The web resource on numerical range and numerical shadow

## Definition

Maximally entangled numerical shadow of a matrix $A$ of size $N=N_1\times N_2$ is defined as a probability distribution $P_A(z)$ on the complex plane, supported on the maximally entangled numerical range $W^\mathrm{ent}(A)$. $$P_A(z) := \int_{\Omega} {\rm d} \mu(\psi) \delta\Bigl( z-\langle \psi|A|\psi\rangle\Bigr),$$ where $\mu(\psi)$ denotes the unique unitarily invariant (Fubini-Study) measure on the set $$\Omega=\{\ket{\psi} \in \mathbb{C}^{N_1\times N_2}: \ket{\psi}=\frac{1}{\sqrt{N_\min}} (U_1\otimes U_2)\sum_{i=1}^{N_\min} \ket{\psi_i^1}\otimes \ket{\psi_i^2} \},$$ where

• $N_\min={\min(N_1,N_2)}$,
• $\ket{\psi_i^1}$, $\ket{\psi_i^2}$ form orthonormal bases in $\mathbb{C}^{N_1}$ and $\mathbb{C}^{N_2}$ respectively,
• $U_1\in SU(N_1)$ and $U_2\in SU(N_2)$.