- #1
munirah
- 31
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OP warned about not using the homework template
Good day,
I try to differentiate GHZ-state and W-state using three tangle. Suppose The value three tangle for GHZ-state equal to 1 while W-state equal to 0.
I used three tangle formula,
$$\tau_{ABC}=\tau_{A(BC)}-\tau_{AB}-\tau_{AC}=2(\lambda^{AB}.\lambda^{AB}+\lambda^{AC}.\lambda^{AC})$$
where the the
$$ \lambda^{AB}.\lambda^{AB}$$ and $$\lambda^{AC}.\lambda^{AC}$$ are eigenvalues of
$$\rho_{AB}.\rho^{\sim}_{AB}=(\sigma_y\otimes\sigma_y\rho^*_{AB}\sigma_y\otimes\sigma_y)$$
where the asterisk denotes complex conjugation in the standard basis and $$\sigma_y=\begin{pmatrix} 0 &-i &
\\i & 0
\end{pmatrix}
$$I measure the W-state and the density matrix of W-state:
$$\rho_w=\begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & \frac{1}3 & \frac{1}3& 0 & \frac{1}3 & 0 & 0 & 0
\\0 & \frac{1}3 & \frac{1}3& 0 & \frac{1}3 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & \frac{1}3 & \frac{1}3& 0 & \frac{1}3 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
\end{pmatrix}$$ and the
$$\rho_{AB}=\rho_{AC}= \begin{pmatrix} \frac{1}3 & 0 & 0 & 0
\\0 & \frac{1}3 & \frac{1}3 & 0 \\0 & \frac{1}3 & \frac{1}3 & 0\\0 & 0& 0 & 0
\end{pmatrix}$$Since it real matrix I assume it a multiplication matrix between $$\rho_{AB}.\rho^{\sim}_{AB}$$.
I get the eigen values= 2/3 and 1/3
and the value of three tangle for W-state = 0.8888... not 0
Where the wrong I did? I'm really stuck. Please help me to show the way to calculate three tangle correctly.
Thank you
I try to differentiate GHZ-state and W-state using three tangle. Suppose The value three tangle for GHZ-state equal to 1 while W-state equal to 0.
I used three tangle formula,
$$\tau_{ABC}=\tau_{A(BC)}-\tau_{AB}-\tau_{AC}=2(\lambda^{AB}.\lambda^{AB}+\lambda^{AC}.\lambda^{AC})$$
where the the
$$ \lambda^{AB}.\lambda^{AB}$$ and $$\lambda^{AC}.\lambda^{AC}$$ are eigenvalues of
$$\rho_{AB}.\rho^{\sim}_{AB}=(\sigma_y\otimes\sigma_y\rho^*_{AB}\sigma_y\otimes\sigma_y)$$
where the asterisk denotes complex conjugation in the standard basis and $$\sigma_y=\begin{pmatrix} 0 &-i &
\\i & 0
\end{pmatrix}
$$I measure the W-state and the density matrix of W-state:
$$\rho_w=\begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & \frac{1}3 & \frac{1}3& 0 & \frac{1}3 & 0 & 0 & 0
\\0 & \frac{1}3 & \frac{1}3& 0 & \frac{1}3 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & \frac{1}3 & \frac{1}3& 0 & \frac{1}3 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
\end{pmatrix}$$ and the
$$\rho_{AB}=\rho_{AC}= \begin{pmatrix} \frac{1}3 & 0 & 0 & 0
\\0 & \frac{1}3 & \frac{1}3 & 0 \\0 & \frac{1}3 & \frac{1}3 & 0\\0 & 0& 0 & 0
\end{pmatrix}$$Since it real matrix I assume it a multiplication matrix between $$\rho_{AB}.\rho^{\sim}_{AB}$$.
I get the eigen values= 2/3 and 1/3
and the value of three tangle for W-state = 0.8888... not 0
Where the wrong I did? I'm really stuck. Please help me to show the way to calculate three tangle correctly.
Thank you