# Numerical Shadow

The web resource on numerical range and numerical shadow

### Site Tools

numerical-range:generalizations:restricted-numerical-range:coherent-numerical-range

# 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 $R_{\vartheta, \varphi}$ $$\ket{\vartheta,\varphi} = R_{\vartheta,\varphi} \ket{\kappa} = (1 + |\gamma|^2)^{-j} e^{\gamma J_-} \ket{j, j}$$ where $R_{\vartheta,\varphi} = \exp[( \ii \vartheta( \cos\varphi J_x - \sin \varphi J_y)]$ , $J_- = J_x - \ii J_y$ and $\gamma = \tan(\vartheta/2)e^{\ii\varphi}$, for $(\vartheta,\varphi) \in S^2$ (we use the spherical coordinates). 