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numerical-range:generalizations:restricted-numerical-range:maximally-entangled-numerical-range

# Maximally entangled numerical range

## Definition

Maximally entangled numerical range $W^{\mathrm{ent}}(A)$ of a square matrix $A$ of size $N = N_1 \times N_2$ is a subset of the complex plane defined as

$$W^{\mathrm{ent}}(A)=\{z \in \mathbb{C}:\ z= \bra{\psi}A\ket{\psi},\ \ket{\psi}\in\mathbb{C}^{N_1\times N_2}_\mathrm{ent} \}.$$

$\mathbb{C}^{N_1\times N_2}_\mathrm{ent}= \{ \ket{\psi} : \ket{\psi}=(U_1\otimes U_2)\sum_{i=1}^{N_\min} \frac{1}{\sqrt{N_\min}} \ket{\psi_i^1}\otimes \ket{\psi_i^2} \}$, 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 $U_2\in SU(N_2)$.