Maximally entangled numerical range
Definition
Maximally entangled numerical range $W^{\mathrm{ent}}(A)$ of a square matrix $A$ of size $d = d_1 \times d_2$ is a subset of the complex plane defined as
\[W^{\mathrm{ent}}(A)=\\\{z \in \mathbb{C}:\\ z= \bra{\psi}A\ket{\psi},\\ \ket{\psi}\in\mathbb{C}^{d_1\times d_2}_\mathrm{ent} \\\}.\]\(\mathbb{C}^{d_1 \times d_2}_\mathrm{ent}= \\\{ \ket{\psi} \coloneqq (U_1 \otimes U_2) \sum_{i=1}^{d_\mathrm{min}} \frac{1}{\sqrt{d_\mathrm{min}}} \ket{\psi_i^1} \otimes \ket{\psi_i^2} \\\}\), 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 $U_2\in SU(d_2)$.