# Numerical Shadow

The web resource on numerical range and numerical shadow

# Hermitian matrices

In the case of Hermitian matrices the numerical shadow is a one dimensional distribution.

## Diagonal matrices

### Example 1

The matrix is $\mathrm{diag}(0,0,4,6)$

Standard numerical shadow with respect to complex states. Numerical shadow with respect to real states.  Numerical shadow with respect to complex product states. Numerical shadow with respect to real product states.  Numerical shadow with respect to complex entangled states. Numerical shadow with respect to real entangled states.  ### Example 2

The matrix is $\mathrm{diag}(0,1,4,6)$

Standard numerical shadow with respect to complex states. Numerical shadow with respect to real states.  Numerical shadow with respect to complex product states. Numerical shadow with respect to real product states.  Numerical shadow with respect to complex entangled states. Numerical shadow with respect to real entangled states.  ### Example 3

The matrix is $\mathrm{diag}(0,2,4,6)$

Standard numerical shadow with respect to complex states. Numerical shadow with respect to real states.  Numerical shadow with respect to complex product states. Numerical shadow with respect to real product states.  Numerical shadow with respect to complex entangled states. Numerical shadow with respect to real entangled states.  ### Example 4

The matrix is $\mathrm{diag}(0,3,4,6)$

Standard numerical shadow with respect to complex states. Numerical shadow with respect to real states.  Numerical shadow with respect to complex product states. Numerical shadow with respect to real product states.  Numerical shadow with respect to complex entangled states. Numerical shadow with respect to real entangled states.  ### Example 5

The matrix is $\mathrm{diag}(0,4,4,6)$

Standard numerical shadow with respect to complex states. Numerical shadow with respect to real states.  Numerical shadow with respect to complex product states. Numerical shadow with respect to real product states.  Numerical shadow with respect to complex entangled states. Numerical shadow with respect to real entangled states.  ### Example 6

The matrix is $\mathrm{diag}(0,4,5,6)$

Standard numerical shadow with respect to complex states. Numerical shadow with respect to real states.  Numerical shadow with respect to complex product states. Numerical shadow with respect to real product states.  Numerical shadow with respect to complex entangled states. Numerical shadow with respect to real entangled states.  ### Example 7

The matrix is $\mathrm{diag}(0,4,6,6)$

Standard numerical shadow with respect to complex states. Numerical shadow with respect to real states.  Numerical shadow with respect to complex product states. Numerical shadow with respect to real product states.  Numerical shadow with respect to complex entangled states. Numerical shadow with respect to real entangled states.  ### Example 8

The matrix is $\mathrm{diag}(0,6,6,6)$

Standard numerical shadow with respect to complex states. Numerical shadow with respect to real states.  Numerical shadow with respect to complex product states. Numerical shadow with respect to real product states.  Numerical shadow with respect to complex entangled states. Numerical shadow with respect to real entangled states.  ### Example 9

The matrix is $\mathrm{diag}(0,0,6,6)$

Standard numerical shadow with respect to complex states. Numerical shadow with respect to real states.  Numerical shadow with respect to complex product states. Numerical shadow with respect to real product states.  Numerical shadow with respect to complex entangled states. Numerical shadow with respect to real entangled states.  # Non-normal matrices

### Example 1

The matrix is TODO:fill. 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 shadow), $|\psi\rangle_{\mathrm{sep}} = U_A \otimes U_B \ket{00}$
3. entangled states, $\ket{\psi} = U_A \otimes U_B(\ket{00} + \ket{11})/\sqrt{2}$, separately for complex and real states.

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

Standard numerical shadow with respect to complex states. Numerical shadow with respect to real states.  Numerical shadow with respect to complex separable states. Numerical shadow with respect to real separable states.  Numerical shadow with respect to complex entangled states. Numerical shadow with respect to real entangled states.  ### Example 2

Even though several results on geometry of numerical range for $N = 4$ are available CITATION, complete classification of numerical ranges in this case is still missing. To provide an overview of the possible structure of the numerical shadow we show the complex numerical shadows of the following matrices of order four,

 $$A^{(4)}_0= \begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & \mathrm{i} & 1 & 1 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & -\mathrm{i} \end{pmatrix},$$ $$A^{(4)}_1= \begin{pmatrix} \mathrm{i} & 0 & -1 & 0 \\ 0 & 0 & -1 & 0 \\ 1 & 1 & 1-\mathrm{i} & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix},$$ $$A^{(4)}_2= \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & \mathrm{i} & 0 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -\mathrm{i} \end{pmatrix},$$ $$A^{(4)}_3= \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & \mathrm{i} & 1 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -\mathrm{i} \end{pmatrix},$$ $$A^{(4)}_4= \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & \mathrm{i} & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -\mathrm{i} \end{pmatrix},$$ $$A^{(4)}_5= \begin{pmatrix} \mathrm{i} & 0 & -1 & 0 \\ 0 & 0 & -1 & 0 \\ 1 & 1 & 1-\mathrm{i} & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix},$$ $$A^{(4)}_6= \begin{pmatrix} 1 & 0 & 1 & 0 \\ 0 & \mathrm{i} & 0 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -\mathrm{i} \end{pmatrix},$$ $$A^{(4)}_7= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \mathrm{i} & 0 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -\mathrm{i} \end{pmatrix},$$ $$A^{(4)}_8= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \mathrm{i} & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -\mathrm{i} \end{pmatrix}.$$

$A_0^{(4)}$ $A_1^{(4)}$  $A_2^{(4)}$ $A_3^{(4)}$  $A_4^{(4)}$ $A_5^{(4)}$  $A_6^{(4)}$ $A_7^{(4)}$  $A_8^{(4)}$  