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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 $d = d_1 \times d_2$ is a subset of the complex plane defined as

$$W^{\mathrm{ent}}(A)=\{z \in \mathbb{C}:\ z= \bra{\psi}A\ket{\psi},\ \ket{\psi}\in\mathbb{C}^{d_1\times d_2}_\mathrm{ent} \}.$$

$\mathbb{C}^{d_1\times d_2}_\mathrm{ent}= \{ \ket{\psi} : \ket{\psi}=(U_1\otimes U_2)\sum_{i=1}^{d_\min} \frac{1}{\sqrt{d_\min}} \ket{\psi_i^1}\otimes \ket{\psi_i^2} \}$, 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 $U_2\in SU(d_2)$.