Real numerical shadow
Definition
Real numerical shadow of a matrix $A$ of dimension $d$ is defined as a probability ditribution $P_A(z)$ on the complex plane, supported on the real numerical range $W^\mathbb{R}(A)$. $P_A(z) := \int_{\Omega} {\rm d} \mu(\psi) \delta\Bigl( z-\langle \psi|A|\psi\rangle\Bigr),$ where $\mu(\psi)$ denotes the unique unitarily invariant (Fubini-Study) measure on the set $\Omega=\{\ket{\psi} \in \mathbb{R}^d: \braket{\psi}{\psi}=1\}.$