Maximally entangled numerical shadow
Definition
Maximally entangled numerical shadow of a matrix $A$ of size $d=d_1\times d_2$ 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 $\Omega=\{\ket{\psi} \in \mathbb{C}^{d_1\times d_2}: \ket{\psi}=\frac{1}{\sqrt{d_\min}} (U_1\otimes U_2)\sum_{i=1}^{d_\min} \ket{\psi_i^1}\otimes \ket{\psi_i^2} \},$ where
- $d_\min={\min(d_1,d_2)}$,
- $\ket{\psi_i^1}$, $\ket{\psi_i^2}$ form orthonormal bases in $\mathbb{C}^{d_1}$ and $\mathbb{C}^{d_2}$ respectively,
- $U_1\in SU(d_1)$ and $U_2\in SU(d_2)$.