How to differentiate GHZ state and W state?

[munirah]

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

 

[mark726]

Thank you for your post. Let me first clarify the difference between GHZ-states and W-states. A GHZ-state is a three-qubit state that is maximally entangled, meaning that it has a three-tangle value of 1. The W-state, on the other hand, is a three-qubit state that is partially entangled, meaning that it has a three-tangle value less than 1.

Now, let's look at your calculations. The three-tangle formula you have used is correct, but there are a few issues with your matrices. First, the GHZ-state and W-state are not the same state, so you cannot use the same density matrix for both. The density matrix for the GHZ-state should be

$$\rho_{GHZ}=\frac{1}{2}\begin{pmatrix} 1 & 0 & 0 & 0
\\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\\0 & 0& 0 & 1
\end{pmatrix}$$

Next, the density matrix you have calculated for the W-state is incorrect. The correct density matrix for the W-state should be

$$\rho_W=\frac{1}{3}\begin{pmatrix} 1 & 0 & 0 & 0
\\0 & 1 & 1 & 0 \\0 & 1 & 1 & 0\\0 & 0& 0 & 1
\end{pmatrix}$$

Using these correct density matrices, you will get the correct eigenvalues and a three-tangle value of 0 for the W-state.

I hope this helps clear up any confusion. Please let me know if you have any further questions. Keep up the good work in your research!