The web resource on numerical range and numerical shadow

# 8x8 matrices

Here we present some example restricted numerical shadow of a 8×8 matrix.

## Diagonal matrices

### Example 1

The matrix is $\mathrm{diag}( 1, \mathrm{e}^\frac{2\mathrm{i}\pi}{3}, \mathrm{e}^\frac{2\mathrm{i}\pi}{3}, \mathrm{e}^\frac{-2\mathrm{i}\pi}{3}, \mathrm{e}^\frac{2\mathrm{i}\pi}{3}, \mathrm{e}^\frac{-2\mathrm{i}\pi}{3}, \mathrm{e}^\frac{-2\mathrm{i}\pi}{3}, 1 )$. We present the (restricted) numerical shadow for the following cases:

1. all pure states (standard shadow), separately for complex and real states
2. product states (product sahdow), $|\psi\rangle_{\mathrm{sep}} = U_A \otimes U_B \otimes U_C|000\rangle$
3. GHZ-entagnled states, $|\psi\rangle_{\mathrm{GHZ}} = U_A \otimes U_B \otimes U_C(|000\rangle + |111\rangle)/\sqrt{2}$, separately for complex and real states
4. so-called W-entangled states, $|\psi\rangle_{\mathrm{W}} = U_A \otimes U_B \otimes U_C(|100\rangle + |010\rangle + |001\rangle)/\sqrt{3}$.

Here $U_A$, $U_B$, $U_C$ are independent random unitary matrices taken from $U(2)$ with respect to the Haar measure.

Standard numerical shadow with respect to complex states.

Standard numerical shadow with respect to real states.

Numerical shadow with respect to real separable states.

Numerical shadow with respect to complex separable states.

Numerical shadow with respect to GHZ-entangled states.

Numerical shadow with respect to W-entangled states.

### Example 2

The matrix is $\mathrm{diag}(1,2,3,4,5,6,7,8)$

Numerical shadow with respect to complex product $2 \times 2 \times 2$ states. Numerical shadow with respect to real product $2 \times 2 \times 2$ states.
Numerical shadow with respect to GHZ states. Numerical shadow with respect to W states.

### Example 3

The matrix is $\mathrm{diag}(1,2,2,4,5,6,7,8)$

Numerical shadow with respect to complex product $2 \times 2 \times 2$ states. Numerical shadow with respect to real product $2 \times 2 \times 2$ states.
Numerical shadow with respect to GHZ states. Numerical shadow with respect to W states.

### Example 4

The matrix is $\mathrm{diag}(1,2,2,4,4,6,7,8)$

Numerical shadow with respect to complex product $2 \times 2 \times 2$ states. Numerical shadow with respect to real product $2 \times 2 \times 2$ states.
Numerical shadow with respect to GHZ states. Numerical shadow with respect to W states.

### Example 5

The matrix is $\mathrm{diag}(1,2,2,2,5,6,7,8)$

Numerical shadow with respect to complex product $2 \times 2 \times 2$ states. Numerical shadow with respect to real product $2 \times 2 \times 2$ states.
Numerical shadow with respect to GHZ states. Numerical shadow with respect to W states.

### Example 6

The matrix is $\mathrm{diag}(1,2,2,2,2,6,7,8)$

Numerical shadow with respect to complex product $2 \times 2 \times 2$ states. Numerical shadow with respect to real product $2 \times 2 \times 2$ states.
Numerical shadow with respect to GHZ states. Numerical shadow with respect to W states.

### Example 7

The matrix is $\mathrm{diag}(2,2,2,6,6,6,7,8)$

Numerical shadow with respect to complex product $2 \times 2 \times 2$ states. Numerical shadow with respect to real product $2 \times 2 \times 2$ states.
Numerical shadow with respect to GHZ states. Numerical shadow with respect to W states.