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

Separable numerical range

Definition

Separable numerical range numerical range $W^{\mathrm{sep}}(A)$ of a square matrix $A$ of size $N = \prod_{i=1}^K N_i$ 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 $\ket{\psi^i_j}\in\mathbb{C}^{N_i}$ 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 of the product numerical range W^\otimes(A) of this matrix $$W^{\mathrm{sep}}(A) = \mathrm{conv}\left(W^\otimes(A)\right)$$

numerical-range/generalizations/restricted-numerical-range/separable-numerical-range.txt · Last modified: 2014/02/04 02:03 by lpawela