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numerical-range:generalizations:restricted-numerical-range:separable-numerical-range

# Separable numerical range

## Definition

Separable numerical range $W^{\mathrm{sep}}(A)$ of a square matrix $A$ of size $d = \prod_{i=1}^K d_i$ is a subset of the complex plane defined as

$$W^{\mathrm{sep}}(A)=\{z \in \mathbb{C}:\ z= \Tr{ \rho A,\ \rho \in \Omega_{\mathrm{sep}} }\},$$ where $\Omega_{\mathrm{sep}}$ is a convex hull of rank one projectors on product vectors i.e. $\rho \in \Omega_{\mathrm{sep}}$ iff:

• $\rho = \sum_j p_j \ket{\psi_j}\bra{\psi_j},\$ $p_j\geq 0 \text{ and } \sum_j p_j=1$,
• $\ket{\psi_j}=\bigotimes_{i=1}^K\ket{\psi^i_j}$,
• for $i=1,\ldots,K$ and all $j$ we have $\ket{\psi^i_j}\in\mathbb{C}^{d_i}$ and $\braket{\psi^i_j}{\psi^i_j}=1$.

### Fact

The separable numerical range of a matrix $A$ $W^{\mathrm{sep}}(A)$ is the convex hull of the product numerical range $W^\otimes(A)$ of this matrix $$W^{\mathrm{sep}}(A) = \mathrm{conv}\left(W^\otimes(A)\right)$$

### Example

Consider family of unitary matrices

$$U_d(\alpha_1, \alpha_2, \alpha_3)= \exp(\ii \sum_{k=1}^3 \alpha_k \sigma_k \otimes \sigma_k).$$

Numerical range (light gray), separable numerical range (dark gray) and product numerical range (black dots) obtained by random sampling of family of matrices $U_d(\alpha, 0, 0) \text{diag}(\ii, -1, -\ii, 1) U_d (\alpha, 0, 0)^\dagger$ for $\alpha = 0, \pi/8, 3 \pi /16, \pi /4$ [1].

1. P. Gawron, Z. Puchała, J.A. Miszczak, Ł. Skowronek, K. Życzkowski, 2010. Restricted numerical range: A versatile tool in the theory of quantum information. Journal of Mathematical Physics, 51, pp.102204.