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numerical-shadow:examples:8x8

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

Standard numerical shadow with respect to real states.
Standard numerical shadow with respect to complex states

Numerical shadow with respect to real separable states.
Numerical shadow with respect to real separable states

Numerical shadow with respect to complex separable states.
Numerical shadow with respect to complex separable states

Numerical shadow with respect to GHZ-entangled states.
Numerical shadow with respect to GHZ-entangled states

Numerical shadow with respect to W-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 complex product states Numerical shadow with respect to real product states
Numerical shadow with respect to GHZ states. Numerical shadow with respect to W 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 complex product states Numerical shadow with respect to real product states
Numerical shadow with respect to GHZ states. Numerical shadow with respect to W 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 complex product states Numerical shadow with respect to real product states
Numerical shadow with respect to GHZ states. Numerical shadow with respect to W 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 complex product states Numerical shadow with respect to real product states
Numerical shadow with respect to GHZ states. Numerical shadow with respect to W 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 complex product states Numerical shadow with respect to real product states
Numerical shadow with respect to GHZ states. Numerical shadow with respect to W 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 complex product states Numerical shadow with respect to real product states
Numerical shadow with respect to GHZ states. Numerical shadow with respect to W states.
Numerical shadow with respect to GHZ states Numerical shadow with respect to W states

numerical-shadow/examples/8x8.txt · Last modified: 2013/12/10 15:46 by lpawela