_y$ and $\gamma=\tan(\vartheta/2) \ee^{\ii \varphi}$, for $(\vartheta, \varphi) \in S^2$ (we use the spherical coordinates).